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with the appropriate scalings. The integrands in (1.25) and (1.24) are bounded by n-independent integrable functions on ]0 ,1], as it can be seen with (2.63), (2.40) and with the bound given in [24, Corollary 2.6]. By Lebesgue’s dominated convergence theorem, the limit n→ ∞ can be taken inside the integrals. Using, Pro... | https://arxiv.org/abs/2501.15765v2 |
much more efficient to use. One can even go further and use the explicit expressions of the Wright generalized Bessel function (2.76) and the Mittag-Leffler function (2.82) to compute the remaining integral term by term. Finally, one can check that Assumptions 2.1 are verified as well. Explicitly, using the asymptotic ... | https://arxiv.org/abs/2501.15765v2 |
Theorem 3.2 For the Jacobi ensemble with P´ olya weight wJac(x) = xα(1− x)β+n−1Θ(1−x), with α, βfixed and α >−1,β >0, the point-wise limit cov(∞) 1,kSH(r;a1, . . . , a k) := lim n→∞n3/2−kcov1,k1 2−r√n; 1−a1 n2, . . . , 1−ak n2 (3.8) of the 1, k-cross-covariance density functions between one squared eigenradius and k ... | https://arxiv.org/abs/2501.15765v2 |
nn−2 ⩽2e−v, (4.6) one has ˆn xdv(1 +v) 1−v nn−2 exp(f(v))⩽2ˆn xdv(1 +v) exp(−v+f(v)) ⩽2ˆ∞ xdv(1 +v) exp(−v+f(v)).(4.7) Due to the asymptotic behavior of f,f(u) =o(u), as u→ ∞ , F(x) := 2ˆ∞ xdv(1 +v) exp(−v+f(v)) (4.8) is the remainder of a converging integral of a continuous integrand on R∗ +. Moreover, F(0) is well... | https://arxiv.org/abs/2501.15765v2 |
0. Finally, to finish the proof, one expands the determinant in (4.1) using Leibniz for- mula, uses the finite limit assumption (2.7) for the terms where none of the arguments are integrated and uses the vanishing limits of I± 0,n(R), I± 1,n(R) and I± 2,n(R) computed above. ■ Proof of Theorem 2.5. Starting from (1.16),... | https://arxiv.org/abs/2501.15765v2 |
=w(∞)(r)∞X k=0rk ˜w(∞)(k+ 1)=ˆ∞ 0dtˆ∞ 0dv vφ(∞)(v, t)K(∞)(rv,−rt). (4.29) Using Lemma 4.3, along with Proposition 2.13, one can interchange the order of inte- gration, as everything is absolutely integrable. This is due to the exponential decay of φ(∞)(2.16) and the last of Assumptions 2.8, which implies the asymptotic... | https://arxiv.org/abs/2501.15765v2 |
. As we focus on the first term of the asymptotic expansion of (5.5) when n→ ∞ , it appears that one needs to compute the following integrals ˆ∞ 0dt K n 1−λ1 n2,−rt (1 +t)−(n+2)(1 +rt)γ, γ = 0,1 (5.6) and ˆr 0dv K n v,1−λ2 n2 1−v rn−2 (1−v)γ, γ = 0,1. (5.7) Unfortunately, none of the factors inside the integral a... | https://arxiv.org/abs/2501.15765v2 |
+ 2 x) 1 +x+O nβ−1En(−x) (1 +x)# . (5.18) Finally, using the following asymptotic equivalent for an(3.3), an∼ n→∞−n 2, (5.19) finishes the proof. ■ Corollary 5.7 For any fixed α >−1, β > 0, the expansion Kn v,1−λ n2 = n→∞−n vα(1−v)β−1h B(α,β) n(λ)P(α,β−1) n (1−2v) +O nβ−1En(v)i . (5.20) holds uniformly for λin co... | https://arxiv.org/abs/2501.15765v2 |
0dt(1 +t)−(n+2)P(α,β−1) n (1 + 2 rt) (1 +rt)1−γ, γ = 0,1, (5.28) one can bound P(α,β−1) n (1+2 rt) by its maximum, which is attained at t=tεand take the very rough upper bound ˆtε 0dt(1 +t)−(n+2)P(α,β−1) n (1 + 2 rt) (1 +rt)1−γ⩽1 n+ 1P(α,β−1) n (1 + 2 rtε), γ = 0,1.(5.29) To estimate the order of the upper bound as n→ ... | https://arxiv.org/abs/2501.15765v2 |
becomes bY(α,β) n,γ(r) =ˆyr−ε yr−1du√n gr,γ(yr)−ug′ r,γ(yr) exp −n fr(yr)−uf′ r(yr) +u2 2f′′ r(yr) (5.40) and can be computed using the following lemma, which is obtained with a simple integration by part. Lemma 5.8 Leta, b, c, d, h, x 0, x1∈R,a̸= 0, ˆx1 x0(d−hx) exp −a 2x2−bx+c dx =h a e−ax2 1 2−bx1+c−e−ax2 0... | https://arxiv.org/abs/2501.15765v2 |
n,0(r)− 1 +1 nr J(α,β) n,1(r) +I(α,β) n,1(r) 1−1 nr J(α,β) n,0(r) +1 nrJ(α,β) n,1(r) .(5.57) The explicit expression of G(α,β) n(r) is rather cumbersome, however, there is some simplification by noting 1 nr−r n J(α,β) n,0(r)− 1 +1 nr J(α,β) n,1(r) =−rα+1(1−r)β+n−1 n 1−r+1 +α−r(α+β−1) n (5.58) and 1−1 nr... | https://arxiv.org/abs/2501.15765v2 |
In the first part of this article, we have exploited the new results of [5], for polynomial ensembles, to study their extension to the large nlimit, which are thus also new. First, we focused on the double scaling limit around the origin for the 1 , k-point correlation function between one eigenradius and ksingular val... | https://arxiv.org/abs/2501.15765v2 |
the main results, we strongly believe the formula (2.15) for the lim- iting 1 , k-point function (equivalently, the 1 , k-cross-covariance) at the origin remains true even in the case the limiting kernel is not a function anymore but a general dis- tribution e.g. a weighted Dirac delta function, as the formula (2.15) s... | https://arxiv.org/abs/2501.15765v2 |
rewrite the sum of polynomials as another polynomial and use the Assumptions 2.8 to show the non-vanishing contribution of the integrand in (A.2) remains in a bounded interval close to the origin as n→ ∞ . The assumption (2.28) excludes a fix number of the first summands. This is not an issue as one can split the sum i... | https://arxiv.org/abs/2501.15765v2 |
bound assumption (2.26) on S(1) n(R), one gets, for all nlarge enough, S(1) n(R)⩽C1 1 +j∗! n−j∗+ 2 R 1−R n+Rn−j∗+1j∗−1X j=0(R|x|)j j!, (A.18) S(2) n(R)⩽C2 1 +Γ(R+ 1) n−R+ 2 R 1−R n+Rn−⌊R⌋+1⌊R⌋−1X j=j∗(R|x|)j j!h(j),(A.19) where C1, C2are some positive constants independent of nandR. As j∗isn- independent, takin... | https://arxiv.org/abs/2501.15765v2 |
47(3), 035305 (2013) https://doi.org/10.1088/1751-8113/47/ 3/035305 arXiv:1308.2609 [11] Bhosale, U.T., Tekur, S.H., Santhanam, M.S.: Scaling in the eigenvalue fluctua- tions of correlation matrices. Phys. Rev. E 98, 052133 (2018) https://doi.org/10. 1103/PhysRevE.98.052133 arXiv:1807.07968 [12] Brunelli, M., Wanjura, ... | https://arxiv.org/abs/2501.15765v2 |
PoS LATTICE2012 , 087 (2012) https://doi.org/10.22323/ 1.164.0087 arXiv:1212.2141 [hep-lat] [32] Kuijlaars, A.B.J., Zhang, L.: Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits. Com- munications in Mathematical Physics 332(2), 759–781 (2014) https://doi... | https://arxiv.org/abs/2501.15765v2 |
arXiv:2501.15926v2 [math.ST] 24 Feb 2025Rates of convergence of a binary classification procedure fo r time-homogeneous S.D.E paths Eddy-Michel Ella-Mintsa(1,2) February 25, 2025 (1) LAMA, Université Gustave Eiffel (2) MIA Paris-Saclay, AgroParisTech Abstract In the context of binary classification of trajectories gene ra... | https://arxiv.org/abs/2501.15926v2 |
study the minimax convergence rate of an emp irical classification procedure /hatwideg, built from the learning sample DNsuch that its excess risk P(/hatwideg(X)∝\e}atio\slash=Y)−P(g∗(X)∝\e}atio\slash=Y) tends to zero as the size of the learning sample Ntends to infinity. Related works. Until now, few papers have tackled... | https://arxiv.org/abs/2501.15926v2 |
estimation of order(Nn)−β/(4β+1)on the Hölder space of smoothness parameter β≥1. In this paper, we study the minimax rate of convergence of the risk of estimation of the drift coefficient on the real li ne over the Hölder space. 2 Main contributions. In Denis et al. (2024), the plug-in-type nonparametric procedure for mu... | https://arxiv.org/abs/2501.15926v2 |
the real line of order exp(c/radicalbig log(N))N−β/(2β+1)over the space of Hölder functions, where the diffusion coeffic ient is unknown. Since the diffusion coefficient is now assumed to be un known, this result extends to a more general framework, the result obtained in Denis et al. (2024), and requires strong assumptions ... | https://arxiv.org/abs/2501.15926v2 |
regression function given for all x∈RbyΦ∗(x) =P(Y= 1|X=x), andΦ∗(X) = P(Y= 1|X)is the probability of belonging of the diffusion process Xto the class 1. The Bayes classifier g∗is the classifier that minimizes the classification error, th at is g∗= argmin g∈GR(g), (4) whereGis a chosen set of classification rules. Since the ... | https://arxiv.org/abs/2501.15926v2 |
.1 and 2.2. First, the diffusion model (2) admits a unique strong solution X= (Xt)t∈[0,1](see Karatzas & Shreve (2014)). Second, the diffusion process X= (Xt)t∈[0,1]admits a transition density (t,x)∝ma√sto→ΓX(t,x)given as follows: ΓX(t,x) =P(Y= 0)Γ0,X(t,x)+P(Y= 1)Γ1,X(t,x), whereΓ0,XandΓ1,Xare the transition densities of... | https://arxiv.org/abs/2501.15926v2 |
Xat discrete time with time-step ∆n= 1/n. We suppose that the diffusion paths ¯Xj, j= 1,...,N are high-frequency observations of the diffusion process X, that is, the time step ∆ntends to zero, which is equivalent to ntends to infinity. From the learning sampleDN, the distribution p∗= (p∗ 0,p∗ 1)of the label Yis estimated... | https://arxiv.org/abs/2501.15926v2 |
(Bℓ)ℓ∈[[−M,KNi−1]], that is SKNi= h=KNi−1/summationdisplay ℓ=−MaℓBℓ,a= (a−M,...,a KNi−1)∈RKNi+M . Then, any element hof the space SKNiis aM−1continuously differentiable function. For more details on the B-spline basis, we refer the reader to Györfi et al. (2006) or De Boor (1978). We propose a projection estimato... | https://arxiv.org/abs/2501.15926v2 |
the projection estimator We study the minimax convergence rate of the nonparametric e stimator/hatwidebiof the drift function b∗ i, i∈ Y, built in Section 3.1.2. The goal is to extend the rate obtain ed in Denis et al. (2024), Theorem 5 in the setting where the diffusion coefficient is assumed to be unk nown. The followin... | https://arxiv.org/abs/2501.15926v2 |
3.3. For each label i∈ Yand on the event {Ni>1}, setANi=/radicalbigg2β 2β+1log(Ni),KNi= N1/(2β+1) ilog−5/2(Ni),/tildewideAN= log(N),/tildewideKN=N1/(2β+1), and suppose that /vextenddouble/vextenddouble/vextenddoubleΨ−1 KNi/vextenddouble/vextenddouble/vextenddouble op≤CNi log2(Ni). Under Assumptions 2.1, 2.2, 2.3 and 2.... | https://arxiv.org/abs/2501.15926v2 |
for instance, a classification procedure that disc riminates high-price financial assets from low-price ones. Back to Model (12) and conditional on the even t{Y=i}, the unique strong solution X= (Xt)t∈[0,1]admits a transition density Γi,Xgiven by Γi,X(t,x) =1√ 2πα∗2texp/parenleftbigg −(x−µit)2 2α∗2t/parenrightbigg , i∈ {... | https://arxiv.org/abs/2501.15926v2 |
Y={0,1}. In this case, we have f∗ 0(x) = µ∗ 0, f∗ 1(x) =µ∗ 1andψ∗(x) =−x, for allx∈R. Thus, the functions b∗ 0andb∗ 1are unbounded and satisfy b∗ 0−b∗ 1=µ∗ 0−µ∗ 1. We establish the following result. Proposition 3.6. Suppose that σ∗= 1. Under Assumptions 2.1, 2.2, 2.4 and 3.5, and for all i,j∈ Y such thati∝\e}atio\slash... | https://arxiv.org/abs/2501.15926v2 |
of the lower bound of the worst excess risk of the plug-in classifier to the multiclass setup. We also think of extendin g these results to mixture models of time- inhomogeneous diffusion processes, and particularly focus on the Gaussian processes. Finally, we can study faster convergence rates of the excess risk of the ... | https://arxiv.org/abs/2501.15926v2 |
of estimation. Upper bound of the risk of estimation Fori∈ Yand on the event {Ni>1}, recall that bANi=b∗ i1[−ANi,ANi]. The risk of estimation of /hatwidebiis given by E/bracketleftbigg/vextenddouble/vextenddouble/vextenddouble/hatwidebi−b∗ i/vextenddouble/vextenddouble/vextenddouble n,i/bracketrightbigg ≤E/bracketleftb... | https://arxiv.org/abs/2501.15926v2 |
exists p∈(0,1)such that inf /hatwidebisup bi∈Σ(β,R)P/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/hatwidebi−b∗ i/vextenddouble/vextenddouble/vextenddouble n,i≥sN/parenrightbigg ≥p. 20 For this purpose, we set m=/ceilingleftig c0N1/(2β+1)/ceilingrightig , h=1 m, xk=k−1/2 m, Kk(x) = 2RhβK/parenleftbiggx−xk h... | https://arxiv.org/abs/2501.15926v2 |
to fullfill the third condition. For each j∈ {0,...,M}, denote by Qa probability under which the process Xis solution of the stochastic differential equation dXt=dWQ t, t∈[0,1], with WQa Brownian motion. Then, from the Girsanov’s theorem (see Rev uz & Yor (2013), Chapter VIII, Theorem 1.7) we obtain dPj dQ(X) := exp/pare... | https://arxiv.org/abs/2501.15926v2 |
is composed of two main parts. The first part focuses on the estab lishment of the upper bound of the average excess risk of the plug-in classifier /hatwideg=g/tildewideb,/tildewideσ2,/hatwidep, and the second part tackles the study of its lower bound. Upper bound of sup g∗∈GβE[R(/hatwideg)−R(g∗)] For any classifier g∗∈ G... | https://arxiv.org/abs/2501.15926v2 |
0b∗ 1(Xs)ds/parenrightbigg . For anyg∗∈ Gβ, we have g∗(X) =1Φ∗(X)≥1/2=1Q1 b∗ 1(X)≥1=gb∗ 1(X). Sinceb∗ 1=σ∗2,b∗ 1satisfies Assumption 2.2. Then, for any nonparametric estim ator/hatwideb1ofb∗ 1, we consider the truncated estimator /tildewideb1given for all x∈Rby /tildewideb1(x) =1 log(N)1/hatwideb1(x)≤1/log(N)+/hatwideb1... | https://arxiv.org/abs/2501.15926v2 |
/hatwideb1sup b∗ 1∈/tildewideΣ(β,R)P/parenleftig/vextendsingle/vextendsingle/vextendsingleΦ/tildewideb1(X)−Φb∗ 1(X)/vextendsingle/vextendsingle/vextendsingle≥ε 2/parenrightig ≥ε Cexp(−c) inf /hatwideb1sup b∗ 1∈/tildewideΣ(β,R)P/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleQ1 /til... | https://arxiv.org/abs/2501.15926v2 |
enough and from Lemma 4.1 in Györfi et al. (2006), E/bracketleftig/vextendsingle/vextendsingle/hatwideα2−α∗2/vextendsingle/vextendsingle21N1>1/bracketrightig ≤4α∗4E/bracketleftbigg1N1>1 N1/bracketrightbigg ≤4α∗4E/bracketleftbigg1N1>0 N1/bracketrightbigg ≤8α∗4 p∗ minN. Then, we obtain, E/bracketleftig/vextendsingle/ve... | https://arxiv.org/abs/2501.15926v2 |
for all t∈[0,1], /angbracketleftbig Mi,j,Mi,j/angbracketrightbig t=/integraldisplayt 0(b∗ i−b∗ j)2(Xs)ds≤C∗2. (56) Fori,j∈ Ysuch thati∝\e}atio\slash=j, under Assumption 2.1, there exists a constant C1>0such that /vextenddouble/vextenddouble/vextenddouble/tildewidebi−b∗ i/vextenddouble/vextenddouble/vextenddouble2 n,j=1... | https://arxiv.org/abs/2501.15926v2 |
>0is a constant, and from the proof of Theorem 6 in Denis et al. (2024), E/bracketleftigg 1Ni>1sup t∈[0,1]P(i)(|Xt|>ANi)/bracketrightigg ≤E/bracketleftigg 1Ni>1 ANiexp/parenleftigg −A2 Ni 3/parenrightigg/bracketrightigg , (64) and finally, P(Ni≤1) = (1−p∗ i)N+Np∗ i(1−p∗ i)N−1≤2N(1−p∗ i)N−1. (65) ForANi=/radicalbig... | https://arxiv.org/abs/2501.15926v2 |
Martingales and BMO pp. 1–24. Lamberton, D. & Lapeyre, B. (2011). Introduction to stochastic calculus applied to finance . Chapman and Hall/CRC. Nagai, T. & Mimura, M. (1983). Asymptotic behavior for a nonl inear degenerate diffusion equation in population dynamics. SIAM Journal on Applied Mathematics 43, 449–464. Revuz,... | https://arxiv.org/abs/2501.15926v2 |
arXiv:2501.15933v2 [math.ST] 24 Feb 2025Minimax rates of convergence for the nonparametric estimat ion of the diffusion coefficient from time-homogeneous SDE paths Eddy-Michel Ella-Mintsa(1,2) February 25, 2025 (1) LAMA, Université Gustave Eiffel (2) MIA Paris-Saclay, AgroParisTech Abstract Consider a diffusion process X= (... | https://arxiv.org/abs/2501.15933v2 |
. First, we investigate the minimax rate of the estimator of σ2from a single diffusion path ( N= 1andn→ ∞). Second, we extend the study to the case of Nindependent discrete observations of the diffusion process X, withN→ ∞. There exists an important literature on statistical infere nce of stochastic differential equations... | https://arxiv.org/abs/2501.15933v2 |
we make the following assumptions. Assumption 2.1. There exists a constant L0>0such that for all x∈R, |b(x)−b(y)|+|σ(x)−σ(y)| ≤L0|x−y|. 2 Assumption 2.2. There exist constants κ0,κ1>0such thatκ0≤1<κ1and ∀x∈R,0<κ0≤σ(x)≤κ1. Assumption 2.3. σ∈ C2(R), and there exist constants γ,C >0such that ∀x∈R,|σ′(x)|+|σ′′(x)| ≤C(1+|x|... | https://arxiv.org/abs/2501.15933v2 |
:=/braceleftig f∈ Cd(I,R),/vextendsingle/vextendsingle/vextendsinglef(d)(x)−f(d)(y)/vextendsingle/vextendsingle/vextendsingle≤R|x−y|β−d, x,y∈R/bracerightig , whered=⌊β⌋is the highest integer strictly smaller than β,I⊂R, and forI=R, we set ΣI(β,R) = Σ(β,R). Finally, from Denis etal. (2021), Lemma D.2 , the approximati... | https://arxiv.org/abs/2501.15933v2 |
a constant C >0such that /vextenddouble/vextenddouble/vextenddoubleΨ−1 KN/vextenddouble/vextenddouble/vextenddouble op≤CKNlog(N) ANexp/parenleftbigg2 3A2 N/parenrightbigg exp/parenleftbigg16−log(N) 24(log(N)−1)A2 N+cAN/parenrightbigg ≤CKNlog(N) ANexp/parenleftbigg2 3A2 N−1 48A2 N/parenrightbigg ≤CN log2(N). In the sequ... | https://arxiv.org/abs/2501.15933v2 |
sk of estimation of σ2. 4 Lower-bound of the risk of estimation of the diffusion coeffic ient In this section, we study the lower bound of the risk of estima tion of the square of the diffusion coefficient. In Section 4.1, we establish the lower bound of th e risk of estimation of σ2in a compact interval[−A,A]that does not d... | https://arxiv.org/abs/2501.15933v2 |
diffusion coe fficient on the real line from i.i.d. diffusion paths This section is devoted to the establishment of a lower bound of the risk of estimation of σ2on the real line Rand on the growing interval [−AN,AN], whereN→+∞. We obtain the following result. Theorem 4.2. Suppose that n> N ,N→ ∞ and grant the assumptions 2... | https://arxiv.org/abs/2501.15933v2 |
the functions S,HandΓ are given by S(x) =/integraldisplayx 01 σ∗(u)du, H(x) =/integraldisplayS(x) 0/parenleftbiggb σ−σ′ 2/parenrightbigg ◦S−1(u)du, x∈R, G=−1 2/bracketleftigg/parenleftbiggb σ−σ′ 2/parenrightbigg2 ◦S−1+σ◦S−1×/parenleftbiggb′σ−bσ′ σ2−σ′′ 2/parenrightbigg ◦S−1/bracketrightigg .(11) Under Assumption 2.4,... | https://arxiv.org/abs/2501.15933v2 |
(20) that /vextenddouble/vextenddouble/hatwideσ2 KN−σ2 AN/vextenddouble/vextenddouble2 n,N≤/vextenddouble/vextenddoubleh−σ2 AN/vextenddouble/vextenddouble2 n,N+2ν/parenleftbig /hatwideσ2 KN−h/parenrightbig +2µ/parenleftbig /hatwideσ2 KN−h/parenrightbig ,withν=ν1+ν2+ν3. Then, it comes, E/bracketleftig/vextenddouble/vex... | https://arxiv.org/abs/2501.15933v2 |
k∆ 2 ≤1 N2n2KN−1/summationdisplay ℓ′=−MN/summationdisplay j=1E /parenleftiggn−1/summationdisplay k=0KN−1/summationdisplay ℓ=−M/bracketleftig Ψ−1/2 KN/bracketrightig ℓ,ℓ′Bℓ(Xj k∆)ζj k∆/parenrightigg2 ≤1 N2n2N/summationdisplay j=1KN−1/summationdisplay ℓ′=−MKN−1/summationdisplay ℓ=−MKN−1/summationdisplay ℓ′′... | https://arxiv.org/abs/2501.15933v2 |
N K2β N+KN Nn+∆2+exp/parenleftig −C1log3/2(N)/parenrightig/bracketrightigg . From Equation (14), and for AN=/radicalbigg1 2β+1log(N), KN∝(Nn)1/(2β+1)withn∝N, there exists a constantC >0depending on β≥1such that E/bracketleftig/vextenddouble/vextenddouble/hatwideσ2 KN−σ2 AN/vextenddouble/vextenddouble2 n/bracketrigh... | https://arxiv.org/abs/2501.15933v2 |
probability measuresPandQby Kdiv(P,Q) = /integraltext logdP dQdP= EP/bracketleftig logdP dQ/bracketrightig ,ifP≪Q, +∞, otherwise. whereEPis the expectation linked to P. Denote by νthe Lebesgue measure and suppose that P≪νandQ≪νand set p=dP dν, q=dQ dν. Ifp>0a.s.andq>0a.s., then the Kullback divergence is give... | https://arxiv.org/abs/2501.15933v2 |
a Brownian bridge such that E(/tildewiderW2 t) =t(1−t), t∈[0,1]. For the case j= 0, we haveσ2 0= 1and the corresponding diffusion process coincides with the s tandard Brownian motion of transition density p0given for all (s,t,x,y)∈[0,1]×[0,1]×R×Rby p0(s,t,x,y) =1/radicalbig 2π(t−s)exp/parenleftbigg −(y−x)2 2(t−s)/parenr... | https://arxiv.org/abs/2501.15933v2 |
deduce that ∀k∈[[0,n−1]],E/bracketleftig ζℓ,1 k∆|Fk∆/bracketrightig =1 ∆E/bracketleftig Mk∆ (k+1)∆|Fk∆/bracketrightig =1 ∆Mk∆ k∆= 0, where(Ft)t∈[0,1]is the natural filtration of the diffusion process X= (Xt)t∈[0,1]. Then for all j∈ {0,...,M}and for allℓ∈ {1,...,N}, we obtain the following likelihood ratio: dPj dP0(¯X... | https://arxiv.org/abs/2501.15933v2 |
k=0G2 m(Xℓ k∆)/bracketrightigg/parenrightigg +Chβ−3∆n−1/summationdisplay k=0EPj/bracketleftig Gm(Xℓ k∆)−G2 m(Xℓ k∆)+O/parenleftig G2 m(Xℓ k∆)/parenrightig/bracketrightig .(60) For allp∈N∗, we have Gp m(Xℓ k∆) = Γp/parenleftig/summationtextm r=1wj rηr(Xℓ k∆)/parenrightigp ≤Γp/parenleftbigg/summationtextm r=1φr/p... | https://arxiv.org/abs/2501.15933v2 |
of a constant pM>0 such that ∀σ2∈ΣM,∀/hatwideσ2:P/parenleftbig/vextenddouble/vextenddouble/hatwideσ2−σ2/vextenddouble/vextenddouble n/parenrightbig ≥pM. (68) For this purpose, we use the set of hypotheses ΣM⊂Σused in the proof of Theorem 4.1 with B=−A= 1,Γ =κ2 1−1 R∝⌊ard⌊lK∝⌊ard⌊l∞, andm=/ceilingleftbig c0(Nn)1/(2β+1)/... | https://arxiv.org/abs/2501.15933v2 |
differential equations in finance. Mathematical finance 7, 1–71. Ella-Mintsa, E. (2024). Nonparametric estimation of the di ffusion coefficient from iid sde paths. Statistical Inference forStochastic Processes pp. 1–56. Florens-Zmirou, D. (1993). On estimating the diffusion coeffi cient from discrete observations. Journal ofapp... | https://arxiv.org/abs/2501.15933v2 |
Statistical Inference for Low-Rank Tensor Models Ke Xu∗1, Elynn Chen†2, and Yuefeng Han‡1 1Department of Applied and Computational Mathematics and Statistics, University of Notre Dame 2Department of Technology, Operations, and Statistics, New York University Abstract Statistical inference for tensors has emerged as a c... | https://arxiv.org/abs/2501.16223v1 |
the goal is to compare the expression levels of gene gin the region sat two different time points t1andt2, the difference of interest can be expressed as ⟨T,A⟩=Tg,s,t 1− Tg,s,t 2. Task 2: Inference of a Subgroup of Entries. In some scenarios, the linear functional ⟨T,A⟩involves a subgroup of entries across one or more ... | https://arxiv.org/abs/2501.16223v1 |
linear functionals under the low-rank matrix trace regression framework with Gaussian design. However, their results required a sample size proportional to p1p2to construct a valid confidence interval for a low-rank parameter matrix M∈Rp1×p2with rank r. This prompts the question: can valid inferences for general linear... | https://arxiv.org/abs/2501.16223v1 |
necessary for valid inference in all scenarios. A detailed discussion on the interplay between the incoherence and alignment conditions is presented in Section 3.6. Inference for General Linear Functionals We establish central limit theorems that facilitate the statistical inference for general linear functionals in bo... | https://arxiv.org/abs/2501.16223v1 |
sampling. Moreover, we relax the dependence on the condition number, ensuring that our analysis remains computationally optimal even as the condition number κof the signal tensor T grows at a rate of O(p1/4) and becomes arbitrarily large. 1.1.1 Tensor Regression Statistical inference in the tensor regression framework,... | https://arxiv.org/abs/2501.16223v1 |
and tensor completion under uniform sampling. Most work in low-rank matrix completion focuses on entrywise inference, typically requiring incoherence conditions. For example, Xia and Yuan [47] employed spectral perturbation techniques, while Chen et al. [11] used a leave-one-out approach for entrywise statistical infer... | https://arxiv.org/abs/2501.16223v1 |
constructing debiased estimators both without sample splitting and with sample splitting, along with their theoretical guarantees for asymptotic normality. This section also explores the relationship between incoherence and alignment conditions and constructs confidence intervals with proven theoretical properties, dem... | https://arxiv.org/abs/2501.16223v1 |
(unfolding) of a tensor, which rearranges the tensor into a matrix by stacking its mode- jfibers as columns. For a tensor T ∈Rp1×p2×p3, the mode- j matricization Mat j(T)∈Rpj×pj+2pj+1is defined as [Mat j(T)]kj,kj+1+pj+1(kj+2−1)=Tk1,k2,k3, where j+1 andj+ 2 are computed modulo 3. For simplicity, we denote the mode- jmat... | https://arxiv.org/abs/2501.16223v1 |
we have ∥[X]j1,j2,j3∥ψ2≤σ, where σis a positive constant, ∥·∥ψ2 denotes the Orcliz ψ2-norm, and there exist positive constants candCsuch that cσ2≤Var(Xj1,j2,j3)≤ Cσ2. Assumption 3 (Subgaussian noise) .In the tensor regression model (7), the noise terms {ξi}n i=1are i.i.d., mean-zero, and σξ-subgaussian. Specifically, w... | https://arxiv.org/abs/2501.16223v1 |
first and second iterations, respectively. For each iteration k= 1,2 and mode j= 1,2,3, the power iteration is performed as follows: For each mode j= 1,2,3,bU(k) jis obtained as the leading r1left singular vectors of Mat j bTunbs×j+1bU(k−1)⊤ j+1 ×j+2bU(k−1)⊤ j+2 = Mat j bTunbs bU(k−1) j+2⊗bU(k−1) j+1 . After complet... | https://arxiv.org/abs/2501.16223v1 |
j=1 PUj⊥AjP(Uj+2⊗Uj+1)G⊤ j F·σ2 ξ σ2 p pr2log(p) nλ+ ∆·pr1/2 nλ! 9 +3X j=1 A ×jUj F·σ2 ξ σ2 r3/2log(p) nλ+ ∆·q pRrlog(p) nλ+ ∆2·pr1/2 nλ + A F·σ3 ξ σ3r3/2log(p)3/2 n3/2λ2+ ∆·p1/2Rr1/2log(p) n3/2λ2+ ∆2·pRr1/2log(p) n3/2λ2+ ∆3·p3/2r1/2 n3/2λ2 , Ω3=3X j=1 PUjAjP(Uj+2⊗Uj+1)G⊤ j F·σ2 ξ σ2·pr1/2 nλ +3X j=1 A ×j+1Uj+1×... | https://arxiv.org/abs/2501.16223v1 |
between incoherence and alignment conditions is provided in Section 3.6. The sub-Gaussian design tensor assumption (Assumption 2) aligns with the sub-Gaussian sampling framework often used in compressed sensing [8]. Notably, our results demonstrate that under sub- Gaussian designs, valid inference for general linear fu... | https://arxiv.org/abs/2501.16223v1 |
leading rjleft singular vectors of Mat j bTunbs,(I) bU(0),(II) j+2⊗bU(0),(II) j+1 ,and Mat j bTunbs,(II) bU(0),(I) j+2⊗bU(0),(I) j+1 , for mode j= 1,2,3, respectively. Unlike the algorithm without sample splitting in Section 3.2, a single iteration suffices due to the independence introduced by sample splitting, wh... | https://arxiv.org/abs/2501.16223v1 |
ξ σ4·p2r1/2 n2λ3! +3X j=1 A × j+1Uj+1×j+2Uj+2 F·σ3 ξ σ3prlog(p)1/2 n3/2λ2 +3X j=1 A × jUj F·σ2 ξ σ2·r3/2log(p) nλ+∥A∥ F·σ3 ξ σ3·r2log(p)3/2 n3/2λ2) , where c >0is a constant, the variance component sAis defined in (2). Statistical optimal sample size w.r.t. p.If the following incoherence condition (3) holds, max j=1,... | https://arxiv.org/abs/2501.16223v1 |
sufficiently aligned with the tangent space of the low-Tucker-rank manifold at T. Specifically, the alignment condi- tions guarantee that the magnitude of the asymptotic normal term ⟨PTTM(r1,r2,r3)(A),PTTM(r1,r2,r3)(bZ(1))⟩ dominates the perturbation terms in the normal space. The tangent space TTM(r1, r2, r3) can be d... | https://arxiv.org/abs/2501.16223v1 |
tensor models. 3.7 Data-driven Inference of Estimated Linear functionals The asymptotic normality of the estimator ⟨bT,A⟩, established in the previous section, provides a foun- dation for statistical inferences about the linear functional ⟨T,A⟩. To construct confidence intervals or perform hypothesis testing in practic... | https://arxiv.org/abs/2501.16223v1 |
us to rigorously show that the proposed procedures attain the fundamental limits of inference accuracy in tensor regression. In what follows, the parameter space Θ( λ, κ) is defined as Θ(λ, κ) := T=G ×1U1×2U2×3U3 G ∈Rr1×r2×r3, Uj∈Opj×rj, λ≤λ≤κλ , (18) where λrepresents any nonzero singular value of Mat j(T) for each m... | https://arxiv.org/abs/2501.16223v1 |
2, andbUinit 3via Higher-Order SVD ( HOSVD , De Lathauwer et al. [12]). For shorthand, we denote bU(0) j:=bUinit jforj= 1,2,3. Since Yis already an unbiased estimator of T,no debiasing step is required in tensor PCA. Assumption 8 (Error Bound for Initial Estimates of Singular Spaces) .We assume that the initial singula... | https://arxiv.org/abs/2501.16223v1 |
linear functionals. Corollary 4.1 (Asymptotic normality of estimated low-Tucker-rank linear functionals) .Under the conditions of Theorem 4.1, assume that the Tucker rank of the loading tensor Asatisfies rank(A) = (R1, R2, R3)is fixed and independent of p. Ifλ≥Cmax{κp1/2,p3/4r1/2log(p)}, and the variance com- ponent sA... | https://arxiv.org/abs/2501.16223v1 |
bσ2in (24) follows the construction in Xia et al. [49] (see their Lemma 1). Its accuracy relies on the assumption that the noise tensor Zhas i.i.d. entries. We further extend the asymptotic normality result to demonstrate that these variance estimates are valid for practical statistical inference. Specifically, the fol... | https://arxiv.org/abs/2501.16223v1 |
while the remaining entries are sampled from a standard normal distribution. In contrast, incoherent singular subspaces are generated by applying SVD to random Gaussian matrices with i.i.d. standard normal entries. Using these subspaces, the signal tensor Tis constructed as T=G ×1U1×2U2×3U3. We set p1=p2=p3=p. We consi... | https://arxiv.org/abs/2501.16223v1 |
via HOSVD for each bTinit,(i), and the debiased estimate is constructed using the data from the alternate subset. For inferring a general linear functional, the sample size nis varied around n=p2r, specifically using n∈ {p7/4r,p2randp9/4r}. For inferring a low-Tucker-rank linear functional, nis varied around the comput... | https://arxiv.org/abs/2501.16223v1 |
models to broaden its applicability. Robust estimation techniques for matrix and tensor parameters have been studied in areas such as low- rank matrix recovery [50], matrix completion [43], and tensor decomposition [36]. However, inference under heavy-tailed noise remains relatively unexplored. Another promising direct... | https://arxiv.org/abs/2501.16223v1 |
– 2439. [8] Carpentier, A., Eisert, J., Gross, D., and Nickl, R. (2019). Uncertainty quantification for matrix compressed sensing and quantum tomography problems. In High Dimensional Probability VIII: The Oaxaca Volume , pages 385–430. Springer. [9] Chen, H., Raskutti, G., and Yuan, M. (2019a). Non-convex projected gra... | https://arxiv.org/abs/2501.16223v1 |
in Biosciences , 10(3):520–545. [28] Li, Z., Suk, H.-I., Shen, D., and Li, L. (2016). Sparse multi-response tensor regression for alzheimer’s disease study with multivariate clinical assessments. IEEE transactions on medical imag- ing, 35(8):1927–1936. [29] Liu, T., Yuan, M., and Zhao, H. (2022a). Characterizing spatio... | https://arxiv.org/abs/2501.16223v1 |
Q., and Zhou, W.-X. (2024). Low-rank matrix recovery under heavy-tailed errors. Bernoulli , 30(3):2326–2345. [51] Zhang, A. (2019). Cross: Efficient low-rank tensor completion. The Annals of Statistics , 47(2):936– 964. [52] Zhang, A. and Han, R. (2019). Optimal sparse singular value decomposition for high-dimensional ... | https://arxiv.org/abs/2501.16223v1 |
of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . 67 H.2 Upper Bound of First-Order Perturbation Terms . . . . . . . . . . . . . . . . . . . . . . . 68 H.3 Upper Bound of Higher-Order Perturbation Terms . . . . . . . . . . . . . . . . . . . . . . 69 H.4 Upper Bound of Leading Terms in the Spectral Representa... | https://arxiv.org/abs/2501.16223v1 |
components of the signal tensor and noise are estimated from observed data. The asymptotic normality of the estimated linear functional with these estimated vari- ances is discussed in Section K, specifically for tensor regression in Section K.1 and for tensor PCA in Section K.2. Furthermore, Section L establishes the ... | https://arxiv.org/abs/2501.16223v1 |
is essential to analyze the leading terms introduced by the Kronecker product across multiple modes. These terms involve complex polyno- mials of bEj,j= 1,2,3. To bound these complex error terms, we derived new concentration inequalities for sub-Gaussian polynomials. Additionally, we developed novel concentration bound... | https://arxiv.org/abs/2501.16223v1 |
F>∆o . With these high-probability events established, we proceed through the following steps to complete the proof of the main theorem. Step 1: Upper Bound of Negligible Terms in ⟨bZ × 1PbU1×2PbU2×3PbU3,A⟩ In Step 3, we establish the asymptotic normality ofD bZ × 1PU1×2PU2×3PU3,AE , by deriving the Berry-Esseen bound ... | https://arxiv.org/abs/2501.16223v1 |
| {z } (170)· PU1⊥bZ1(PU3⊗ PU2) F| {z } (261) + (PU3⊗ PU2)A⊤ 1PU1 F· PU1+∞X k1=3SG1,k1 bE1 PU1⊥ · PU1⊥bZ1(PU3⊗ PU2) F| {z } (261) 5 ≲∥A × 1U1×2U2×3U3∥F·" σ3 ξr1/2 λ2σ3· pp rlog(p) n3/2+ ∆·p3/2 n3/2!# . (37) It implies that III≲ PU1A1P(U3⊗U2)G⊤ 1 F·σ2 ξr1/2 λσ2·p n+∥A × 1U1×2U2×3U3∥F·" σ2 ξr1/2 λσ2 p plog(p) n+ ∆·p n!... | https://arxiv.org/abs/2501.16223v1 |
{z } (166)∥A × 2U2×3U3∥F ≲∥A × 2U2×3U3∥F·" σ4 ξr1/2 λ3σ4pRlog(p) n2+ ∆2·p2 n2# . (51) For the term II .IV in the fourth term (48), we have II.IV≤ sup W1∈Rp1×2r1,∥W1∥=1 W2∈Rp2×2r2,∥W2∥=1 PU3bZ3(W2⊗W1) F PU1⊥ PbU1− PU1 PU1⊥V1 | {z } (166) PU2⊥ PbU1− PU1 PU2 ∥A × 2U2×3U3∥F ≲∥A × 2U2×3U3∥F·σ4 ξr1/2 λ3σ4· p3/2q R... | https://arxiv.org/abs/2501.16223v1 |
1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,A × 1PU1⊥×2PU2×3PU3E , (64) III = D bZ × 1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,A × 1PU1×2PU2⊥×3PU3E , (65) IV = D bZ × 1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,A × 1PU1⊥×2PU2⊥×3PU3⊥E .(66) We first consider the first term I (63) I≤ sup Wj∈Rpj×2rj,∥Wj∥=1 j=1,2,3 W⊤... | https://arxiv.org/abs/2501.16223v1 |
PbU3− PU3 PU3 | {z } (168) · PU2⊥ PbU2− PU2 PU2⊥⊗ PU1⊥ PbU1− PU1 PU1⊥ A3PU3 F| {z } (164) ≲∥A × 3U3∥F· σ7 ξr1/2 λ6σ7 rp5/2log(p) n7/2+ ∆·p3q Rlog(p) n7/2+ ∆2·p7/2 n7/2 . (73) 10 Finally, consider III.VIII≤ PU3 PbU3− PU3 PU3⊥ F· sup W1∈Rp1×2r1,∥W1∥=1 W2∈Rp2×2r2,∥W2∥=1 PU3⊥bZ3(W2⊗W1) · PU2⊥ PbU2− PU... | https://arxiv.org/abs/2501.16223v1 |
PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,AE ≲(67) + (68) + (75) + (80) ≲∥A × 1U1×2U2×3U3∥F·σ4 ξr1/2 λ3σ4·p2 n2+3X j=1∥A × j+1Uj+1×j+2Uj+2∥F" σ4 ξr1/2 λ3σ4 rp3/2p log(p) n2+ ∆·p2 n2!# +3X j=1∥A × jUj∥F· σ5 ξr1/2 λ4σ5· rp3/2log(p) n5/2+ ∆·p2q Rlog(p) n5/2+ ∆2·p5/2 n5/2 +∥A∥F·" σ4 ξr1/2 λ3σ4· r2log(p)2 n2+ ∆·r1... | https://arxiv.org/abs/2501.16223v1 |
2=1 nσ2Pn i=1ξiMat 2(Xi) andbZ(1) 2=1 nσ2Pn i=1hD Xi,b∆E Mat 2(Xi)−σ2·b∆i . Here, we have I.II.I.I.V + I .II.I.I.VI + I .II.I.I.VII≲ PU1⊥A1P(U3⊗U2)G⊤ 1 F·σ3 ξr1/2 λ2σ3·p3/2 n3/2. (93) It remains to find upper bounds for I .II.I.I.I (86), I .II.I.I.II (87), I .II.I.I.III (88) and I .II.I.I.IV (89). By Lemma F.7, it foll... | https://arxiv.org/abs/2501.16223v1 |
· sup W2∈Rp2×2r2,∥W2∥=1 W3∈Rp3×2r3,∥W3∥=1 PU1bZ1(W3⊗W2) F ≲ PU1⊥A1P(U3⊗U2)G⊤ 1 F·σ4 ξr1/2 λ3σ4·p2 n2.. (106) It further implies that I.V≲(104) + (105) + (106) ≲ PU1⊥A1P(U3⊗U2)G⊤ 1 F·" σ2 ξr1/2 λσ2·rlog(p) n+ ∆2·p n# . (107) Combining the results above, we have I≲(102) + (103) + (107) ≲ PU1⊥A1P(U3⊗U2)G⊤ 1 F·" σ2 ξr1/2... | https://arxiv.org/abs/2501.16223v1 |
} (168)·∥(PU3⊗ PU2)A1PU1∥F· PU1 PbU1− PU1 U1G1(U3⊗U2)⊤ F| {z } (168) ≲∥A × 1U1×2U2×3U3∥F·σ4 ξr1/2 λ3σ4·p2 n2. (118) Step 2.3: Upper Bound of ⟨T × 1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,A⟩ By similar arguments, we have D T × 1 PbU1− PU1 ×2 PbU2− PU2 ×3 PbU3− PU3 ,AE ≤ trh PU3 PbU3− PU3 PU3⊗ PU2 PbU2− P... | https://arxiv.org/abs/2501.16223v1 |
Recall that bZj=1 nσ2nX i=1ξiMat j(Xi) | {z } bZ(1) j+1 nσ2nX i=1⟨∆,Xi⟩Mat j(Xi)−σ2·Mat j(∆) | {z } bZ(1) j. LetPTTM(r1,r2,r3)(A) =A× 1PU1×2PU2×3PU3+P3 j=1Mat−1 j PUj⊥AjP(Uj+2⊗Uj+1)Gj . Then, we can write D bZ,PTTM(r1,r2,r3)(A)E =D bZ,A × 1PU1×2PU2×3PU3E +3X j=1D bZj,PUj⊥AjP(Uj+2⊗Uj+1)GjE =D bZ(1),PTTM(r1,r2,r3)(A)E ... | https://arxiv.org/abs/2501.16223v1 |
Since the proof of Theorem 3.2 is similar to the proof of the Theorem 3.1, we will focus on the parts that differ. For identical or repetitive steps, such as the decomposition of certain terms, we will provide a concise description to maintain textual conciseness. First, for any j= 1,2,3, we assume that the following e... | https://arxiv.org/abs/2501.16223v1 |
proof of (37), we have III.II≲∥A × 1U1×2U2×3U3∥F· σ3 ξr1/2 λ2σ3·pp rlog(p) n3/2+σ4 ξr1/2 λ3σ4·p2 n2! . (134) It implies that III≲ PU1A1P(U3⊗U2)G⊤ 1 Fσ2 ξr1/2 λσ2·p n +∥A × 1U1×2U2×3U3∥F" σ2 ξr1/2 λσ2 p rplog(p) n+∆2log(p)3/2 √n! +σ4 ξr1/2 λ3σ4·p2 n2# . (135) For the fourth term IV in (130), we have IV≲∥A × 2U2×3U3∥F·" ... | https://arxiv.org/abs/2501.16223v1 |
1− PU1 ×2 PbU(I) 2− PU2 ×3PU3,AE (143) (Step 2.3) :D T × 1 PbU(I) 1− PU1 ×2 PbU(I) 2− PU2 ×3 PbU(I) 3− PU3 ,AE . (144) Step 2.1: Upper Bound of Negligible Terms in ⟨T × 1 PbU(I) 1− PU1 ×2PU2×3PU3⟩ Note that D T × 1 PbU(I) 1− PU1 ×2PU2×3PU3,AE =D T × 1SG1,1 bE(I) 1 ×2PU2×3PU3,AE | {z } I+* T × 1+∞X k1=2SG... | https://arxiv.org/abs/2501.16223v1 |
PbU(I) 1− PU1 ×2 PbU(I) 2− PU2 ×3PU3,AE ≤I + II + III + IV . Applying similar arguments, we can show I≲∥A × 3U3∥F· σ2 ξr1/2 λσ2·rlog(p) n+σ2 ξr1/2 λσ2·∆·Rlog(p) n! , (151) II≲∥A × 2U2×3U3∥F· σ3 ξr1/2 λ2σ3·pp rlog(p) n3/2! , (152) III≲∥A × 1U1×3U3∥F· σ3 ξr1/2 λ2σ3·pp rlog(p) n3/2! , (153) IV≲∥A × 1U1×2U2×3U3∥F· σ3 ξr... | https://arxiv.org/abs/2501.16223v1 |
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