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5ce061975eeee98a8dbbd6a9fbb9d5151d7b279c | subsection | 81 | 84 | Exchange identities | The equivalence between the LHS and RHS in the identities above can be shown by simply shifting the contour of integration from being first along the \hat{\rho }_2 axis then in the \rho _1 direction to being first along the \rho _1-axis then in the \hat{\rho }_2 direction when applying the map (REF ) (this time using t... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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a03cf8b97229aa785914f26d3ca7d602d1ae22af | subsection | 82 | 84 | Explicit Results | In this section, we are presenting explicitly the NLLA perturbative coefficients \tilde{g} and \tilde{h} at two loops both in the MHV case and the NMHV case for the helicity configurations -++ and -+-. Together with target-projectile symmetry and conjugation, these span the full set of two-loop perturbative coefficient... | {
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} | 10.1007/JHEP06(2018)116 | 1801.10605 | The seven-gluon amplitude in multi-Regge kinematics beyond leading
logarithmic accuracy | [
"Vittorio Del Duca",
"Stefan Druc",
"James Drummond",
"Claude Duhr",
"Falko Dulat",
"Robin Marzucca",
"Georgios Papathanasiou",
"Bram Verbeek"
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4385db54d9481d0a5c96a2ea922dc6f5bed8eda1 | subsection | 83 | 84 | Explicit Results | In the following, for compactness of the results, we will use the notation\begin{}G\end{}_{\vec{a}}^i \equiv \begin{}G\end{}_{\vec{a}}(\rho _i) \,.The leading singularities R_{bac} introduced in and correspond toR_{234} &= \frac{\rho _1 (\rho _2-1)}{(\rho _1-1) \rho _2}
&R_{235} &= \frac{\rho _1}{\rho _1-1}
&R_{345} &=... | {
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cfb2e42f1429af2dc555b03f7f47c40fab4067d6 | abstract | 0 | 127 | Abstract | Consider an affine Gaussian field X : R 2 $\rightarrow$ R, that is a process
equal in law to Z(At), where Z is isotropic and A : R2 $\rightarrow$ R2 is a
self-adjoint definite positive matrix. Denote 0 < $\lambda$ = $\lambda$\_2 /
$\lambda$\_1 \le 1 the ratio of the eigenvalues of A. This paper is aimed at
testing the ... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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3515aca662b3336a2ff733f22c55c4c97d212c8f | subsection | 1 | 127 | Introduction | The aim of the present paper is to test the null hypothesis that a given Gaussian process X indexed in \mathbb {R}^2 and living in the class of affine processes is isotropic.
We assume that X is partially observed through some level functionals of its level curve \textrm {C}_{T, X}(u) for a fixed level u, say \textrm {... | {
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e14b3a2921eb0ef14cea355dee54a1b8ccc2b24a | subsection | 2 | 127 | Introduction | Wschebor proposed probability consistent estimators of anisotropy directions \theta _o, say \widehat{\theta }_{o, n} and also of its value \lambda , say \widehat{\lambda }_n, based on the observation of the ratio of functionals J_{f^{\star }}^{(n)}(u) and J_{\bf 1}^{(n)}(u) where J_{f^{\star }}^{(n)}(u) is the integral... | {
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"raw": "Mario Wschebor, Surfaces aléatoires: mesure géométrique des ensembles de niveau, vol. 1147, Springer, 1985. MR 0871689",
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a6d6c355808babf167d535420d5974d29d551a00 | subsection | 3 | 127 | Main contribution of the paper | In the present work following the way opened by Wschebor, we consider his proposed estimators (\widehat{\lambda }_n, \widehat{\theta }_{o, n}).
Our main contribution has consisted in the one hand, by using the Birkhoff- Kintchine ergodic therorem of Cramér and Leadbetter and Rice formula (see the seminal work of Rice )... | {
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434fe507f896bb787c295f51ef0349f556e00b52 | subsection | 4 | 127 | Main contribution of the paper | In this work the authors show a CLT for the Euler characteristic of the excursions above u of the field X on T as T grows to \mathbb {R}^d, X being a stationary Gaussian isotropic process indexed in \mathbb {R}^d.Our real contribution for proving the CLT, apart from showing the non degeneration of the asymptotic limit ... | {
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22cc327cbe3df6f2bbb57c3b87aadca686ef3bbd | subsection | 5 | 127 | Outline of the paper | Section contains some definitions, assumptions and notations, among others definitions of an affine process, of an isotropic process and explicit the type of general functionals on the level set u, say J^{(n)}_f(u), we are looking for.Section is devoted to establish a Rice formula for such functionals, in other words t... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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c267fd3bf2d979ebaed2ecdf80f8e2423e2e0a4d | subsection | 6 | 127 | Hypothesis and notations | Let us give some definitions, assumptions and notations.A process (Z(t), t \in \mathbb {R}^2) is said to be isotropic if it is a stationnary process and if for any isometry U in \mathbb {R}^2, k \in \mathbb {N} and t_1, \dots , t_k \in \mathbb {R}^2, the joint laws of (Z(t_1), \dots , Z(t_k)) and (Z(U(t_1)), \dots , Z(... | {
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"Corinne Berzin"
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fd761cefbfd16d9d231488165928fb4646f7d5ce | subsection | 7 | 127 | Assumptions on the covariance | For any multidimensional index {{m}}= (i_1, \dots , i_k) with 0 \leqslant k \leqslant 2 and 1 \leqslant i_j \leqslant 2, we write
\dfrac{\partial ^{{{m}}}r_{z}}{\partial t^{{{m}}}}(t)= \dfrac{\partial ^{k}r_{z}}{\partial t_{i_{1}}\cdots \partial t_{i_{k}}}(t).Let \Psi (t) = \max \left\lbrace \left| {\dfrac{\partial ^{{... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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c8346dab996a7fb7f19ee2efdd035b0bfb53b8b7 | subsection | 8 | 127 | Level set | For u \in \mathbb {R} we define the level set at u as:\textrm {C}_{T, X}(u)=\left\lbrace t \in T: X(t)=u \right\rbraceand we denote \textrm {D}^{\textrm {r}}_{\textrm {X}} the following set\textrm {D}^{\textrm {r}}_{\textrm {X}}=\left\lbrace t \in \mathbb {R}^2 : \nabla X(t)\,\,\mbox{is of rank } 1
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\lef... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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115710527cfdce4f8bbda5d9e001f7d201b3e7c2 | subsection | 9 | 127 | General level functionals | S^{1} is the boundary of the unit ball of \mathbb {R} and
for d=1, 2, \sigma _d denotes the Lebesgue measure on
\mathbb {R}^d.For f: S^{1} \rightarrow \mathbb {R}^{d} a continuous and bounded function, we define the following general functional J_{f}^{(n)}(u) of the fixed level u by:J_{f}^{(n)}(u)= \frac{1}{\sigma _{2}... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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55ad459f6322763d5821479067f68c717687eac4 | subsection | 10 | 127 | Hermite polynomials | We use the Hermite polynomials (H_n)_{n \in \mathbb {N}} defined byH_n(x)= (-1)^n e^{x^2/2} \displaystyle \frac{{\rm \,d}^n}{{\rm \,d}x^n}(e^{-x^2/2} ).They provide an orthogonal basis of L^2(\mathbb {R}, \phi (x) {\rm \,d}n) where \phi denotes the standard Gaussian density on \mathbb {R}.
We also denote by \phi _m the... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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7909099f219ba75a7d52b31c39c9418454d19822 | subsection | 11 | 127 | Rice formula | Let u a fixed level in \mathbb {R}. | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
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2cac8ceee525f1072654e61be6a9f562dbd2416b | subsection | 12 | 127 | Almost sure convergence for | For f: S^{1} \rightarrow \mathbb {R} a continuous and bounded function, we show that process X and f(\nu _{X}) verify the assumptions of Theorem 3.3.1 of Berzin et al. .
Then the one order Rice formula for the general functional J_{f}^{(n)}(u) is valid.
More precisely we have the
{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{... | {
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0ac4df7a7820a2a3e2a4964712ab87eaf0ace440 | subsection | 13 | 127 | Almost sure convergence for | Let us show that process X verifies assumptions of Remark 3.3.1 of Theorem 3.3.1 of .First X/T: \Omega \times T \rightarrow \mathbb {R} is a stationary Gaussian field belonging to C^2(T) and T is a bounded convex open set of \mathbb {R}^2.Furthermore by using results given in Section 4.3 of Azaïs and Wschebor , one can... | {
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fd89540de98661fc83c9296585d3b25b67a0ae2f | subsection | 14 | 127 | Almost sure convergence for | And since process Z is an isotropic process, it is such that the random variable Z(0), \frac{\partial Z}{\partial t_1}(0) and \frac{\partial Z}{\partial t_2}(0) are mutually independent.The non-degeneration of processes Z(0) and \nabla Z(0) provides from the assumptions made on the covariance r_z, more precisely from t... | {
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"Corinne Berzin"
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c55ce9194c0009b20611b614a22f88245116fe57 | subsection | 15 | 127 | Almost sure convergence for | Let v^{\star } \in S^{1} a fixed vector and consider\begin{aligned}S^{1}& \longrightarrow S^{1}\\
\theta & \mapsto f^{\star }(\theta )= \theta \times (1_{\lbrace \langle \theta , v^{\star }\rangle \geqslant 0\rbrace }-1_{\lbrace \langle \theta , v^{\star }\rangle < 0\rbrace }).
\end{aligned}Applying the Rice formula fo... | {
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0.013449824415147305,
0.026426654309034348,
-0.03555086627602577,
0.0025099217891693115,
0.016173357143998146,
-0.002... | |
cecc67bfe8f960a41d8779e8b93105d2c40cca89 | subsection | 16 | 127 | Almost sure convergence for | We denote it by g(\left\Vert {v} \right\Vert _{2}).Thus if f: S^1 \rightarrow \mathbb {R}^2 is a continuous and bounded function,{\rm E}_{{}}\hspace{-2.0pt}\left[{f(\nu _{X}(0)) \left\Vert {\nabla X(0)} \right\Vert _{2}}\right]
&= {\rm E}_{{}}\hspace{-2.0pt}\left[{f\left(\tfrac{A\nabla Z(0)}{\left\Vert {A\nabla Z(0)} \... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.010744636878371239,
0.036721017211675644,
0.008989476598799229,
0.007737970445305109,
0.028448868542909622,
-0.02115350402891636,
-0.015384366735816002,
0.017047954723238945,
0.012888985686004162,
0.025854283943772316,
-0.03275282680988312,
0.011202504858374596,
-0.00401016091927886,
0.... | |
f0bdeebad088d348b634955e259378b6e6fcbefa | subsection | 17 | 127 | Almost sure convergence for | \BoxNow by applying an ergodic theorem for stationnary processes (Cramér and Leadbetter , §7.11), we shall show the following general almost sure convergence theorem.
For f: S^{1} \rightarrow \mathbb {R} a continuous and bounded function,J_{f}^{(n)}(u) \xrightarrow[n\rightarrow +\infty ]{a.s.}{\rm E}_{{}}\hspace{-2.0... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0416911281645298,
0.06180420145392418,
-0.045964013785123825,
-0.000002861305574697326,
0.03256547078490257,
-0.03232130780816078,
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0.010407522320747375,
0.03897479549050331,
0.025683078914880753,
-0.013436690904200077,
-0.020967647433280945,
0.0014516357332468033,
... | |
629113a106bf87a5ec864a6443110132f7145617 | subsection | 18 | 127 | Almost sure convergence for | Lemma REF is proved in Appendix .\int _{0}^{n-1} \int _{0}^{n-1} \int \limits _{\textrm {C}_{]t\,, \,t+1[\times ]s\, ,\,s +1[\,,\, X}(u)} f(\nu _{X}(x)) {\rm \,d}\sigma _{1}(x)\, {\rm \,d}t {\rm \,d}s \\
\hfill - \int _{0}^{1}\int _{0}^{1} \int _{\textrm {C}_{]0\,,\,t[\times ]0\,,\,s[\,, \,X}(u)} f(\nu _{X}(x)) {\rm \,... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.016021784394979477,
0.04266372323036194,
-0.024520959705114365,
-0.030105695128440857,
0.0019435950089246035,
-0.0005826970445923507,
0.02146919071674347,
0.029556376859545708,
0.026657195761799812,
0.018905704841017723,
0.004394546616822481,
-0.02786264568567276,
-0.0027999975718557835,
... | |
e498ef7d07c2250046ccd66da152db8f8346d5e3 | subsection | 19 | 127 | Almost sure convergence for | On the other hand, noting by\xi (t, s)= \int \limits _{\textrm {C}_{]t\,, \,t+1[\times ]s\, ,\,s +1[\,,\, X}(u)} f(\nu _{X}(x)) {\rm \,d}\sigma _{1}(x),one has\frac{1}{(2n)^2} \int _{0}^{n-1} \int _{0}^{n-1} \int \limits _{\textrm {C}_{]t\,, \,t+1[\times ]s\, ,\,s +1[\,,\, X}(u)} f(\nu _{X}(x)) {\rm \,d}\sigma _{1}(x)\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2987169",
"end": 722,
"openalex_id": "https://openalex.org/W2795404169",
"raw": "Harald Cramér and M. Ross Leadbetter, Stationary and related stochastic processes, John Wiley & Sons, 1967. MR 0217860",
"source_ref_id": "6b6acd... | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.023921623826026917,
0.013257583603262901,
-0.03499757871031761,
0.059865083545446396,
-0.0006941542378626764,
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0.015217997133731842,
-0.014447562396526337,
-0.023052023723721504,
0.003917013294994831,... | |
23e6488aca389d10e098259d585134f2060be83b | subsection | 20 | 127 | Almost sure convergence for | By the Birkhoff-Kintchine ergodic Theorem (Cramér and Leadbetter ) and last proposition, we deduce that\left({\frac{n-1}{2n} }\right)^2 {\frac{1}{(n-1)^2} \int _{0}^{n-1} \int _{0}^{n-1} \xi (t, s) {\rm \,d}t {\rm \,d}s}\xrightarrow[n\rightarrow +\infty ]{a.s.} \frac{1}{4} \,{\rm E}_{{}}\hspace{-2.0pt}\left[{\xi (0,0)}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2987169",
"end": 402,
"openalex_id": "https://openalex.org/W2795404169",
"raw": "Harald Cramér and M. Ross Leadbetter, Stationary and related stochastic processes, John Wiley & Sons, 1967. MR 0217860",
"source_ref_id": "6b6acd... | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.022352151572704315,
0.0384487509727478,
-0.0005917026428505778,
0.031170953065156937,
0.004458987154066563,
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0.034725937992334366,
-0.029446862637996674,
0.026670007035136223,
-0.020490743219852448,
... | |
41198ed2ce3747ce3e8dffad37bd2ec2b4fe6bfe | subsection | 21 | 127 | Almost sure convergence for | Finally, if f is a positive function, by using the linearity of the interest functional one have proved thatJ_{f}^{(n)}(u) \xrightarrow[n\rightarrow +\infty ]{a.s.} 4 \times \frac{1}{4}\, {\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f}^{(1)}(u)}\right]={\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f}^{(1)}(u)}\right].To conclude the pr... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.037564150989055634,
-0.005477469880133867,
0.006583644077181816,
-0.023237287998199463,
0.003217441262677312,
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0.014120886102318764,
0.02010948583483696,
0.00240306812338531,
-0.027356833219528198,
-0.018797334283590317,
-... | |
3784868bd49de6a9dc4d013e8737f43a4d307b1e | subsection | 22 | 127 | The affinity parameters | Theorem REF applied to the particular functions f^{\star } and {\bf 1} and the result of convergence of Corollary REF imply that,\frac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)}\xrightarrow[n\rightarrow +\infty ]{a.s.} \frac{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f^{\star }}^{(1)}(u)}\right]}{{\rm E}_{{}}\hspace{-2.0pt... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02747291699051857,
0.008592917583882809,
-0.0065629747696220875,
-0.0058837831020355225,
0.033150654286146164,
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0.021428875625133514,
0.015308519825339317,
0.013156474567949772,
-0.021612027660012245,
0.02295514941215515,
-0.027701858431100845,
... | |
58205a5968baab9eba09a0a0139fb0017964b820 | subsection | 23 | 127 | The affinity parameters | The vector v^{\star } can always be written in this basis:v^{\star }= \cos (\theta _{o}) v_1+ \sin (\theta _{o}) v_2.It is always possible to choose -\frac{\pi }{2} < \theta _{o} \leqslant \frac{\pi }{2}.Indeed, \theta _o could be the angle between v^\star and the eigenvector corresponding to the highest eigenvalue, be... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02177709899842739,
0.02574489451944828,
-0.0038953078910708427,
0.02481398917734623,
0.04373732581734657,
0.012002584524452686,
-0.009331951849162579,
0.013063207268714905,
0.03821293264627457,
0.015146300196647644,
-0.017153089866042137,
0.04343210905790329,
-0.0009814572986215353,
0.0... | |
da304534fa8851adfdf8c19e16a51bf60cb8ceef | subsection | 24 | 127 | The affinity parameters | One has:A^2v^{\star }= \lambda _{1}^2\left[{\cos (\theta _{o})\,v_1+\lambda ^2 \sin (\theta _{o})\,v_2
}\right],and\left\Vert {Av^{\star }} \right\Vert _{2}= \lambda _{1}\left[{\cos ^2(\theta _{o})+\lambda ^2 \sin ^2(\theta _{o})
}\right]^{\frac{1}{2}}.Also\int _{S^1} \left\Vert {A\alpha } \right\Vert _{2}\, {\rm \,d}\... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.008193835616111755,
0.02064480446279049,
0.021163593977689743,
0.01620456948876381,
0.02543598599731922,
-0.01789826713502407,
-0.0038623178843408823,
0.0049704695120453835,
0.033233098685741425,
0.009879904799163342,
-0.04266287758946419,
0.03436223044991493,
-0.01398445200175047,
0.007... | |
762de704268f428de0387a79343830691fc620a0 | subsection | 25 | 127 | The affinity parameters | \end{array}\right.}That yields Corollary REF .
\BoxWritting the observed ratio of functionals \frac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)} as\frac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)}= X_n v^{\star } + Y_n v^{\star \star },we shall show the following proposition.
Let consider the following system of equ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06848418712615967,
0.012917082756757736,
-0.030595097690820694,
-0.014450651593506336,
0.03314341604709625,
0.016144445165991783,
-0.02050863392651081,
0.03769071772694588,
0.008781785145401955,
0.0025254306383430958,
-0.02545267902314663,
-0.006355540361255407,
-0.01573244109749794,
0.... | |
ef7e4757c2fbb21dd0465c22a13d7cb96f8d8c0b | subsection | 26 | 127 | The affinity parameters | Let us consider the following system of equations{\left\lbrace \begin{array}{ll}
X_n &= F_1(\lambda , \theta _o)\\
Y_n & = F_2(\lambda , \theta _o)
\end{array}\right.}If the system admits a solution \widehat{\lambda }_{n}, this solution ought to verify the following equation in \lambda :X_n^2\, I^4(\lambda ) \,(X_n^2+Y... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.05594870075583458,
0.002043134765699506,
-0.018145937472581863,
-0.00009425175812793896,
0.0169860627502203,
-0.01588723435997963,
0.030782468616962433,
0.04444152116775513,
0.010873827151954174,
0.001885750563815236,
-0.053964704275131226,
-0.020129933953285217,
-0.02864585816860199,
0... | |
2bcb4905af72a686a058f8ce245372900f32a037 | subsection | 27 | 127 | The affinity parameters | By summarizing the situation we know that f_1-f_2 is continuous on {]}{0},{1}{]}, strictly decreasing and such that
(f_1-f_2)(0^+)=+\infty and (f_1-f_2)(1)=\displaystyle \frac{X_n^2\, Y_n^2\, (\frac{\pi }{2})^4 + (X_n^2\, (\frac{\pi }{2})^2-1)^2}{(X_n^2+Y_n^2)\,(\frac{\pi }{2})^2 -1} <0.
Thus there exists 0< \lambda <1... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.10194076597690582,
0.008301762863993645,
-0.004776565823704004,
0.0027774281334131956,
0.017305513843894005,
-0.0215326976031065,
0.04370633885264397,
0.05405302345752716,
0.0022509375121444464,
0.027926886454224586,
-0.07434961199760437,
-0.031009526923298836,
-0.037571582943201065,
0.... | |
9b3704598ee1763229c3091faba549fed3c75a24 | subsection | 28 | 127 | The affinity parameters | On the other side, still if Y_n \ne 0 and if \lambda _0 <1, we have (f_1-f_2)(\lambda _0^+)=+\infty and (f_1-f_2)(1^-) >0.
Then there is no more solution of (REF ) into interval ]\lambda _0^+, 1].
Thus we have proved that in the case where Y_n \ne 0, if (REF ) admits a solution \widehat{\lambda }_{n} \ne \lambda _0, th... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04257817193865776,
0.007016241550445557,
-0.009324466809630394,
-0.005715242121368647,
0.021243302151560783,
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0.005589338485151529,
0.03925127536058426,
0.024554938077926636,
-0.006653793156147003,
-0.03754204139113426,
-0.0031819171272218227,
-0.014200353994965553,
... | |
3bd99b1c50dfcf93482504beae7725f8c912b5d8 | subsection | 29 | 127 | The affinity parameters | Or (X_n^2+Y_n^2)(\frac{\pi }{2})^2 -1 \geqslant 0 and Y_n \ne 0, and this solution 0 < \widehat{\lambda }_{n} <1 verifies equation f_1(\lambda )=f_2(\lambda ).
Or X_n^2 (\frac{\pi }{2})^2 >1 and Y_n = 0, and this solution 0 < \widehat{\lambda }_{n} <1 is \widehat{\lambda }_{n} =I^{-1}(\frac{1}{X_n}).
Or X_n^2 (\frac{... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06267377734184265,
0.006747195031493902,
-0.008276660926640034,
0.0027862214483320713,
0.011190655641257763,
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0.023952731862664223,
0.03563922643661499,
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-0.0003072757972404361,
-0.05284857749938965,
-0.012449318543076515,
-0.027446474879980087,
... | |
46527bb076f5474ed194b9b2ed5ba5cb90584aa7 | subsection | 30 | 127 | The affinity parameters | We have the following equivalence:\left({\displaystyle \frac{\frac{1}{\widehat{\lambda }_{n}^2}-f_1(\widehat{\lambda }_{n})}{\frac{1}{\widehat{\lambda }_{n}^2}-1} \leqslant 1}\right) \iff \left({f_1(\widehat{\lambda }_{n}) \geqslant 1}\right).In the case where X_n^2 (\frac{\pi }{2})^2 >1 and Y_n = 0, \widehat{\lambda }... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04109761118888855,
0.017223162576556206,
-0.02495756931602955,
-0.002551438519731164,
0.02777978777885437,
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0.010495603084564209,
0.041250161826610565,
-0.008756506256759167,
0.0073797209188342094,
-0.056322336196899414,
-0.009534523822367191,
-0.04600979760289192,
... | |
89e8697288c1b757162f8cc43ab93191619b30bc | subsection | 31 | 127 | The affinity parameters | To verify last inequality, remark that\left({\widehat{\theta }_{o,n}=-\frac{\pi }{2}}\right) \iff \left({Y_n>0 \mbox{ and \,} X_n^2 I^2(\widehat{\lambda }_n)=\widehat{\lambda }_n^2}\right)\\
\Longrightarrow \left({Y_n>0 \mbox{ and \,} f_1(\widehat{\lambda }_{n})=f_2(\widehat{\lambda }_{n})=1}\right)
\Longrightarrow \le... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04430647939443588,
0.021832842379808426,
-0.007304313592612743,
-0.02039867825806141,
0.019223885610699654,
-0.025372477248311043,
0.008940632455050945,
0.028790056705474854,
0.008643119595944881,
0.014311115257441998,
-0.03362654149532318,
0.0036426212172955275,
-0.010809621773660183,
... | |
6fc1ecd0ea0407dd473a782c50417340ea6d9ac5 | subsection | 32 | 127 | The affinity parameters | In the other cases, one has f_1(\widehat{\lambda }_{n})= f_2(\widehat{\lambda }_{n}), and since X_n^2\, I^2(\widehat{\lambda }_{n})=\cos ^2(\widehat{\theta }_{o,n})+ \widehat{\lambda }_{n}^2 \sin ^2(\widehat{\theta }_{o,n}) one has{\left({f_1(\widehat{\lambda }_{n})= f_2(\widehat{\lambda }_{n})}\right)}\\
& \iff \left(... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.021754350513219833,
0.01755908504128456,
-0.008131183683872223,
-0.016567477956414223,
0.020854275673627853,
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0.03566737100481987,
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0.02724633179605007,
-0.05083134025335312,
-0.022212015464901924,
-0.046620823442935944,
0... | |
704325b5c5b9909d29736cf14997cccb81e4e8a2 | subsection | 33 | 127 | The affinity parameters | In this case X_n= \frac{2}{\pi }=F_1(1, \theta _o) and Y_n = 0=F_2(1, \theta _o), for all -\frac{\pi }{2} <\theta _o \leqslant \frac{\pi }{2}.This yields the proposition.
We have proved that F is a one to one function from ]0, 1[ \times {]}{-\frac{\pi }{2}},{\frac{\pi }{2}}{]} onto \lbrace (X, Y) \in \mathbb {R}^2, X... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.045350078493356705,
0.03179998695850372,
-0.012901578098535538,
0.03582839295268059,
0.02533012442290783,
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0.04852397367358208,
0.004589177668094635,
0.004844767972826958,
-0.018112564459443092,
0.026032043620944023,
0.013588238507509232,
0.0... | |
11c54ed2c9f290ab78d5a13eb24e7612725aeacd | subsection | 34 | 127 | Consistency for the parameters | Now we are ready to state the following results of consistency for the two proposed estimators \widehat{\lambda }_{n} and \widehat{\theta }_{o,n}.For 0 < \lambda \leqslant 1 and -\frac{\pi }{2} < \theta _{o} \leqslant \frac{\pi }{2}, one has
\widehat{\lambda }_{n} \xrightarrow[n\rightarrow +\infty ]{a.s.} \lambda .
... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03536924719810486,
-0.010452086105942726,
-0.05602928623557091,
0.022689418867230415,
0.04229661822319031,
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0.018554359674453735,
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0.021163567900657654,
-0.03420960158109665,
0.0073393480852246284,
-0.01428197417408228,
0.000... | |
5e4b6e966a6ab12fde5652a085f81965d06c2416 | subsection | 35 | 127 | Consistency for the parameters | If F^{-1} denotes the inverse function of F, thus since F^{-1} is continuous from V to U, one deduces that almost surely F^{-1}(X_n, Y_n)=(\widehat{\lambda }_{n}, \widehat{\theta }_{o,n}) converges to F^{-1}(X, Y)=(\lambda , \theta _{o}).It remains to consider two cases, case where \lambda =1 and -\frac{\pi }{2} < \the... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.052504196763038635,
0.004998582880944014,
-0.02761049196124077,
0.03250986710190773,
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960ffde29fc838bf33c8d126aea751c1cf216bb9 | subsection | 36 | 127 | Consistency for the parameters | By Remark REF stated at the end of Proposition REF proof, the estimator \widehat{\lambda }_{n} has to verify the following equationX_n^2\, I^4(\widehat{\lambda }_{n}) \,(X_n^2+Y_n^2)-X_n^2\,I^2(\widehat{\lambda }_{n})\,(\widehat{\lambda }_{n}^2+1)+\widehat{\lambda }_{n}^2=0.Thus\left({X_n^2 -(\tfrac{2}{\pi })^2}\right)... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b04eca547ca591c2618208ee0f02e266e16024a6 | subsection | 37 | 127 | Consistency for the parameters | We deduce that\widehat{\lambda }_{n} \xrightarrow[n\rightarrow +\infty ]{a.s.} \lambda ,this yields Theorem REF .
\BoxWe thus set up a first approach to detect if the process X is isotropic or not. | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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242f33118050d734b8f38c851a501b1db97d4fe6 | subsection | 38 | 127 | Convergence in law for | We built consistent estimators, \widehat{\lambda }_{n} and \widehat{\theta }_{o,n} for parameters \lambda and \theta _o, by using Theorem REF applied in the particular cases where f=f^{\star } and f={\bf 1}.
In other words, we used the almost sure convergence of J_{f^{\star }}^{(n)}(u) and of J_{\bf 1}^{(n)}(u).
In ord... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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59c4b63e0478d54200c8e97c9f1e4dccd8e12847 | subsection | 39 | 127 | Hermite expansion for | Our objective is to decompose the random variable \xi _{f}^{(n)}(u) as a sum of multiple Itô integrals.
In this aim, the idea consists in approaching functionals J_{f}^{(n)}(u) by other functionals, say J_{f}^{(n)}(u, \sigma ) (\sigma \rightarrow 0), in such a way that the last ones be expressed into the Itô-Wiener cha... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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15b0d12cf4e3209459db9a72dd653b68137ac5d8 | subsection | 40 | 127 | Coarea formula | For \sigma >0, we define an approximation of J_{f}^{(n)}(u) given byJ_{f}^{(n)}(u, \sigma )= \frac{1}{\sigma } \int _{-\infty }^{+\infty } K(\tfrac{u-v}{\sigma })\, J_{f}^{(n)}(v) {\rm \,d}v,where K is a continuous density function with a compact support in [-1, 1].By applying Corollary 2.1.1.
of to the function h: \ma... | {
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e43d34273a0ea0126c131406e837e1c924038c54 | subsection | 41 | 127 | Coarea formula | We can write, for any fixed t \in \mathbb {R}^2, {{X}}(t)= \Delta Y(t), where Y(t) is a 3-dimensional standard Gaussian vector.With these notations and if Y(t)= (Y_{i}(t))_{1 \leqslant i \leqslant 3}, one obtains:J_{f}^{(n)}(u, \sigma ) = \frac{\sqrt{\mu }}{\sigma _{2}(T)} \frac{1}{\sigma }
\int \limits _{T}\,f\left({\... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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943c8ae7ea0f1742ca9db99e142450e72a27863a | subsection | 42 | 127 | Coarea formula | This fact comes from the way we obtained
this expansion, the series\sum _{k_{3}=0}^{\infty } a^2(k_{3}, u) k_{3}! being equal to +\infty as the Hermite development inL^2(\mathbb {R}, \phi (x) {\rm \,d}n) of delta's Dirac function in point u.
So now the idea consists in proving that J_{f}^{(n)}(u, \sigma ) tends in L^2... | {
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d1c5c6e3459eaddd49ffc018e1d276aec2788096 | subsection | 43 | 127 | Body | We demonstrate Proposition REF .
One has the following convergence,J_{f}^{(n)}(u, \sigma ) \xrightarrow[\delta \rightarrow 0]{L^2(\Omega )} J_{f}^{(n)}(u).Proof of Proposition REF .
One can easily see that the proof of this proposition follows immediately from that of Theorem REF .\BoxThe function(x, y) \mapsto {\rm ... | {
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3d1b3122254fa1b378a19b2fea66baac867537e8 | subsection | 44 | 127 | Body | Now Y verifies condition (3.18) of since Y(t)=G(t, \nabla X(t)), where\begin{aligned}G: \mathbb {R}^2 \times \mathfrak {L}(\mathbb {R}^{2}, \mathbb {R})& \rightarrow \mathbb {R}\\
(t, A) & \mapsto G(t, A),
\end{aligned}is a bounded continuous function of its arguments.
Y is thus a continuous process on T and we already... | {
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"Corinne Berzin"
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9bf37a3190281e0a39447dc70feb9fc83bd06fd2 | subsection | 45 | 127 | Body | Then applying the second order Rice formula and using similar arguments, working with Y instead of \left| {Y} \right|, we can prove that y \mapsto {\rm E}_{{}}\hspace{-2.0pt}\left[{\int _{\textrm {C}_{T,\, X}(y)} Y(t) {\rm \,d}\sigma _1(t)}\right]^2 is still continuous.
This would achieve the proof of Lemma REF .In thi... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
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edf1cf6cade00d79fd88c21431a18c97dfc38333 | subsection | 46 | 127 | Body | 60) that is, for \tau \in T-T\nabla X(0)&=& \xi + (X(0) \alpha + X(\tau )\, \beta )\\
\nabla X(\tau )&=& \xi ^{\star } - (X(\tau ) \alpha + X(0)\, \beta ),where \xi and \xi ^{\star } are centered Gaussian vectors taking values in \mathbb {R}^2, with joint Gaussian distribution, each of them independent of (X(0), X(\tau... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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6262c7521c25e19459ccb0d7c93b07e98e05c95b | subsection | 47 | 127 | Body | \exists \,A >0, \forall \, \tau \in \overline{T} - \overline{T}, \left({\left\Vert {\tau } \right\Vert _{2} \leqslant A \Longrightarrow r^2_x(0)-r^2_x(\tau ) \geqslant A \left\Vert {\tau } \right\Vert _{2}^2 }\right)
Now let us choose A>0 small enough such that for all \tau \in T - T with \left\Vert {\tau } \right\Ver... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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abcb425af7a0ff904b52b6af67d3cda88661dc67 | subsection | 48 | 127 | Body | If \left\Vert {\tau } \right\Vert _{2} \geqslant A, since r_x is continuous on the compact set K= \overline{T} - \overline{T} \cap \lbrace \left\Vert {\tau } \right\Vert _{2} \geqslant A \rbrace and r_x^2(0)-r^2_x(\tau ) \ne 0 for \tau \in K, thus there exists E>0 such for all \left\Vert {\tau } \right\Vert _{2} \geqsl... | {
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"Corinne Berzin"
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f61c120f44276c0f41295cec0e29074d023a79ff | subsection | 49 | 127 | Body | For all y_1, y_2 \in \mathbb {R},H(y_1, y_2)=\\
\int _{T \times T}
{\rm E}_{{}}\hspace{-2.0pt}\left[{\left| {Y(t_1)} \right| \left| {Y(t_2)} \right| \left\Vert {\nabla X(t_1)} \right\Vert _{2} \left\Vert {\nabla X(t_2)} \right\Vert _{2}|X(t_1)=y_1, X(t_2)=y_2}\right] \\
\times {p}_{X(t_1), X(t_2)}(y_1, y_2) {\rm \,d}t_... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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4ceb5947984d21ce4740a5d65d75479671264d28 | subsection | 50 | 127 | Body | Then for all n \in \mathbb {N}^{\star },\left\Vert {I_{f}^{(n)}(y_k)- I_{f}^{(n)}(y)} \right\Vert _{L^2(\Omega )}\xrightarrow[k \rightarrow +\infty ]{} 0,where \left\Vert {\cdot } \right\Vert _{L^2(\Omega )} stands for the L^2(\Omega )-norm.
Note that Lemma REF imply Theorem REF .Proof of Lemma REF .
Let us give an o... | {
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7453c9adf878c20b0d62f3e7d90ab4c6a7b8feb1 | subsection | 51 | 127 | Body | Furthermore, assertion (1) implies that for all m \in \mathbb {N}^{\star },\left\Vert {I_{f, m}^{(n)}(y_k)} \right\Vert _{L^2(\Omega )} \xrightarrow[k \rightarrow +\infty ]{} \left\Vert {I_{f, m}^{(n)}(y)} \right\Vert _{L^2(\Omega )},so that Scheffé's lemma allows to conclude that for all m \in \mathbb {N}^{\star },\li... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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e9791bc263eff64c1c38fe6bdafb0427f659c7ab | subsection | 52 | 127 | Body | The closed sets T^{2m} and T^{(m)} are defined byT^{2m}=\left\lbrace x \in \mathbb {R}^2, d( x , T^{c}) \leqslant \tfrac{1}{2m}\right\rbrace \mbox{ and } T^{(m)}=\left\lbrace x \in \mathbb {R}^2, d( x , T^{c}) \geqslant \tfrac{1}{m}\right\rbrace ,T^c denoting the complement of T on \mathbb {R}^2.We have shown in (, Lem... | {
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5ef881d06746b547c332177505d5b862ea72d37e | subsection | 53 | 127 | Body | Now going back to the upper bound given in (REF ), let us look closer at its first and third terms.By using assertion (1) one proves that Z_m=Y-Y_m verifies assumptions of Lemma REF and then, one has\lim _{k \rightarrow +\infty } \left\Vert {I_{f}^{(n)}(y_k)- I_{f, m}^{(n)}(y_k)} \right\Vert _{L^2(\Omega )}= \left\Vert... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
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0239a281a55926b3350286013e85b15b6c7a4457 | subsection | 54 | 127 | Body | Thus by using inequality (REF ) and convergences obtained in (REF ) and (REF ), one concludes that\underset{k \rightarrow +\infty }{\overline{\lim }} \left\Vert {I_{f}^{(n)}(y_k)- I_{f}^{(n)}(y)} \right\Vert _{L^2(\Omega )} =0,yielding Lemma REF .
\BoxTheorem REF follows from last lemma.
\BoxWe are now ready to propose... | {
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} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
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be6ee27f57ecb8d8e81030cd3f60cca90a122aca | subsection | 55 | 127 | The functional | One has the following expansion in L^2(\Omega ),\xi _{f}^{(n)}(u) = \frac{1}{\sqrt{\sigma _{2}(T)}} \sum _{q=1}^{\infty } \sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \right|=q
}
} a_{f}({{k}}, u)\, \int _{T} \widetilde{H}_{{{k}}}(Y(t))\, {\rm \,d}t,where coefficients a_{f}({{k}}, u) have been defined by (REF ).... | {
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bf96cb48f8279f479a208a8c6679135ab751eefb | subsection | 56 | 127 | The functional | Let X=(X_i)_{i=1,\,2,\,3} and Y=(Y_j)_{j=1,\,2,\,3} be two centered standard Gaussian vectors in \mathbb {R}^3 such that for 1 \leqslant i, j \leqslant 3, {\rm E}_{{}}\hspace{-2.0pt}\left[{X_iY_j}\right]=\rho _{ij}, then for {{k}}, {{m}} \in \mathbb {N}^3, one has{\rm E}_{{}}\hspace{-2.0pt}\left[{\widetilde{H}_{{{k}}}(... | {
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"Corinne Berzin"
] | [
"math.PR"
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... | |
3b8bc252d7fef764bbf5bb5592cc23e20d118ab1 | subsection | 57 | 127 | The functional | Remember that coefficients a_{f, \sigma }({{k}}, u) have been defined by (REF ).Applying the Fatou's lemma, since \lim _{\sigma \rightarrow 0}a_{f, \sigma }({{k}}, u)=a_{f}({{k}}, u), we obtain{\rm E}_{{}}\hspace{-2.0pt}\left[{\pi ^Q(\eta (T))}\right]^2 &\leqslant \mathop {\underline{\lim }}\limits _{\sigma \rightarrow... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
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73f8b85ea4635950b6381bedd08bc7487d44d0a0 | subsection | 58 | 127 | The functional | \BoxIt remains to prove that J_{f}^{(n)}(u)= \eta (T) in L^2(\Omega ).As in the proof of Theorem REF , we write \left\Vert {\cdot } \right\Vert _{L^2(\Omega )} for the L^2(\Omega )-norm.For fixed Q \in \mathbb {N} and \sigma >0, one has the following inequalities\left\Vert {J_{f}^{(n)}(u)-\eta (T)} \right\Vert _{L^2(\O... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.02314809523522854,
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8c71542ab25010bfe74bf0cb7ce572bbdd5908e3 | subsection | 59 | 127 | The functional | Indeed it is enough to remark that a(0, u)= \phi \bigg (\dfrac{u}{\sqrt{r_z(0)}}\bigg )\dfrac{1}{\sqrt{r_z(0)}}=p_{X(0)}(u) and that
a_{f}(0, 0)={\rm E}_{{}}\hspace{-2.0pt}\left[{f\left(\dfrac{\nabla X(0)}{\left\Vert {\nabla X(0)} \right\Vert _{2}}\right) \left\Vert {\nabla X(0)} \right\Vert _{2}}\right], since\left\Ve... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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2a40e8f5384a0c03e695235611fc66f958b1cc67 | subsection | 60 | 127 | Asymptotic variance for | The functionals \xi _{f}^{(n)}(u) are also orthogonal in L^2(\Omega ).
This is a crucial fact for computing its variance.
Using the Arcones inequality (see , Lemma 1, p. 2245), we deduce the asymptotic variance of \xi _{f}^{(n)}(u) as T grows to \mathbb {R}^2, this variance depending on the level u as follows.
We hav... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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8a51896f1ac911c1e9807afbee794eabfa1de257 | subsection | 61 | 127 | Asymptotic variance for | \xi _{f}^{(n)}(u)=2n\left(J_{f}^{(n)}(u)-{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f}^{(n)}(u)}\right]\right).Since the random variable \xi _{f}^{(n)}(u) is a centered one, using equality given in (REF ), we obtain{\rm Var}_{{}}\hspace{-2.0pt}\left[{\xi _{f}^{(n)}(u)}\right]={\rm E}_{{}}\hspace{-2.0pt}\left[{\xi _{f}^{(n)}(... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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6d59819f0b9dc3b90917f5f8dcdf2354b3fc7d53 | subsection | 62 | 127 | Asymptotic variance for | \prod _{1 \leqslant i, j \leqslant 3} \frac{(\mathit {\Gamma }^Y_{ij}(v))^{d_{ij}}}{d_{ij}!},where\mathit {\Gamma }^Y_{ij}(v)= {\rm E}_{{}}\hspace{-2.0pt}\left[{Y_i(0)Y_j(v)}\right].Since\mathit {\Gamma }^Y(v)&=(\mathit {\Gamma }^Y_{ij}(v))_{1\leqslant i, j \leqslant 3}\\
&=
\begin{pmatrix}
-\displaystyle \frac{1}{\mu ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.0271... | |
6f600235fd9034c19587f9ce908d0830d600755d | subsection | 63 | 127 | Asymptotic variance for | We can apply the Lebesgue convergence theorem and obtain, for {{k}}, {{m}} \in (\mathbb {N}^{3})^{\star },\lim _{n \rightarrow +\infty }R_n({{k}}, {{m}}) = R({{k}}, {{m}})= \int _{\mathbb {R}^2} {\rm E}_{{}}\hspace{-2.0pt}\left[{\widetilde{H}_{{{k}}}(Y(0)) \widetilde{H}_{{{m}}}(Y(v))}\right] {\rm \,d}v.Now turning back... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... | |
5209f80f25627eb676e7e9d4fa2c14c03f9a1acf | subsection | 64 | 127 | Asymptotic variance for | First, let us remark that the convergence in (REF ) is equivalent to the following one:\lim _{Q \rightarrow +\infty } {\rm Var}_{{}}\hspace{-2.0pt}\left[{\pi _Q(\xi _{f}^{(n)}(u))}\right]=0,uniformly with respect to n, where \pi _Q stands for the projection onto the chaos of strictly upper order in Q.For the sake of si... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.0... | |
bf64c9e8125e05ceabc43f4618211a318d005823 | subsection | 65 | 127 | Asymptotic variance for | Let us also introduce the set of indices I_n=[-n, n[^2 \,\cap \,\mathbb {Z}^2, clearly we have\pi _Q(\xi _{f}^{(n)}(u))= \frac{1}{2n}\, \sum _{s \in I_n} \theta _s(\pi _Q(\xi _{f,1}(u))),where the random variable \xi _{f,1}(u) is\xi _{f,1}(u)= \sum _{q=1}^{\infty } \sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \r... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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5b1534050994624a0a009fc956a70e30a33f0afc | subsection | 66 | 127 | Asymptotic variance for | By Schwarz inequality and since \alpha _s(n) \leqslant (2n)^2, using the stationarity of X one has the following upper bound,\left| {V_{n, Q}^{(1)}} \right| \leqslant (2(a+3))^2\, {\rm E}_{{}}\hspace{-2.0pt}\left[{\pi _Q(\xi _{f,1}(u))}\right]^2,which goes to zero as Q goes to infinity uniformly with respect to n, sinc... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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7f540946ae8e8d9c1165e636b84b70d0b7881912 | subsection | 67 | 127 | Asymptotic variance for | So let us suppose that \sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \right|=q
}
} a^2_{f}({{k}}, u) {{k}}! \ne 0.We are going to apply Arcones inequality (see , Lemma 1 p.
2245).
By using notations of this lemma, we apply it to f=F_q and to X=(X^{(j)})_{1\leqslant j \leqslant 3}= Y(t) and Y=(Y^{(k)})_{1\leqslant... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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... | |
387be629c94e046b05940d4457334c7c627184ea | subsection | 68 | 127 | Asymptotic variance for | By Lemma REF , {\rm E}_{{}}\hspace{-2.0pt}\left[{F_q(X)\widetilde{H}_{{{m}}}(X)}\right]= \sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \right|=q
}
} a_{f}({{k}}, u) {{k}}! 1_{\left| {{{k}}} \right|=\left| {{{m}}} \right|}, which implies {\left| {{m}} \right|}=q and rank F_q= q.Thus we have all the ingredients to ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.... | |
10f2fbc51942ee2ed4a9290e8ac93262a2ba1dcc | subsection | 69 | 127 | Asymptotic variance for | For q \geqslant 2, using inequality given in (REF ) we get the bound{\rm E}_{{}}\hspace{-2.0pt}\left[{F_q(Y(t))F_q(Y(s+v))}\right] \leqslant \psi ^q\, {\rm E}_{{}}\hspace{-2.0pt}\left[{F_q(Y(t))}\right]^2\\
\leqslant \rho ^{q-2}\, (3{\bf L})^2\, \Psi ^2(A(s-t+v))\, (\sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0... | |
a43a75e02feebd2cefb85f9790438fdca31620d2 | subsection | 70 | 127 | Asymptotic variance for | \leqslant {\bf C}\, \sum _{q=2}^{\infty } \rho ^{q-2}\, (q+1)^2\, {\bf L}^q(u)< +\infty ,last finiteness providing from inequality (REF ).This yields Lemma REF .
\BoxProposition REF ensues.
\BoxNow, we have got all the tools to prove that the random variable \xi _f^{(n)}(u) converges in law as n tends to infinity to a ... | {
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"Corinne Berzin"
] | [
"math.PR"
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69b4c8b4b6b492fba70b8ec282ef4890a8b76aa9 | subsection | 71 | 127 | Convergence in law of | Using the Peccati and Tudor theorem (see ), we obtain the following theorem.\xi _{f}^{(n)}(u) \xrightarrow[n \rightarrow +\infty ]{Law} \mathcal {N}(0; \mathit {\Sigma }_{f, f}(u)).If f is a function with constant sign, then \mathit {\Sigma }_{f, f}(u) >0.For example if f \equiv 1, we find that the curve length converg... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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858ea8a720d6d99f309c3ab524cacb2cfce5d8b6 | subsection | 72 | 127 | Convergence in law of | For this purpose and in order to apply the Peccati and Tudor theorem (see , Theorem 1), we will give an expansion of this random variable into the Wiener-Itô chaos of order less or equal to Q.To this end, remember that for any t \in \mathbb {R}^2 one has defined{{X}}(t)= \left({\nabla X(t), X(t) }\right)^t,and Y(t)= \D... | {
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] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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9cc4679472e95ca9da8342c774b8decb9e31f12e | subsection | 73 | 127 | Convergence in law of | 30), for fixed {{k}}=(k_i)_{1 \leqslant i \leqslant 3}^t \in \mathbb {N}^3 such that \left| {{{k}}} \right|=q,\widetilde{H}_{{{k}}}(Y(t))
&= \prod _{j=1}^3 H_{k_j}(Y(t))\\
&= \int _{\mathbb {R}^{2q}} \varphi _{t, 1}^{\otimes k_1} \otimes \varphi _{t, 2}^{\otimes k_2} \otimes \varphi _{t, 3}^{\otimes k_3} (\lambda _1, \... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.01972624845802784,
-0.007673861924558878,
0... | |
ca1277b78efd294c49461ead45eeec74376af80f | subsection | 74 | 127 | Convergence in law of | So we are going to write function f_q^{(n)} in another way.For {{k}} =(k_i)_{1 \leqslant i \leqslant 3}^t such that \left| {{{k}}} \right|=q, we define\mathcal {A}_{{{k}}}= \lbrace m=(m_1, \dots , m_q) \in \lbrace 1, 2, 3\rbrace ^q,
\forall i=1, 2, 3,\, \sum _{j=1}^q 1_{\lbrace i\rbrace }(m_j)=k_i \rbrace ,one has \ope... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.044407907873392105,
-0.00120080669876188... | |
825bf4996bd333e2c31e1fa29a133fcfe24d4f87 | subsection | 75 | 127 | Convergence in law of | Thus{\sum _{
{
{{k}} \in \mathbb {N}^3 \\ \left| {{{k}}} \right|=q
}
} a_{f}({{k}}, u)\, \operatorname{sym}(\varphi _{t, 1}^{\otimes k_1} \otimes \varphi _{t, 2}^{\otimes k_2} \otimes \varphi _{t, 3}^{\otimes k_3})}\\
&\qquad =
\operatorname{sym}\Big (\sum _{m \in \lbrace 1, 2, 3\rbrace ^q} b_{f}(m, u) \varphi _{t, m_1... | {
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"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.0077499... | |
59f270be37c594074674f9b7bfafbb2e076b5480 | subsection | 76 | 127 | Convergence in law of | LetC_n=\int _{(\mathbb {R}^2)^{2(q-p)}} \left| {f_q^{(n)} \otimes _p f_q^{(n)}(\lambda _1, \dots , \lambda _{q-p}, \mu _1, \dots , \mu _{q-p})} \right|^2\, \\
{\rm \,d}\lambda _1 \dots {\rm \,d}\lambda _{q-p} {\rm \,d}\mu _{1}\dots {\rm \,d}\mu _{q-p}.Straightforwards calculations show thatC_n= \left(\frac{1}{2n}\right... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.027652902528643608,
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-0.003956410102546215,
... | |
9a20e9a5b6f8b7d693295c5332ceefddc6e49a88 | subsection | 77 | 127 | Convergence in law of | \BoxHence we proved that for all Q \geqslant 1,\pi ^Q(\xi _f^{(n)}(u)) \xrightarrow[n \rightarrow +\infty ]{Law} N(0; \sum _{q=1}^Q V_q(u)),
where V_q(u) has been defined by (REF ).
On the other hand we proved in Lemma REF that
for all n \geqslant 1,
\pi ^Q(\xi _f^{(n)}(u)) \xrightarrow[Q \rightarrow +\infty ]{L^2(\... | {
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{
"arxiv_id": "",
"doi": "10.1214/aop/1176991884",
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"raw": "E. B. Dynkin, Self-intersection gauge for random walks and for brownian motion, The Annals of Probability 16 (1988), 1–57. MR 0920254",
"s... | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.015451312065124512,
-0.01716812513768673,
... | |
8ff4599c8094c200880ea5ac13203ff6d82b0811 | subsection | 78 | 127 | Convergence in law of | Thus\mathit {\Sigma }_{f, f}(u) \geqslant V_1(u)+ V_2(u).By using Lemma REF and the inversion formula, a computation gives that for \left| {{{k}}} \right|= \left| {{{m}}} \right|=1, R({{k}}, {{m}})=0 except when {k}={m}= (0, 0, 1) and in this case one hasR((0, 0, 1), (0, 0, 1))= \frac{1}{r_x(0)} \int _{\mathbb {R}^2} r... | {
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{
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"doi": "",
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"raw": "Marie F. Kratz and José R. León, Central limit theorems for level functionals of stationary gaussian processes and fields, Journal of Theoretical Probability 14 (2001), no. 3, 639–672. MR 1860517",... | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0.002069737296551466,
0.030155213549733162,
0.00... | |
5d0db3383088ddddc94bfd1bdb45f282c631ad73 | subsection | 79 | 127 | Convergence in law of | Since \det (D) \ne 0, one gets the following equivalence:(V_2(u)=0) \iff (a_f({{k}}, u)=0, \mbox{\, for all\, } {{k}} \in \mathbb {N}^{3} \mbox{\, such that } \left| {\bf {k}} \right|=2)In particular, since f has a constant sign, a_f(0,0) \ne 0 so that a_f((0, 0, 2), 0) \ne 0 and V_2(0) >0.Finally we proved that for u ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.024152124300599098,
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... | |
1411c8309994835fc86ea7cb94e6d94d7777efe2 | subsection | 80 | 127 | Towards a test of isotropy | Let u be a fixed level in \mathbb {R}. | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.026078928261995316,
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0.023683147504925728,
0.019593531265854836,
0.016297422349452972,
0.00... | |
8bf7622418a3fb756f9ef79435fe1f91e1728b15 | subsection | 81 | 127 | Convergence in law for the affinity parameters | First, we apply results of Section REF .Recalling that f^{\star }=(f_1^{\star }, f_2^{\star }) has been defined in (REF ), by Theorems REF and REF one can show the following proposition.2n\left({
\tfrac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)} - \tfrac{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f^{\star }}^{(n)}(u)}\righ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.032658930867910385,
0.004204455763101578,
-0.028797851875424385,
-0.028492627665400505,
0.023578526452183723,
0.02872154489159584,
-0.005146833602339029,
0.05051451176404953,
0.018313419073820114,
0.022815467789769173,
-0.010698088444769382,
0.026218710467219353,
-0.020968863740563393,
... | |
ac802ccb314ee3581fe59511c23a8892d3c7a2f8 | subsection | 82 | 127 | Convergence in law for the affinity parameters | Since {\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f}^{(n)}(u)}\right] =a_f({\bf 0}, u) the following decomposition ensues2n\left({
\tfrac{J_{f}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)} - \tfrac{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f}^{(n)}(u)}\right]}{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{\bf 1}^{(n)}(u)}\right]}
}\right) =
\tfrac{1}{a_{... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.0027547383215278387,
0.015154875814914703,
-0.022007383406162262,
-0.03574291989207268,
0.036139726638793945,
0.035254545509815216,
-0.007604145910590887,
0.03317895531654358,
0.010210082866251469,
0.01736782304942608,
-0.011423388496041298,
0.0011369972489774227,
-0.03116440773010254,
-... | |
96016780a4165c28eb97f08c0f866f6883806378 | subsection | 83 | 127 | Convergence in law for the affinity parameters | In this aim, for f_1, f_2:S^1 \rightarrow \mathbb {R} continuous and bounded functions, let us note for q \in \mathbb {N}^{\star },\left({\mathit {\Sigma }_{f_1, f_2}(u)}\right)_q=\sum _{
{
{{k}, {m}} \in \mathbb {N}^3 \\ \left| {{{k}}} \right|= \left| {{{m}}} \right|=q
}
} a_{f_1}({{k}}, u)\, a_{f_2}({{m}}, u)\, R({{k... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06797904521226883,
0.0034134453162550926,
-0.03951205685734749,
-0.0652940571308136,
0.01347834151238203,
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0.020564576610922813,
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-0.010465355589985847,
0.016521837562322617,
-0.00661712558940053,
0.... | |
0cf2a2eb062509985775567db6915b6a93aee267 | subsection | 84 | 127 | Convergence in law for the affinity parameters | Using arguments similar to those given in the proof of Remark REF , simple calculations give\det \left({\mathit {\Sigma }^{\star }(u)}\right) \geqslant 4 \times (2\pi )^2 \int _{\mathbb {R}^4} f^2_x(s) f^2_x(t) \left\lbrace {g^2(s, t) -g(s, t)g(t, s) }\right\rbrace {\rm \,d}s\, {\rm \,d}t \geqslant 0,whereg(s, t)= s_1^... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.036783065646886826,
0.020955665037035942,
-0.03217373788356781,
-0.02220720425248146,
0.006177571136504412,
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0.04328497126698494,
0.013705889694392681,
0.032875820994377136,
-0.0022474301513284445,
-0.035531528294086456,
0.0059333681128919125,
0.021474596112966537,
... | |
85ac5cca140f9f2dd59684b4e84f0fdfc4c839e3 | subsection | 85 | 127 | Convergence in law for the affinity parameters | We deduce that \mathit {\Sigma }^{\star }(u) will be strictly positive if (A, B, C) \ne (0, 0, 0).Let us see that C \ne 0.By using Lemmas and , straightforward computations show that,C= -\frac{\det (P)}{a_{\bf 1}^2(0, 0)} \times \frac{\lambda _1 \lambda _2 \mu }{\pi (\lambda _1^2 (\omega ^{\star }_1)^2+\lambda _2^2 (\o... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.025428306311368942,
0.023779893293976784,
-0.03556299954652786,
-0.003189985640347004,
0.019613070413470268,
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-0.016575714573264122,
0.0074789137579500675,
0.044049277901649475,
0.01772044599056244,
-0.03345669060945511,
0.0020567013416439295,
0.017064133659005165,
... | |
04962c998aabb11166a709093b8931cdcf928020 | subsection | 86 | 127 | Convergence in law for the affinity parameters | For 0 < \lambda < 1 and -\frac{\pi }{2} < \theta _{o} < \frac{\pi }{2}, one has2n \left({
\widehat{\lambda }_{n} - \lambda ; \,\widehat{\theta }_{o,n} - \theta _{o}
}\right)
\xrightarrow[n \rightarrow +\infty ]{Law} \mathcal {N}(0; \mathit {\Sigma }_{\lambda , \theta _{o}}(u)),where \mathit {\Sigma }_{\lambda , \theta ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.053717125207185745,
0.030353227630257607,
-0.010445844382047653,
0.018160050734877586,
0.01210924331098795,
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-0.014116005040705204,
0.016969727352261543,
-0.008568797260522842,
0.03093312680721283,
-0.03259652853012085,
0.0020964175928384066,
-0.020037097856402397,
... | |
516cb2d4dc27f22628240c856735a65a3e307671 | subsection | 87 | 127 | Convergence in law for the affinity parameters | The decomposition given in (REF ), Proposition REF , Corollary REF and Proposition REF imply that2n\left({
\frac{J_{f^{\star }}^{(n)}(u)}{J_{\bf 1}^{(n)}(u)} - \frac{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{f^{\star }}^{(n)}(u)}\right]}{{\rm E}_{{}}\hspace{-2.0pt}\left[{J_{\bf 1}^{(n)}(u)}\right]}
}\right) =
2n \left({
F_1... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.010276184417307377,
-0.0017279552994295955,
-0.03716820478439331,
0.01342693716287613,
0.02808976359665394,
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0.016020776703953743,
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0.001187254092656076,
-0.02490086480975151,
0.... | |
bbec4f64838e92bb37fa066ac74f590fa271413e | subsection | 88 | 127 | Convergence in law for the affinity parameters | Using a second order Taylor-Young expansion of F^{-1} about F(\lambda , \theta _{o}), we get2n \left({
\widehat{\lambda }_{n} - \lambda ; \,\widehat{\theta }_{o,n} - \theta _{o}
}\right)=\\
\sum _{j=1}^{2} \dfrac{\partial F^{-1}}{\partial x_j}(F(\lambda , \theta _o)) \times 2n(F_j(\widehat{\lambda }_{n} , \widehat{\the... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.044302891939878464,
0.032647449523210526,
-0.030618425458669662,
0.007482978515326977,
0.013585305772721767,
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0.03365433216094971,
-0.018368003889918327,
0.027796098962426186,
-0.07646215707063675,
0.0... | |
c6903497a8b6b8a1779bd6a73a9c62d4a4cd84ba | subsection | 89 | 127 | Confidence intervals for | In this section, we suppose that parameters \lambda and \theta _o are such that 0 < \lambda <1, -\frac{\pi }{2} < \theta _{o} < \frac{\pi }{2}.
We will also assume that the covariance function r_z is a known function.One can build confidence intervals for parameters (\lambda , \,\theta _{o}).
We will show that\mathit {... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02877557836472988,
0.031033683568239212,
-0.036648429930210114,
0.021101074293255806,
0.029263818636536598,
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0.00257660448551178,
-0.009139221161603928,
0.021390965208411217,
0.0008467892766930163,
-0.... | |
f1436d532ca5c26eebb235b2d290fe8ecf45431d | subsection | 90 | 127 | Confidence intervals for | We propose \widehat{P}_n= (\widehat{v}_{1,n}, \widehat{v}_{2,n}) as estimator of P=(v_1, v_2), the orthonormal basis of eigenvectors of matrix A, with:{\left\lbrace \begin{array}{ll}
\widehat{v}_{1,n}= \cos (\widehat{\theta }_{o,n}) v^{\star } - \sin (\widehat{\theta }_{o,n}) v^{\star \star }\\
\widehat{v}_{2,n} = \sin... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.006386261433362961,
0.0038569357711821795,
-0.023546000942587852,
-0.022798266261816025,
0.03613540530204773,
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0.029466835781931877,
0.004917497746646404,
0.03808867186307907,
0.03586072847247124,
-0.01724366843700409,
0.02427847497165203,
-0.022752488031983376,
0.0... | |
e1d4b49011670d554b0fca314f03d14227db5bd9 | subsection | 91 | 127 | Confidence intervals for | Thus let:D_n(u)= (C^{-1}(\widehat{\lambda }_{n}, \widehat{\theta }_{o,n}))^t Q R_n (\mathit {\Gamma }_{n}^{\star })^{-\frac{1}{2}} R_n^t.Theorem REF implies Corollary REF .
For 0 < \lambda < 1 and -\frac{\pi }{2} < \theta _{o} < \frac{\pi }{2}, one has2n \left({
\widehat{\lambda }_{n} - \lambda ; \,\widehat{\theta }_... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.057412195950746536,
0.034245871007442474,
-0.02128921076655388,
-0.03363542631268501,
0.007744999602437019,
0.003504326334223151,
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0.03339124843478203,
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0.007058251183480024,
-0.02788199856877327,
0.006340980064123869,
0.002430327469483018,
0.0... | |
cf2b371d8c26c6dbf60b7a0e4f51bd065d38f232 | subsection | 92 | 127 | Confidence intervals for | By defining W(v) as the 3-dimensional vector defined as
W(v) = \begin{pmatrix}
\frac{P^t}{\sqrt{\mu }} \nabla Z(v),
\frac{Z(v)}{\sqrt{r_z(0)}}
\end{pmatrix}^t
,
one hasR({{k}}, {{m}})= \int _{\mathbb {R}^2} {\rm E}_{{}}\hspace{-2.0pt}\left[{\widetilde{H}_{{{k}}}(Y(0)) \widetilde{H}_{{{m}}}(Y(v))}\right] {\rm \,d}v\\
= ... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.028062744066119194,
-0.009781521745026112,
0.0166942048817873,
0.009957009926438332,
0.03482282906770706,
0.008820155635476112,
0.027360795065760612,
0.0017090959008783102,
0.016785763204097748,
0.025743257254362106,
-0.007057650480419397,
0.01776238903403282,
0.03277801722288132,
-0.00... | |
3c4e2de248b70a98cef3522d7ab481a59f9bd20c | subsection | 93 | 127 | Confidence intervals for | In this aim, let us compute \mathit {\Sigma }_{f_1, f_1}, similar arguments would be raised for \mathit {\Sigma }_{f_2, f_2} and for \mathit {\Sigma }_{f_1, f_2}, where
f_i= {\frac{f_i^{\star }}{a_{\bf 1}({\bf 0}, u)}- \frac{a_{f_{i}^{\star }}({\bf 0}, u)}{a_{\bf 1}^2({\bf 0}, u)} {\bf 1}}
, i=1, 2.Applying Propositio... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0613480769097805,
-0.008195004425942898,
-0.010919041931629181,
-0.04169227555394173,
0.037724487483501434,
-0.015207302756607533,
-0.012696915306150913,
0.04642309620976448,
0.0004065549874212593,
0.018297597765922546,
-0.03595424443483353,
-0.003614882007241249,
-0.02031201310455799,
... | |
b3e4cd60d1527a3eec523d5cdc0b298f2dca38b5 | subsection | 94 | 127 | Confidence intervals for | Let A=R \mathit {\Gamma } R^t a definite positive matrix such that R is an unitary matrix, while \mathit {\Gamma } is a diagonal one.
Let also (A_n)_n be an approximation of matrix A, i.e.
\lim _{n}A_n=A, such that A_n=R_n \mathit {\Gamma }_n R_n^t with R_n an unitary matrix and \mathit {\Gamma }_n a diagonal one.
Cons... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04717802628874779,
0.02876150794327259,
-0.018187647685408592,
-0.04534705728292465,
-0.0007848382228985429,
-0.03607013449072838,
-0.02599979192018509,
0.02870047464966774,
0.021590203046798706,
0.02560308203101158,
-0.047055963426828384,
0.01974397338926792,
0.0079418383538723,
0.0029... | |
fbd831876a46c45a0add95c6a340af04d3ff3b54 | subsection | 95 | 127 | Complementary results for estimating the parameter | We emphasize that convergence result in Theorem REF is valid under the assumption that
0 < \lambda < 1 and -\frac{\pi }{2} < \theta _{o} < \frac{\pi }{2}.
However, we can better elaborate what is happening to \widehat{\lambda }_{n} in the isotropic case, when \lambda =1 (and -\frac{\pi }{2} < \theta _{o} \leqslant \fra... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.029847413301467896,
0.014122587628662586,
-0.004131486173719168,
0.009964397177100182,
0.0009832784999161959,
-0.015839271247386932,
-0.012985761277377605,
-0.00026560918195173144,
0.0021935408003628254,
0.004532046150416136,
-0.051180075854063034,
0.00679806899279356,
-0.0068514766171574... | |
bee462a730efdd20a0569af6e49e670cb93bb92a | subsection | 96 | 127 | Complementary results for estimating the parameter | For \lambda =1,
2n\, (1-\widehat{\lambda }_{n})
\xrightarrow[n \rightarrow +\infty ]{Law} \sqrt{U},
where the density f_U(t) of the positive random variable U is given by:
f_U(t)= \left({\displaystyle \frac{1}{4\pi } \int _{0}^{2\pi } e^{\ -t \frac{(\cos (\theta )-a \sin (\theta ))^2}{2\, \sigma _{11}^2\,(1+a^2)}}\,... | {
"cite_spans": []
} | 1801.03760 | Estimation of Local Anisotropy Based on Level Sets | [
"Corinne Berzin"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03320230171084404,
-0.012489008717238903,
-0.01853896863758564,
-0.026793768629431725,
0.02041575312614441,
-0.04684332013130188,
-0.013145120814442635,
0.008193766698241234,
0.00440205167979002,
0.0485522635281086,
-0.03356850519776344,
0.009971371851861477,
-0.021010830998420715,
0.00... |
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