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Thompson and Fox-Kean (2005)). The following theoretical results describe the asymptotic behavior of the success-probability Pt,h and of the number St,hof successes along time, for each Bernoulli process h, with h= 1, . . . , N . The rigorous proofs of all the theorems are collected in Appendix B. Theorem 3.1. Under as... | https://arxiv.org/abs/2505.13364v1 |
the number of past successes observed in many processes who themselves have high scores. The uncertainty—or volatility—of the innovation process within technological domains, with higher variance often associated with emerging or rapidly evolving fields, has been a subject of debate (Jalonen and Lehtonen 2011, Jalonen ... | https://arxiv.org/abs/2505.13364v1 |
eigenvector centrality scores (with respect to Γ⊤) of the two nodes (processes) handjand the share of successes observed for process hconverges almost surely to the absolute eigenvector centrality score of h. It is interesting to note that, in the innovation framework, Theorem 3.3 highlights how the long- run proportio... | https://arxiv.org/abs/2505.13364v1 |
note that the slope of the two lines and the distance between their intercepts are the same before and after the shock, because the interaction matrix Γ does not change: indeed, the slope is equal to γ∗and the distance between the intercepts corresponds to the quantity log10(u1/u2) = log10(u1)−log10(u2). 3.1.Model impl... | https://arxiv.org/abs/2505.13364v1 |
database8. More specifically, we collected all the patents published in the period [1980 −2018], with their exact (full) date of publication and their CPC-1 category. Moreover, for each patent n, we know if it has been cited by subsequent patents (published in the considered period) and which are the citing patents. We... | https://arxiv.org/abs/2505.13364v1 |
to bγ∗= 0.689. Indeed, the goodness of fit R2index obtained imposing a common slope is 0 .969 and so basically equal to the one obtained allowing the slopes to be different across categories, i.e. 0 .972. The emergence of a common Heaps’ exponent across all categories supports the fact that these innovation processes a... | https://arxiv.org/abs/2505.13364v1 |
equi- librium configuration of the subset of Γ related to such technologies; and (ii) the stabilization in the rate of recombination of existing knowledge, as captured by the behavior of Pt,hover time. Moreover, the limit corresponds to the ratio of their respective components of the eigenvector-centrality scores u (wi... | https://arxiv.org/abs/2505.13364v1 |
t, for all the pairs ( k, h) of categories ( k=sub-figure, h=color), with k̸=H. Category His arbitrarily chosen to be the baseline category. The horizontal dashed red lines represent the value 10 to the power of the differences between the intercepts of the regression lines in Figure 4 for the pairs of categories ( k, ... | https://arxiv.org/abs/2505.13364v1 |
and Kager (2009). Broadly speaking, the transformation of general capabilities into core capabilities occurs through a process of smart specialization. In this process, technological capabilities must adapt to multidisciplinary stimuli originating from other technologies, translate them into their own technological lan... | https://arxiv.org/abs/2505.13364v1 |
Acknowledgments. Giacomo Aletti is a member of the Italian Group “Gruppo Nazionale per il Calcolo Scientifico” of the Italian Institute “Istituto Nazionale di Alta Matematica”. Irene Crimaldi is a member of the Italian Group “Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni” of the Ita... | https://arxiv.org/abs/2505.13364v1 |
had as much time to accumulate citations.Citations Trajtenberg et al. (1997) (Modified better version)GX= 1−PMi j=1 1 NPN i=1Tn ji Tn i2Xis the focal patent with Yipatents citing the focal patent X, with i= 1,·, N. Tn iis the total number of IPC n-digit classes in yi Tn jiis the total number of IPC n-digit classes in... | https://arxiv.org/abs/2505.13364v1 |
cited patent can be an indication of a core technology”. This literature have been recently criticized. In Abbas et al. (2014), for instance, the authors primarily discuss patent novelty in the context of citation analysis, classification systems, and keywords. They highlight how backward citations (patents cited by MO... | https://arxiv.org/abs/2505.13364v1 |
a developing method, with an emphasis on how linguistic analysis (e.g., analyzing keywords, abstracts) is becoming crucial in identifying novel patents. But, their definition of novelty primarily revolves around citation gaps14and technological distance15(e.g., patents in new technological classes). More recently Kelly... | https://arxiv.org/abs/2505.13364v1 |
the Squicciarini et al. (2013)’s index offers a better replicability, and ease of modification in a cross-category setting as ours. Notice, however that our setting is general enough to allow for other measures of patent novelty as well. Appendix B.Proofs of the main results and technical details for the statistical te... | https://arxiv.org/abs/2505.13364v1 |
use (Aletti et al. 2023a, Theorem S1.3). Indeed, setting ePt=v⊤Pt, we have (B.4)eP0=v⊤P0=1 cv⊤θ>0 ePt+1= 1−1 t+ 1 ePt+γ∗ t+ 1eXt+1+O(1/t2), t≥0, witheXt+1=v⊤Xt+1. Hence, if we define the stochastic process V= (Vt)t≥0, taking values in the interval [0 ,1], as V0=eP0>0, Vt+1= 1−1 t+ 2 Vt+1 t+ 2Yt+1, t≥0, where Yt+1=γ... | https://arxiv.org/abs/2505.13364v1 |
proofs, we have by ( A3) E[Xt+1,hYt+1,k|Ft] =E[Yt+1,k]E[Xt+1,h|Ft] =πkPt,h a.s.∼t−(1−γ∗)πkP∞,h=t−(1−γ∗)πkγ∗S∞,h and so it is enough to apply (Williams 1991, sec. 12.15) in order to obtain St,k,h=Pt n=1Xn,hYn,ka.s.∼ πkS∞,htγ∗. □ □ Technical details for the statistical inference (Subsec. 4.1). We here use the same choice... | https://arxiv.org/abs/2505.13364v1 |
DYNAMICS 27 where ∆ fMt+1=v⊤∆Mt+1. Setting ˘Mt=Pt n=11 n∆fMnandeRA,t+1=v⊤RA,t+1=O(1/t1+β) (by (ii)), we have (B.9) eAt+1−eA0=tX n=0(eAn+1−eAn) =ϕ∗˘Mt+1+tX n=0eRA,n+1. Hence, since suptE[|eAt|]<+∞(by (iii)) andP t1/t1+β<+∞, we also have suptE[|˘Mt|]<+∞. Therefore, ( ˘Mt)tis a martingale bounded in L1and so it converges ... | https://arxiv.org/abs/2505.13364v1 |
(Aletti et al. 2017, Lemma A.4)) |Ak+1,t|=O|pt(α∗ 2)| |pk(α∗ 2)| =Ok ta∗ 2 fork=m0, . . . , t −1, and simply |At+1,t|=O(1) for k=t. Moreover, recalling that RA,t+1=O(t−(1+β))1for some β >0, we have |ρt+1|= tX k=m0Ck+1,tRA,k+1 =Ot−1X k=m0k ta∗ 21 k1+β +O(1/t1+β) =O1 ta∗ 2t−1X k=m0ka∗ 2−1−β +O(1/t1+β)→0, beca... | https://arxiv.org/abs/2505.13364v1 |
we have aϕ,t+1= 1−1 t+ 1(ϕ∗−ϕ) aϕ,t+ϕ t+ 1v⊤ ϕ∆Mt+1. Then, using (B.14), we have that E[|aϕ,t+1|2|Ft]≤ 1−ϕ∗ t+ 1+ϕ t+ 1 2 |aϕ,t|2+|ϕ|2 (t+ 1)2NX j=1|vj|2E[(∆Mj,t+1)2|Ft] ≤ 1−ϕ∗ t+ 1+ϕ t+ 1 2 |aϕ,t|2+|ϕ|2 (t+ 1)2max j{|vj|2}Wt. MODELING INNOVATION DYNAMICS 31 Then, regarding the first term we have that 1−ϕ∗ t+ 1+ϕ t+ ... | https://arxiv.org/abs/2505.13364v1 |
γ rare the distinct eigenvalues of Γ, p1, . . . , p rare the sizes of the corresponding Jordan blocks and ciare suitable vectors related to cand to the generalized eigenvectors of Γ. Indeed, we can write Γ as PJP−1, where Jis its canonical Jordan form and Pis a suitable invertible matrix of generalized eigenvectors. Th... | https://arxiv.org/abs/2505.13364v1 |
higher than 34 G. ALETTI, I. CRIMALDI, A. GHIGLIETTI, AND F. NUTARELLI category (In,h>0.1)% ( In,h>0.3)% ( In,h>0.5)% ( In,h>0.7)% ( In,h>0.9)% A 1.640 % 0.185 % 0.045 % 0.017 % 0.008 % B 2.440 % 0.250 % 0.059 % 0.021 % 0.008 % C 1.180 % 0.154 % 0.042 % 0.017 % 0.008 % D 0.554 % 0.1410 % 0.031 % 0.017 % 0.011 % E 1.070... | https://arxiv.org/abs/2505.13364v1 |
and innovation. Economics of Innovation and New Technology 22(7):702– 725. Alves J, Marques MJ, Saur I, Marques P (2007) Creativity and innovation through multidisciplinary and multi- sectoral cooperation. Creativity and innovation management 16(1):27–34. and MN (2005) Power laws, pareto distributions and zipf’s law. C... | https://arxiv.org/abs/2505.13364v1 |
science quarterly 35(1):128–152. Colladon AF, Guardabascio B, Venturini F (2025) A new mapping of technological interdependence. Research Policy 54(1):105126. Corrocher N, Mancusi ML (2021) International collaborations in green energy technologies: What is the role of distance in environmental policy stringency? Energy... | https://arxiv.org/abs/2505.13364v1 |
resource-based view: Capability lifecycles. Strategic management journal 24(10):997–1010. Higham K, Contisciani M, De Bacco C (2022) Multilayer patent citation networks: A comprehensive analytical framework for studying explicit technological relationships. Technological forecasting and social change 179:121628. Higham... | https://arxiv.org/abs/2505.13364v1 |
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URL http://dx.doi.org/10.1214/07-PS094 . Perri A, Silvestri D, Zirpoli F (2020) Change and stability in the automotive industry: a patent analysis. Technical Report 5/2020, Department of Management, Universit` a Ca’Foscari Venezia, URL https://hdl.handle. net/10278/3752090 . Pichler A, Lafond F, Farmer JD (2020) Techno... | https://arxiv.org/abs/2505.13364v1 |
1(1):239–263. Weitzman ML (1998) Recombinant growth. The quarterly journal of economics 113(2):331–360. Williams D (1991) Probability with Martingales (Cambridge: Cambridge University Press). Yang S (2023) Predictive patentomics: Forecasting innovation success and valuation with chatgpt. arXiv preprint arXiv:2307.01202... | https://arxiv.org/abs/2505.13364v1 |
arXiv:2505.14051v1 [math.ST] 20 May 2025Information bounds for inference in stochastic evolution equations observed under noise Gregor Pasemann∗Markus Reiß∗ May 21, 2025 Abstract We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs)... | https://arxiv.org/abs/2505.14051v1 |
dY(t, y) =BX(t, y)dt+εdV(t, y), t∈[0, T], y∈Λ, (1.3) where Vis a cylindrical Brownian motion on L2(Λ), independent of W,Bis an observation operator and ε⩾0 is the noise level. This formulation is standard for stochastic filtering problems in dynamical systems, see e.g. Bain & Crisan (2009). From an alternative point of... | https://arxiv.org/abs/2505.14051v1 |
in different coefficients as a function of observation time T, static noise level ε, dimension dand diffusivity ν(dropping a log factor in the last row for d= 2). In Section 4 we construct estimators in a general parametric setting with real parameter ϑ and commuting operators which under quite general conditions attai... | https://arxiv.org/abs/2505.14051v1 |
in classical nonparametric regression whenever Tis not growing too fast (or is constant) as ε↓0, while they are completely different for T→ ∞ with T⩾ε−pfor certain powers p. Table 2 reports where (at which p) the ellbow occurs and states the result for T→ ∞ and noise levels εof order one. Detailed results are given in ... | https://arxiv.org/abs/2505.14051v1 |
find two parameters (real values or functions) under which the observation laws satisfy a non-trivial Hellinger bound and which at the same time have a large (Euclidean or functional) distance. We shall apply the lower bound for sequences of models, which become more informative in n∈N, and try to find the largest δ=δn... | https://arxiv.org/abs/2505.14051v1 |
2.3 will allow us to obtain feasible expressions also for non-commuting covariance operators. 5 Proof. Following the proof of the Feldman-H´ ajek Theorem (step 2 for Thm. 2.25 in Da Prato & Zabczyk (2014)) we let ( ek)k⩾1be the orthonormal basis of eigenvectors of R:=Q−1/2 0Q1/2 1(Q−1/2 0Q1/2 1)∗ with corresponding pos... | https://arxiv.org/abs/2505.14051v1 |
(3.1) defines for each ta cylindrical Gaussian measure on Hvia⟨Xt, z⟩=Rt 0⟨eA∗ ϑ(t−s)z, dW s⟩,z∈H, which solves for z∈dom( A∗ ϑ) the weak formulation (Da Prato & Zabczyk, 2014, Thm. 5.4) d⟨z, X t⟩=⟨A∗ ϑz, X t⟩dt+⟨z, dW t⟩, t∈[0, T],with⟨z, X 0⟩= 0. Let us stress that we do not assume commutativity of the operators ( Aϑ... | https://arxiv.org/abs/2505.14051v1 |
the complex Hilbert space Hsatisfy dom( Aϑ) = dom( A∗ ϑ) and can be decomposed by two self-adjoint operators Rϑ: dom( Rϑ)→H,Jϑ: dom( Jϑ)→Hwith dom( Aϑ) = dom( Rϑ)∩dom( Jϑ) such that Aϑv=Rϑv+iJϑv, A∗ ϑv=Rϑv−iJϑv,∥Aϑv∥2=∥Rϑv∥2+∥Jϑv∥2 for all v∈dom( Aϑ). Note that any two operators from Aϑ, A∗ ϑ, Rϑ, Jϑcommute. Since Aϑis... | https://arxiv.org/abs/2505.14051v1 |
the time derivative. Then we have the identities (∂t−Aϑ)Sϑ= (−∂t−A∗ ϑ)S∗ ϑ= IdonH1(H)andSϑ(∂t−Aϑ) =S∗ ϑ(−∂t−A∗ ϑ) = Id onH1 0(H)∩L2([0, T]; dom( Aϑ)). (c) We have on L2(H) S1−S0=S0(A1−A0)S1=S1(A1−A0)S0. Proof. See Section B.1. The perturbation property in (c) will be essential for us. The assumption dom( Aϑ) = dom( Rϑ)... | https://arxiv.org/abs/2505.14051v1 |
1+ Id)−1 dudt = trace (A1−A0)∗(ε2 0R2 0¯B−2 0+ Id)−1(A1−A0)T2f(2TR1)2(ε2 1R2 1¯B−2 1+ Id)−1 =T2∥(ε2 0R2 0¯B−2 0+ Id)−1/2(A1−A0)f(2TR1)(ε2 1R2 1¯B−2 1+ Id)−1/2∥2 HS. Changing the roles of ϑ0andϑ1and using ∥F∥2 HS=∥−F∗∥2 HSfor Hilbert–Schmidt operators F, we obtain an analogous bound for the second summand in (3.5). T... | https://arxiv.org/abs/2505.14051v1 |
Gaussian approach. Our estimation procedure is based on a preaveraging method, where the observational noise is first reduced by local averaging and then a regression-type estimator is applied to the averaged data. A similar approach is employed in nonparametric drift estimation for diffusions under noise by Schmisser ... | https://arxiv.org/abs/2505.14051v1 |
ε4 n|¯Rn|4 T−1 nB−4 n+ Id−1|Rϑ|−1 Tn . The minimax optimality follows directly from the lower bound in Theorem 4.2 because |Rϑ|has the same order for all ϑ∈[ϑn,¯ϑn] by¯ϑn≲ϑn. Using that |¯Rn|is of the same order as |¯An|and|Rϑ|≼|¯Rn|, the left-hand side of Condition (4.5) is at most of order Tntrace |Λ|4 ε4 n|¯Rn|4... | https://arxiv.org/abs/2505.14051v1 |
constant noise intensity, which for fixed ϑis also well known (Kutoyants, 2004, Example 3.3). In view of the spectral approach to SPDE statistics (Huebner & Rozovskii, 1995), where the operator Aϑis unbounded and each Fourier mode forms an Ornstein-Uhlenbeck process with drift given by the eigenvalues of Aϑ, we note th... | https://arxiv.org/abs/2505.14051v1 |
β)ρand the asymptotics vn:=T−1/2 nε(2ρ+d)/(4ρ(1+β)) n →0asn→ ∞ . Then uniformly over ϑ∈[ϑ,¯ϑ]the estimator ˆϑnfrom Definition 4.3 satisfies ˆϑn−ϑ=OPϑ(vn) and the rate vnis minimax optimal. The rate for εn→0 is becoming faster with the dimension dand slower with the fractional index ρand the dynamic correlation index β.... | https://arxiv.org/abs/2505.14051v1 |
nk∗ n∼Tnν−1 n (νnεn)−d/(2+2β)∧(νnε4 n)−d/(5+8β) , where we used that the summands are of order 1 for k∼k∗ n. It remains to check Condition (4.5) in Theorem 4.4. A simple bound of its left-hand side is In(ϑ)∥|Λ|2|Rϑ|−2 Tn∥≲TnIn(ϑ), which for εnνn→0 iso(In(ϑ)2) as required. Otherwise, we have νn, εn∼1 and Tn→ ∞ . In th... | https://arxiv.org/abs/2505.14051v1 |
ϑ∈[ϑ,¯ϑ] ˆϑn−ϑ=OPϑ(vn) and the rate vnis minimax optimal. 5.8 Remark. If the noise covariance operator is Bn= (Id −∆)−β, that is not depending onνn, then similar calculations for εn≲νβ nyield the same rate for d∈ {1,2}, but vn= T−1/2 nν(d+2β)/(4+4β) n ε(d−2)/(4+4β) n for 3⩽d < 10 + 8 β. This is slower in νnthan before ... | https://arxiv.org/abs/2505.14051v1 |
λ7→λ2+2βis not operator monotone, see Bhatia (2013, Chapter V) for this and more results on operator monotonicity for matrices which directly extend to linear operators. Based on perturbation ideas from Engel & Nagel (2000, Prop. VI.5.24), however, we are able to establish ( ε2(−∆ϑ)2+2β+ Id)−1≼Cd,β(ε2(−∆)2+2β+ Id)−1for... | https://arxiv.org/abs/2505.14051v1 |
β) (which suffices also for fixed Tn) and is always satisfied if α⩾5/2. Proof. For simplicity we drop the index natεn, Tn. We transfer the problem into the spectral domain by the Fourier transform Ff(u) =R ei⟨u,x⟩f(x)dx. In particular, we have F(∆ϑg)(u) = (2π)−dM[−iu⊤]C[Fϑ(u)]M[−iu]Fg(u) for g∈ H2(Rd), where M[f]g=fgan... | https://arxiv.org/abs/2505.14051v1 |
the condition is equivalent to T≲ε−((10+4 β)α+5)/((1+β)(5−2α)), which is equivalent to (6.5) by Lemma C.7 below. 6.2 Space-dependent transport For each n∈Nconsider the observations ( dYt, t∈[0, Tn]) given by dYt=Xtdt+εndVtwith dXt= (νn∆Xt+∇•(ϑ(x)Xt))dt+dWt (6.9) andX0= 0, Tn⩾1,εn∈[0,1] and νn∈(0,1]. The Laplace operato... | https://arxiv.org/abs/2505.14051v1 |
1, . . . , d andx∈Rd. Then Kis divergence-free and 24 FK(u) =Fφ(u)A(−iu) has the desired properties. Following the proof of Theorem 6.4 and with ∥A∇φ∥Cαsufficiently small, we deduce by Lemma C.6(a) below H2(Ncyl(0, Q0),Ncyl(0, Q1)) ≲Tε−d/2ν−(d+2)/2h2α+dZ Rd(|u|4∧ |u|−2d)(1 + ε1/2ν1/2h−1|u|)d−5du ≲Tε−d/2ν−(d+2)/2h2α+dZ∞... | https://arxiv.org/abs/2505.14051v1 |
Tn≲ε−(11.5∧(1.5d+1)) n ind⩾2. In particular, it is true if Tnis fixed. Proof. We follow the road exposed in Theorem 6.4 and also drop the index n. Using A0andA1 as in Proposition 6.9, for which the conditions will be discussed below, we find from Theorem 3.9, writing δ=ϑ−1 1 4H2(Ncyl(0, Q0),Ncyl(0, Q1)) ⩽T∥(ε2(ν∆−Id)2+... | https://arxiv.org/abs/2505.14051v1 |
operator on Hand by im( Q) the range or image of an operator Q. For (possibly unbounded) self-adjoint operators A, B on a Hilbert space Hwrite A≼Band B≽Aif dom( B)⊆dom( A) and ⟨(B−A)v, v⟩⩾0 for all v∈dom( B). ForT >0 letL2([0, T];H) =L2(H) be the Hilbert space of all Borel-measurable f: [0, T]→H with∥f∥2 L2(H):=RT 0∥f(... | https://arxiv.org/abs/2505.14051v1 |
Rϑf⟩L2(H) ⩽1 2 2 +1 2(e2∥(Rϑ)+∥T−1) ∥g∥2 L2(H)= 3 4+1 4e2∥(Rϑ)+∥T ∥f∥2 L2(H). 29 Since L2([0, T]; dom( Aϑ)) is dense in L2(H), this shows that S∗ ϑRϑand its adjoint RϑSϑextend to bounded linear operators on L2(H). Proof of Proposition 3.7. We present the proof only for Sϑ, that for S∗ ϑis completely analogous. We p... | https://arxiv.org/abs/2505.14051v1 |
inner integral is adapted due to supp( K(n)(t,•))⊆[0, t], we have − ⟨(∂t+A∗ ϑ)ΛK(n)˙Y , B nX⟩L2 =−ZTn 0ZTn 0⟨∂t+A∗ ϑ)ΛK(n)(t, s)dYs, BnXtdt⟩ =ZTn 0Zt 0 ⟨ΛK(n)(t, s)dYs, BndXt⟩ − ⟨ΛK(n)(t, s)dYs, AϑBnXtdt⟩ =ZTn 0ZTn 0⟨ΛK(n)(t, s)dYs, BndWt⟩=⟨ΛK(n)˙Y , B n˙W⟩L2, by commutativity of AϑandBn. This gives Zn−ϑNn=−⟨(∂t+A∗ ϑ... | https://arxiv.org/abs/2505.14051v1 |
Lemma C.4 and Lemma C.3 below Var(Nn)⩽2ZTn 0ZTn 0∥((ε2 nId +B2 nCϑ)|Λ|2K(n))(t, s)∥2 HSdsdt in terms of the corresponding operator kernel. We bound the kernel of CϑK(n), using the kernel ofCϑfrom Lemma 3.4, (B.8) and (B.7): |(CϑK(n))(t, s)|≼1 2ZTn 0Zt+u |t−u|eRϑvdv Knψ(n) u−s,s(¯An)du =1 2KneRϑ|t−s|ZTn 0Zt+u−|t−s| |... | https://arxiv.org/abs/2505.14051v1 |
−√ 2dh−1/2∥∇L∥∞∥∆∆ϑu∥3/4∥∆ϑu∥1/4 ⩾1 4∥∆∆ϑu∥ −27d4h−2∥∇L∥4 ∞∥∆ϑu∥. (B.15) Further, by expanding ∆∆ ϑand using the inverse triangle inequality, Lemma C.1 below together with the bound ∥∆L∥2 ∞⩽d∥∇2L∥2 ∞, (A.1) and AαB1−α⩽αA+(1−α)BforA, B⩾0,α∈[0,1], 1 2∥∆2u∥ − ∥ ∆∆ϑu∥ ⩽∥3⟨∇ϑ,∇∆u⟩+ (∆ ϑ)(∆u) + 2⟨∇2ϑ,∇2u⟩HS+⟨∇∆ϑ,∇u⟩∥ ⩽3(2d2h... | https://arxiv.org/abs/2505.14051v1 |
∥⟨∇2ϑ,∇2u⟩HS∥⩽(2d4h)1/2∥∇2ϑ∥∞∥∆u∥1/2∥∇∆u∥1/2, u∈ H3(Rd). (C.6) Proof. To prove (C.1), use integration by parts and obtain ∥ϑu∥2=Z Rdϑ2(x)u2(x)dx = −Z RdZxi∧(h/2) −h/2ϑ2(x1, . . . , x i−1, y, x i+1, . . . , x d)dy 2u(x)∂iu(x)dx ⩽2h∥ϑ∥2 ∞Z Rd|u(x)||∂iu(x)|dx⩽2h∥ϑ∥2 ∞∥u∥∥∂iu∥. All other estimates follow by replacing uan... | https://arxiv.org/abs/2505.14051v1 |
0(za∧zb) 1+tz 1+tcdz⩽Z∞ 0(za∧zb)(1∨zc)dz, and for a > b this is finite if and only if ( a+c)∧a >−1 and ( b+c)∨b <−1. For 0 ⩽t⩽2 the claim in (b) reduces to 1 ∼1, so assume t⩾2. We split the integralR∞ 0=R1 0+Rt−1 1+Rt+1 t−1+R∞ t+1and treat each part separately. First,R1 0za(t−z)cdz≲(t−1)c≲tc. Next, Zt−1 1zb(t−z)cdz≲t1... | https://arxiv.org/abs/2505.14051v1 |
Operators, (pp. 271–280). Institute of Mathematics of the National Academy of Sciences of Ukraine. Pasemann, G., Flemming, S., Alonso, S., Beta, C., & Stannat, W. (2020). Diffusivity estimation for activator–inhibitor models: Theory and application to intracellular dynamics of the actin cytoskeleton. Journal of Nonline... | https://arxiv.org/abs/2505.14051v1 |
arXiv:2505.14058v1 [math.ST] 20 May 2025A Characterization of a Subclass of Separate Ratio-Type Copulas ∗Ziad Adwan†Nicola Sottocornola May 24, 2025 Abstract Copulas are essential tools in statistics and probability theory, enabling the study of the dependence structure between random variables independently of their m... | https://arxiv.org/abs/2505.14058v1 |
ϕ(u, v)is expressed as a separable product of two univariate functions that we assume differentiable a.e.: Dθ(u, v) =uv 1−θf(u)g(v), 0≤u, v≤1, θ∈R. (2) Now let’s consider the function Gdefined on Sin this way: G= (f−uf′)(g−vg′)−2uvf′g′(3) and define α1=min S(G) α2=max S(G). (4) In a nice paper published in 2024 [4], El... | https://arxiv.org/abs/2505.14058v1 |
(5) with θ=−30onS(left) and zoomed around the minimum point (right). In order to avoid these situations we restrict our analysis to the case where fandgare concave down. Therefore we will use the following assumptions: B1.f(0) = g(0) = 1 , f(1) = g(1) = 0. B2.fandgare strictly decreasing functions. B3.fandgare concave ... | https://arxiv.org/abs/2505.14058v1 |
further it could be useful to check if this new condition is reasonable, testing some simple examples where the maximum of Gis clearly reached on the boundary ofS. We use as a benchmark the examples provided in Table 1 in [4]. The results in Table 1 (see Appendix) should convince the reader that B4 is a reasonable choi... | https://arxiv.org/abs/2505.14058v1 |
logb(v+b(1−v)) cos( πv/2) 1 < b T T T T T (1−u)ecu(1−v)ecv0≤c≤1 T T T T T eau−ea 1−eaeav−ea 1−ea0< a≤3.7T T T T T 3.7< a T T F T T hM(u) hM(v) 1<M<2 T T T T F Table 1: Testing conditions B1,...,B4 (T=True, F=False). We added A3 to show how restrictive this condition is. 11 References [1] Chesneau C. (2022). Some new ra... | https://arxiv.org/abs/2505.14058v1 |
arXiv:2505.14138v1 [math.ST] 20 May 2025Sample Complexity of Correlation Detection in the Gaussian Wigner Model Dong Huang and Pengkun Yang∗ May 21, 2025 Abstract Correlation analysis is a fundamental step in uncovering meaningful insights from complex datasets. In this paper, we study the problem of detecting correlat... | https://arxiv.org/abs/2505.14138v1 |
a subset of vertices and edges from the graph—becomes a powerful approach for exploring graph structure. This technique has been widely used in various settings, as it allows for inference about the graph without needing full observation [LF06, HL13]. In fact, there are several motivations leading us to consider the gr... | https://arxiv.org/abs/2505.14138v1 |
dQ|is the total variation distance between PandQ. Thus strong and weak detection are equivalent to lim n→∞TV(P,Q) = 1 and lim n→∞TV(P,Q)>0, respectively. We then establish the main results of correlation detection in the Gaussian Wigner model. Theorem 1. There exist constants C, C such that, for any 0< ρ < 1, ifs2≥C n... | https://arxiv.org/abs/2505.14138v1 |
[BES80, Bol82, DCKG19, GM20, DMWX21, MRT23, DL23, MWXY23, ABT24, DL24, MWXY24, GMS24, MS24]. The low-degree likelihood ratio [HS17, Hop18] has emerged as a framework for studying com- putational hardness in high-dimensional statistical inference. It conjectures that polynomial-time algorithms succeed only in regimes wh... | https://arxiv.org/abs/2505.14138v1 |
Under the alternative hypothesis H1, recall that π∗denotes the latent bijective mapping from V(G1) to V(G2). For the induced subgraphs G1, G2sampled from G1,G2, we denote the set of common vertices as Sπ∗≜V(G1)∩(π∗)−1(V(G2)), (3) Tπ∗≜π∗(V(G1))∩V(G2). (4) We note that the restriction of π∗toSπ∗is a bijective mapping bet... | https://arxiv.org/abs/2505.14138v1 |
subgraphs of size minG1andG2, respectively. Indeed, the expected mean- square error for a correlated pair Eh βe(G1)−βπ∗(e)(G2)2i is 2(1 −ρ), while it stays bounded away from 1 for an uncorrelated pair. As a result, the choice of feffectively distinguishes between H0andH1under strong signal condition. We now state the... | https://arxiv.org/abs/2505.14138v1 |
tight when the correlation is weak. We will establish the remaining regimes by the conditional second moment method. For notational simplicity, we use TV(P,Q) to denote TV(P(G1, G2),Q(G1, G2)) in this paper. By [Tsy09, Equation 2.27], the total variation distance between PandQcan be upper bounded by the second moment: ... | https://arxiv.org/abs/2505.14138v1 |
parts. To formally describe all correlation relationships, we use the correlated functional digraph of two mappings πand ˜πbetween a pair of graphs introduced in [HSY24]. Definition 3 (Correlated functional digraph) .Letπand ˜πbe two bijective mappings between V(G1) and V(G2) and Sπ, Tπ, S˜π, T˜πbe the sets of common v... | https://arxiv.org/abs/2505.14138v1 |
3, we finish the proof of Proposition 4. In fact, under the strong correlation condition, detecting π∗is no longer the bottleneck. We instead use a more delicate analysis based on the conditional second moment method. By (36) in Lemma 6, there exists C≤1 8such that, when s2≤Cn, we have P[|Sπ∗|= 0]≥0.9, which implies th... | https://arxiv.org/abs/2505.14138v1 |
with |U|=K2, letVU≜∪j∈UVj. We say Uiscompatible if for any v∈VU,πj(v)’s are identical for all j∈Usuch that v∈Vij. Let I(U) denote the indicator function of compatible setU. IfI(U) = 1, we define πUas the union of πjfor any j∈U. Specifically, πU(v) =πj(v) such that v∈Vij,for any v∈VU. The seed is then defined as π0= arg... | https://arxiv.org/abs/2505.14138v1 |
mdo 7:forv1∈V(G1)\S0andv2∈V(G2)\T0do 8: ComputeP v∈S0f βv1v(G1), βv2π0(v)(G2) . 9:end for 10: Find the pair ( v1, v2) for the maximal value ofP v∈S0f βv1v(G1), βv2π0(v)(G2) and add v17→v2intoπ0. 11:end while 12:ComputeP e∈(S0 2)βe Hf π0 , output H1if it exceeds τ, otherwise output H0. to identify the seeds. While... | https://arxiv.org/abs/2505.14138v1 |
n= 50, s= 40, K1= 4, K2= 3, N1= 10000 , N2= 500 , ϵ= 0.01, and vary ρ∈ {0.95,0.96,0.97,0.98,0.99}, with m=j (1−ϵ)s2 nk = 31. We observe that as ρincreases, the ROC curve is moving toward the upper left corner, and the AUC increases from 0 .55 to 1, indicating an improvement in the performance of our test statistic as t... | https://arxiv.org/abs/2505.14138v1 |
with respect to the sample size for the correlation detection problem using the low-degree conjecture. 14 •Other graph models. The sample complexity for correlation detection remains unknown for many models (e.g., the stochastic block model, the graphon model). A natural next step is to explore whether our results can ... | https://arxiv.org/abs/2505.14138v1 |
with non-vanishing correlation. arXiv preprint arXiv:2306.00266 , 2023. [DL24] Jian Ding and Zhangsong Li. A polynomial time iterative algorithm for matching gaus- sian matrices with non-vanishing correlation. Foundations of Computational Mathe- matics , pages 1–58, 2024. [DMWX21] Jian Ding, Zongming Ma, Yihong Wu, and... | https://arxiv.org/abs/2505.14138v1 |
In ISAAC Congress (International Society for Analysis, its Applications and Computa- tion) , pages 1–50. Springer, 2019. [LF06] Jure Leskovec and Christos Faloutsos. Sampling from large graphs. In Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining , pages 631–636, 2006. [... | https://arxiv.org/abs/2505.14138v1 |
Estimation . Springer Verlag, 2009. [VCL+15] Joshua T Vogelstein, John M Conroy, Vince Lyzinski, Louis J Podrazik, Steven G Kratzer, Eric T Harley, Donniell E Fishkind, R Jacob Vogelstein, and Carey E Priebe. Fast approximate quadratic programming for graph matching. PLOS ONE , 10(4):1– 17, 2015. 18 [WCA+16] Yanhong Wu... | https://arxiv.org/abs/2505.14138v1 |
pairs of standard normals with correlation coefficient ρ. Then, by the Chernoff bound, Qh e Hf π ≥τi ≤exp (−λτ)Eh exp λe Hf πi (14) = exp ( −λτ)E"mY i=1exp (λAiBi)# (a) ≤exp −λm 2ρ 2−1 2m 2 log 1−λ2 = exp −m 2ρ2 4+1 2log 1−ρ2 4(b) ≤exp −1 12m 2 ρ2 , (15) where (a) is because E[λAiBi] =1√ 1−λ2f... | https://arxiv.org/abs/2505.14138v1 |
−1 3log 1 1−ρ ≤ −1 3log 1 1−ρ . Then, applying the union bound yields that Q(T ≥τ)≤ |S s,m|Qh e Hf π ≥τi ≤exp mlogen 1−ϵ −1 3log1 1−ρm 2 , 21 where the last inequality is because |Ss,m|= s m2m!≤ es mmsm= en 1−ϵm . Therefore, when m−1≥6(1+ϵ) log(en 1+ϵ) log(1 /(1−ρ)), we have Q(T ≥τ)≤exp −ϵmlog en 1−... | https://arxiv.org/abs/2505.14138v1 |
the last equality follows from (20), (21) and the fact that TV(X⊗Z, Y⊗Z) =TV(X, Y) for any distributions X, Y, Z such that Zis independent with XandY. For the random graphs G1[S] and G2[T] with S⊆V(G1), T⊆V(G2), and |S|=|T|, they follow the correlated Gaussian Wigner model with node set size |S|under ˜P, while they are... | https://arxiv.org/abs/2505.14138v1 |
x≥1. Since P[|I∗|=t]≤ s n2tby Lemma 3 and |I∗| ≤ |V(G1)∩π−1(V(G2))| ≤ 24 (1+ϵ)s2 nif (G1, G2, π),(G1, G2,˜π)∈ E, we obtain EQP′(G1, G2) Q(G1, G2)2 ≤(1 +o(1))Eπ⊥˜π" 1(G1,G2,π)∈E 1(G1,G2,˜π)∈E1 1−ρ2|I∗|(|I∗|−1)/2# = (1 + o(1))(1+ϵ)s2 nX t=0P[|I∗|=t]1 1−ρ2t(t−1)/2 ≤(1 +o(1))(1+ϵ)s2 nX t=0s n2t1 1−ρ2t(t−1)/2 . (... | https://arxiv.org/abs/2505.14138v1 |
1 ≤t≤k. Then [C1, C2,···, Ck−1, Ck]⊤=Jk[B1, B2,···, Bk−1, Bk]⊤, where Jk≜ 1−ρ0··· 0 0 1 −ρ··· 0 0 0 1 ··· 0 ............... −ρ0··· 0 1 and thus det ( Jk) = 1−ρk(see, e.g., [Dav79, Section 3.2]). Then, we obtain that EQ[LC] =1 (2π(1−ρ2))k/2det (Jk)Z ···Z exp Pk t=1−c2 t 2(1−ρ2)! dc1···dck=1 1−ρk=1 1−ρ2|C|... | https://arxiv.org/abs/2505.14138v1 |
(35) Proof. The results follow from Theorems 1 and 2 in [Gho21]. 28 D.3 Concentration Inequalities for Hypergeometric Distribution Lemma 6 (Concentration inequalities for Hypergeometric distribution) .Forη∼HG(n, s, s )and anyϵ >0, we have P η≥(1 +ϵ)s2 n ≤exp −ϵ2s2 (2 +ϵ)n ∧exp −ϵ2s3 n2 , (36) P η≤(1−ϵ)s2 n ≤exp... | https://arxiv.org/abs/2505.14138v1 |
arXiv:2505.14458v1 [math.ST] 20 May 2025Adaptive Estimation of the Transition Density of Controlled Markov Chains Imon Banerjee1, Vinayak Rao2, and Harsha Honnappa3 1Department of Industrial Engineering and Management Sciences, Northwestern University 2Department of Statistics, Purdue University 3Edwardson School of In... | https://arxiv.org/abs/2505.14458v1 |
. . 14 4.3 Estimating the Transition Density of Fully Connected non-Markovian CMC’s . . . . . . . . 15 5 Conclusions 16 1 A Sketch of Proof of Proposition 2 20 B Proofs 22 B.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 B.2 Proof of Proposition 2 . . . . . . . .... | https://arxiv.org/abs/2505.14458v1 |
. . . . . . . . . . . . 39 B.12 Proposition 19 and proof of its upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B.13 Proof of the lower bound of Proposition 19 . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B.14 Proof of the upper bound in Lemma 15 . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2505.14458v1 |
. . . . . . . 58 1 Introduction A stochastic process {(Xi, ai)}is called a controlled Markov chain (CMC) [18] if the next “state” Xi+1 depends only on the current state Xiand the current “control” ai. Informally, this means: P Xi+1∈dy|X0, a0, . . . , X i, ai =P Xi+1∈dy|Xi=xi, ai=li =s(xi, li, y)µχ(dy), where s(xi, ... | https://arxiv.org/abs/2505.14458v1 |
to be σ-H¨older continuous for all values of l. To avoid such strong assumptions, we rely upon the recent and rapidly evolving techniques of adaptive density estimation . This technique was pioneered by [12] and has been further developed in [42, 8, 10, 11, 17, 52]. In this paper, our objective is to adapt this techniq... | https://arxiv.org/abs/2505.14458v1 |
even when us- ing the traditional Hellinger loss, the assumption of stationarity—though convenient (Theorem 2)—is not necessary (Theorem 3). A careful analysis reveals a deeper connection with the return times of the stochastic process {(Xi, ai)}. Key in making this connection is a Kac-type lower bound (Lemma 25) for r... | https://arxiv.org/abs/2505.14458v1 |
histogram estimator ˆsm(·,·,·)of s(we will just use the term estimator ) is defined as ˆsm(·,·,·) :=X k∈mPn−1 i=0 1k(Xi, ai, Xi+1)Pn−1 i=0R χ1k(Xi, ai, y)dµχ(y)1k(·,·,·). (2.1) For any two bounded positive functions f1andf2(not necessarily densities) define the square of the empirical Hellinger distance H2as H2(f1, f2)... | https://arxiv.org/abs/2505.14458v1 |
of finding ˆmisO nl(d1+d2) +l2(l+1)(d1+d2) . See [52, Proposition A.1] or [10, Section 3.2.4] for details. Observe that ˆmdepends solely on{(X0, a0), . . . , (Xn, an)},l, andL. We define the estimator ˆs:= ˆsˆm and highlight its dependence on landL, although we omit these details in the notation for brevity. Theorem ... | https://arxiv.org/abs/2505.14458v1 |
second inequality follows from the definition of m†. The third inequality follows from the definition of H2(s, Vm)in Proposition 1. The final equality follows by observing that H2(s,0∅) = 1 /2and by substituting the value of pen(∅). Substituting the value of pen(m†)from eq. (2.2) we now get |m†| ≤2 +n/(Llogn) Recall fr... | https://arxiv.org/abs/2505.14458v1 |
can be viewed as a generalization of stationarity, while the latter dispenses with stationar- ity altogether. Proposition 5 provides a simple example showing that a sharper bound can be derived by incorporating the ergodic occupation measure than by ignoring it. 3.1 Ergodic Occupation Measure Exists The ergodic occupat... | https://arxiv.org/abs/2505.14458v1 |
that we recover a sharper bound for R(n)due to our use of the Bernstein’s inequality (see Section 3.2 for details). In particular, when d1=d2, we show that R(n)≤ O 22ldexp −Cpk0n C∆22ld+3(logn)2 , whereas [52] obtains the bound O n223ld+1exp −s nk0 (40×2ld)!! which is larger for sufficiently large n. We now turn to... | https://arxiv.org/abs/2505.14458v1 |
f1(x, l, y )−p f2(x, l, y )2 µχ(dy)νn(dx, dl ). Choose a depth l≤nand let m(2) refbe the partition of χ×Iinto uniform cubes of edge length 2−l. To avoid trivialities, we implicitly assume throughout the rest of this section that T(S)<∞for any S ∈m(2) ref. We interpret this condition to mean that the controlled Markov ... | https://arxiv.org/abs/2505.14458v1 |
n. This completes the proof. A final question concerns whether the utility of considering the ergodic occupation measure described in Section 3.1 when Theorem 3 proves a risk bound under a more general setting. The benefit is in the inherent tightness that an average case object like the ergodic occupation measure prov... | https://arxiv.org/abs/2505.14458v1 |
1. Let p∈(2d/(d+ 1),∞),σ∈(2d(1/p−1/2)+,1), and√s∈Bσ(Lp(A)). 2. For each i,(Xi, ai)admits the density Φisuch that Φi(x, l)≤Γfor all (x, l)∈χ×I. Recall from the Section 1 the definition of Vol(·). Then we have the following corollary. Corollary 3. Under Assumption 2, the estimator ˆs= ˆs(L0,∞)satisfies C′E H2(s,ˆs) ≤ΓV... | https://arxiv.org/abs/2505.14458v1 |
Its proof is in Section B.7. Proposition 7. For all (i,S)∈N×χ×I, it holds P-almost everywhere that Eh τ(i) S FPi−1 p=1τ(p) Si <T(⋆)(S) ε3 0Vol(Sχ). (4.1) Remark 8. The bound in eq. (4.1) can be improved by a more careful (but considerably more tedious) bookkeeping, but this is sufficient for the purposes of illustratio... | https://arxiv.org/abs/2505.14458v1 |
data-dependent risk bounds, and risk bounds for controlled Markov chains with continuous state-control spaces. To conclude, we list a few directions for future research. Our deterministic guarantees rely on geometric α-mixing; existing concentration technology does not yet deliver comparably sharp bounds under summable... | https://arxiv.org/abs/2505.14458v1 |
: 1432- 2064. DOI:10.1007/s004400050210 . [13] J ¨oran Bergh and J ¨orgen L ¨ofstr ¨om.Interpolation Spaces: An Introduction . en. Ed. by S. S. Chern et al. V ol. 223. Grundlehren der mathematischen Wissenschaften. Berlin, Heidelberg: Springer, 1976. ISBN : 978-3-642-66453-3 978-3-642-66451-9. DOI:10.1007/978-3-642-664... | https://arxiv.org/abs/2505.14458v1 |
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