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2024. [3] O. E. Barndorff-Nielsen and D. R. Cox. Inference and Asymptotics . Chapman & Hall, London, 1994. [4] D. W. Berreman. Optics in stratified and anisotropic media: 4 ×4-matrix formulation. J. Opt. Soc. Am., 62(4):502–510, Apr 1972. [5] P. Billingsley. Probability and Measure . John Wiley & Sons, Inc., New York, ... | https://arxiv.org/abs/2504.16029v1 |
Majumdar and A. Zarnescu. Landau-De Gennes theory of nematic liquid crystals: the Oseen- Frank limit and beyond. Arch. Ration. Mech. Anal. , 196(1):227–280, 2010. [29] N. J. Mottram and C. J. P. Newton. Introduction to q-tensor theory. arXiv:1409.3542 , 2014. [30] H.Niederreiter. RandomNumberGenerationandQuasi-MonteCar... | https://arxiv.org/abs/2504.16029v1 |
arXiv:2504.16279v1 [math.ST] 22 Apr 2025Detecting Correlation between Multiple Unlabeled Gaussian Networks Taha Ameen and Bruce Hajek Department of Electrical and Computer Engineering Coordinated Science Laboratory University of Illinois Urbana-Champaign Urbana, IL 61801, USA Email:{tahaa3,b-hajek}@illinois.edu Abstrac... | https://arxiv.org/abs/2504.16279v1 |
Lelarge and Massoulié [14] and later by Ganassali, Massouli é and Semerjian [15]. Another example is the work of Rácz and Sridhar [16], where a pair of graphs are either independent, or grow together until a time t∗and independently afterwards according to an appropriate evolution model. The correlation detection probl... | https://arxiv.org/abs/2504.16279v1 |
graphs are correlated up to an (unknown) latent permutation . Under the null hypothesis H0, the vectors X1,···,Xmare independent. Under the alternative hypothesis H1, there exist uniformly random permutations π∗ 12,···,π∗ 1mon[n]such that (X1 ij,X2 π∗ 12(i),π∗ 12(j),···,Xm π∗ 1m(i),π∗ 1m(j))1≤i<j≤n are independent tupl... | https://arxiv.org/abs/2504.16279v1 |
for some ε >0, ρ2≤/parenleftbigg4 m−1−ε/parenrightbigglogn n. (3) Then weak detection is impossible, i.e. δ(P,Q) =o(1). The positive result in Theorem 2 establishes that for each m >2, there is a region in parameter space where weak detection is impossible with 2graphs alone, but using m graphs as side information allo... | https://arxiv.org/abs/2504.16279v1 |
loss of generality, we may assume that the underlying permutatio ns π∗ 1iare all the identity permutation id. It then follows from the definition of TthatP(T≤τ)≤P(T∗≤τ)whereT∗is the log-likelihood for the identity permutation profile: T∗=/summationdisplay 1≤k<ℓ≤m/summationdisplay 1≤i<j≤nXk ijXℓ ij UnderP,T∗is the sum of/... | https://arxiv.org/abs/2504.16279v1 |
for m= 3,and briefly explain at the end how the proof for general m≥3can be given. So suppose m= 3 and suppose ρ2≤(2−ǫ)logn n. To begin, we let Pdenote the joint distribution of X1,X2,X3and the unobserved random permutation profile π∗= (π∗ kℓ)k,ℓ∈[m]under hypothesis H1andQdenote the joint distribution of X1,X2,X3underH0.... | https://arxiv.org/abs/2504.16279v1 |
the same under PandQ. By the Markov prop- erty (under P) and independence property (under Q) discussed in the previous paragraph, adding in conditioning on X1does not change the conditional distributions. In other words, t he conditional distribution (π∗ 23,X2,X3|S= 1,X23,X1)is the same under PandQ. That is: Pπ∗ 23,X2,... | https://arxiv.org/abs/2504.16279v1 |
A. Haghighi, A. Y . Ng, and C. D. Manning, “Robust textual i nference via graph matching,” in Proceedings of Human Language Technology Conference and Conference on Empirical Methods in Natural L anguage Processing , 2005, pp. 387–394. [9] J. Ding and H. Du, “Detection threshold for correlated Er d˝os-Rényi graphs via d... | https://arxiv.org/abs/2504.16279v1 |
k-core estimator,” in 2023 IEEE International Symposium on Information Theory (ISIT) . IEEE, 2023, pp. 2499–2504. [27] T. Ameen and B. Hajek, “Robust graph matching when nodes are corrupt,” arXiv preprint arXiv:2310.18543 , 2023. [28] D. Huang, X. Song, and P. Yang, “Information-theoretic thresholds for the alignments ... | https://arxiv.org/abs/2504.16279v1 |
than or equal to the corresponding norms of AwhileZis the same for A replaced by its symmetrization. Since A=UΛU⊤for some orthonormal matrix UandΛbeing the diagonal matrix of eigenvalues, we have Z=/summationtext kλkW2 kwhereW=U⊤X so that the Wkare independent standard Gaussian random variables. Also, ∝bardblA∝bardblF=... | https://arxiv.org/abs/2504.16279v1 |
Linear Regression Using Hilbert-Space-Valued Covariates with Unknown Reproducing Kernel Xinyi Lia, Margaret Hochb, and Michael R. Kosorokb aClemson University,bUniversity of North Carolina at Chapel Hill Abstract: We present a new method of linear regression based on principal components using Hilbert-space- valued cov... | https://arxiv.org/abs/2504.16780v1 |
develop a new method for ob- taining PCA representations in such settings and establish asymptotic theory. Specifically, we achieve the dimensionality reduction for Hilbert space elements using Karhunen-Lo `eve expansions, derive the statistical learning properties (Donsker class results) ensuring uniform convergence, ... | https://arxiv.org/abs/2504.16780v1 |
we introduce the proposed regression model, outlining key methodological principles and estimation strategies. Section 4 discusses the prac- tical inference techniques using bootstrap methods and presents the corresponding asymptotic proper- ties of the bootstrap estimators. Section 5 details computational aspects and ... | https://arxiv.org/abs/2504.16780v1 |
Zk=P∞ j=1λ1/2 kjUkjϕkj, where ∞> λk1≥λk2≥ ··· ≥ 0,{ϕk1, ϕk2,···} is an orthonormal basis on Hk,EUkj= 0,EU2 kj= 1, and EUkjUkj′= 0, for all 1≤j̸=j′<∞and all 1≤k≤K. Denote Hk0⊂ H kas the separable subspace spanned 4 by{ϕk1, ϕk2,···}, and let Bk0=Bk∩ H k0,1≤k≤K. Since ⟨ak, Zk⟩k= 0 almost surely for all ak∈ H k\Hk0, define... | https://arxiv.org/abs/2504.16780v1 |
{a(rk) k=Prk j=1akjϕkj:ak=P∞ j=1akjϕkj∈ Bk0}for1≤k≤K. Since B∗ 0(r)is compact, we have that there exists a finite subset Tδ⊂ B∗ 0(r)such that supa∈B∗ 0(r)infb∈TδPK k=1∥ak−bk∥k≤δ, where a= (a1,···, aK)andb= (b1,···, bK). For the given set TδofK-tuples of finite sequences, define T∗ δas: T∗ δ={a∗=(a∗ 1, . . . , a∗ K) :∃a... | https://arxiv.org/abs/2504.16780v1 |
product ⟨·,·⟩kand norm ∥ · ∥ k, and random variable Zksatisfy As- sumption (A1), for each 1≤k≤Kfor some K <∞. Let µk=EZk,1≤k≤K, and assume thatPK k=1P r∈NkE(PK k′=1∥Zk′−µk′∥2rk′ k′)<∞, where Nk={r= (r1,···, rK)∈ {0,1}K:PK k′=1rk′=k}. Then F={f(Z) =QK k=1⟨hk, Zk⟩k:hk∈ Bk,1≤k≤K}is Donsker. Proof. We have KY k=1⟨hk, Zk⟩k=... | https://arxiv.org/abs/2504.16780v1 |
in ℓ∞(B × B ). This means the norm we want to compose for operators of this type is ∥V∥B×B= supa1,a2∈B|a⊤ 1V a2|, forV∈ℓ∞(B×B ). We also remind the reader that a sequence {Xn} converges outer almost surely to Xif there exists a sequence {∆n}of measurable random variables satisfying ∥Xn−X∥ ≤∆nfor all nand Pr (lim supn→∞... | https://arxiv.org/abs/2504.16780v1 |
the generality we are working in and goes beyond the scope of our paper. 11 (A3) For some 1≤m <∞,∞> λ1> λ2>···> λm> λm+1≥0. Theorem 4. Let Assumptions (A1)–(A3) be satisfied for a random variable Z∈ H and for some 1≤m <∞. Then (i)max 1≤j≤m+1|bλj−λj|as∗− − →0. (ii)max 1≤j≤m∥bϕj−ϕj∥as∗− − →0. Proof. As a consequence of P... | https://arxiv.org/abs/2504.16780v1 |
follows from Theorem 3. It is also the case that (2.4) implies V0(bϕj−ϕj)=(bϕj−ϕj)λj−(bVn−V0)ϕj−(bVn−V0) (bϕj−ϕj)+(bϕj−ϕj) (bλj−λj)+ϕj(bλj−λj) ⇒(V0−λjI)(bϕj−ϕj) =−(bVn−V0)ϕj+oP(n−1/2) +ϕjϕ⊤ j(bVn−V0)ϕj+oP(n−1/2) ⇒(V0−λjI+ϕjϕ⊤ j)(bϕj−ϕj)=−(bVn−V0)ϕj+ϕjϕ⊤ j(bVn−V0)ϕj+ϕjϕ⊤ j(bϕj−ϕj)+oP(n−1/2). 13 Since ϕjϕ⊤ j(bϕj−ϕj) =OP(... | https://arxiv.org/abs/2504.16780v1 |
n−δn|/σ0>−c1)≥1−ϵ/2. Letη2=ϵ/2; then η=p ϵ/2and this implies thatlim inf n→∞Pr{(bS2 n+ 1/n)−1/2(σ2 0+ 1/n)1/2)> η} ≥1−ϵ/2. We can now pick k1such thatp ϵ/2 −p 2/ϵ+k1 ≥kand thus k1≥kp 2/ϵ+p 2/ϵ. As long as (2.5) holds, this implies that lim inf n→∞Pr bTn≥k ≥lim inf n→∞Pr −√n|δ∗ n−δn| σ0>−c1, bS2 n+1 n σ2 0+1 n!−... | https://arxiv.org/abs/2504.16780v1 |
we have 1{Pm−1 j=1bλj<(1−α)Pn∥Z−¯Zn∥2<Pm j=1bλj}as∗− − → 1, asn→ ∞ , and the desired result is proved. Remark 2. Theorem 4 actually also gives us that bλm+1→λm+1; however, we will also need consis- tency of the accompanying eigenfunctions to be able to estimate all principal components for down- stream analysis. 3. Lin... | https://arxiv.org/abs/2504.16780v1 |
−1 Pn 1 X bZ∗ Y−bα∗ n− bβ∗ n⊤ X−(bγ∗ n)⊤bZ∗ = Pn 1 X bZ∗ ⊗2 −1 √n Pn 1 X bZ∗ Y−bα∗ n− bβ∗ n⊤ X−(bγ∗ n)⊤bZ∗ −P 1 X bZ∗ Y−bα∗ n− bβ∗ n⊤ X−(bγ∗ n)⊤bZ∗ ! = Pn 1 X bZ∗ ⊗2 −1 √n(Pn−P) 1 X bZ∗ Y−bα∗ n− bβ∗ n... | https://arxiv.org/abs/2504.16780v1 |
with mean and variance 1, and ¯ξn=n−1Pn i=1ξi, with∥ξ∥2,1=R∞ 0p Pr(|ξ|> x)dx <∞(the wild bootstrap). By Theorem 2.7 of Kosorok (2008), since Fis Donsker and P∗{supf∈F(f(x)−Pf)2}<∞, then eGnas− →Ginℓ∞(F). LeteV∗ n= G⊤ Nn−1ePn{(Z−eZ∗ n)(Z−eZ∗ n)⊤}GN, witheZ∗ n=ePnZ. We remind the reader that for a sequence of bootstrappe... | https://arxiv.org/abs/2504.16780v1 |
−1 ePn 1 X eZ∗ n Y−eα∗ n−(eβ∗ n)⊤X−(eγ∗ n)⊤eZ∗o = ePn 1 X eZ∗ ⊗2 −1 √n ePn 1 X eZ∗ n Y−eα∗ n−(eβ∗ n)⊤X−(eγ∗ n)⊤eZ∗o −Pn 1 X eZ∗ n Y−eα∗ n−(eβ∗ n)⊤X−(eγ∗ n)⊤eZ∗o ! = Pn 1 X eZ∗ ⊗2 −1 √n(ePn−Pn) 1 X eZ∗ n Y−eα∗ n−(eβ∗ n)... | https://arxiv.org/abs/2504.16780v1 |
to approximate the domain of interest; and second, constructing multivariate splines based on this triangulation. A multivariate spline over a triangulation is defined as a piecewise polynomial function on a triangulated domain that ensures smooth connections across simplex boundaries. The degree of smoothness is typic... | https://arxiv.org/abs/2504.16780v1 |
shown below. In all simulations, we fix d= 4,α0= 1, and β0= (1,1,1,1)⊤. We examine two correlation scenarios: r= 0 (independent covariates) and r= 0.5(moderately correlated covariates), and three sample sizes: n= 100 ,500, and 2000 . We conduct 100Monte Carlo replications for each setting, 27 compare the estimated basi... | https://arxiv.org/abs/2504.16780v1 |
n λ 1 λ2 λ3 λ4 λ5 λ6bmPVE nbmPA VE n 0 100 39.66 9.04 6.82 7.58 4.57 4.24 5.98 4.58 500 4.77 3.67 2.41 1.42 0.83 0.45 6.00 5.20 2000 1.25 0.97 0.64 0.45 0.18 0.09 6.00 5.53 0.5 100 39.66 9.04 6.82 7.58 4.57 4.24 5.98 4.58 500 4.77 3.67 2.41 1.42 0.83 0.45 6.00 5.20 2000 1.25 0.97 0.64 0.45 0.18 0.09 6.00 5.53 the concl... | https://arxiv.org/abs/2504.16780v1 |
(97%) (100%) (98%) (100%) (99%) (99%) 0.5 100 15.48 14.96 22.90 22.77 14.25 19.62 36.26 48.92 39.14 33.31 (98%) (98%) (97%) (95%) (97%) (100%) (100%) (100%) (100%) (100%) 500 1.96 2.46 3.32 3.74 2.62 3.42 4.02 6.76 6.24 4.91 (96%) (96%) (94%) (93%) (96%) (100%) (100%) (99%) (99%) (98%) 2000 0.54 0.68 0.95 0.69 0.66 0.8... | https://arxiv.org/abs/2504.16780v1 |
the true basis functions, which further support the conclusions in Theorem 4. Table 6.4 reports the MSEs for the estimates of eigenvalues λ1andλ2, regression coefficients α, βandγ, and the estimated number of PCA components bmn, under both PVE and PA VE criteria As sample size increases from n= 100 ton= 2000 , MSEs dec... | https://arxiv.org/abs/2504.16780v1 |
disease progression. Furthermore, we extend our modeling framework to a precision medicine setting, where imaging biomarkers, along with environmental and genetic risk factors, inform optimal treatment decisions for Alzheimer’s disease. This approach enables patient-specific intervention and treatment strategies based ... | https://arxiv.org/abs/2504.16780v1 |
VE. Right: Esti- mated coefficient bγbased on the six leading PCs. Table 7.1 presents the estimated coefficients for the nonfunctional predictors, along with the cor- responding 95% bootstrap confidence intervals. With the increase of age, the general MMSE scores decrease significantly; and with the increase of educati... | https://arxiv.org/abs/2504.16780v1 |
in MMSE by around 9.85units on average. The conclusions for the main effect generally agree with what we observed in Section 7.1. Table 7.2: Estimated coefficients and 95% bootstrap confidence intervals for the linear covariates αand β. Term Estimate 95% Bootstrap CI Term Estimate 95% Bootstrap CI Intercept 12.488 (4.5... | https://arxiv.org/abs/2504.16780v1 |
R., Ringman, J. M., Rossor, M. N., Schofield, P. R., Sperling, R. A., Salloway, S., and Morris, J. C. (2012), “Clinical and biomarker changes in dominantly inherited Alzheimer’s disease,” New England Journal of Medicine , 367, 795–804. Chen, Y ., Goldsmith, J., and Ogden, R. T. (2019), “Functional data analysis of dyna... | https://arxiv.org/abs/2504.16780v1 |
2, 321– 359. Perry, R., Panigrahi, S., Bien, J., and Witten, D. (2024), “Inference on the proportion of variance explained in principal component analysis,” arXiv preprint arXiv:2402.16725 . Petersen, R. C., Aisen, P., Beckett, L. A., Donohue, M., Gamst, A., Harvey, D. J., Jack, C., Jagust, W., Shaw, L., Toga, A., and ... | https://arxiv.org/abs/2504.16780v1 |
Estimation and Inference for the Average Treatment Effect in a Score-Explained Heterogeneous Treatment Effect Model Kevin Christian Wibisono University of Michigan, Statistics kwib@umich.eduDebarghya Mukherjee Boston University, Statistics mdeb@bu.edu Moulinath Banerjee University of Michigan, Statistics moulib@umich.e... | https://arxiv.org/abs/2504.17126v1 |
treatment effect (ITE) which can potentially depend on the background information and score variable. However, as is true for most real world applications, the score variable Qand the unobserved error νcan be correlated through some unobserved confounders . In the exam-based scholarship example, νmay encode students’ i... | https://arxiv.org/abs/2504.17126v1 |
does not hold as argued earlier. Motivated by these observations, Mukherjee et al. [2021] proposed an efficient√n-rate estimator of the treatment effect in the presence of endogeneity that uses all observations. Their approach assumes a homogeneous treatment effect model, where the response variable Yis modeled as Y=α0... | https://arxiv.org/abs/2504.17126v1 |
our model (with X=Z) is pictorially presented in Figure 1. X Y Qη ϵ Figure 1: A graphical representation of the variables of Equations (1.3) and (1.4), with X=Z. In this paper, our primary goal is to estimate the average treatment effect on the treated (ATT), defined as θ0=E[α0(X, η)|Q≥τ0]. (1.5) As elaborated previous... | https://arxiv.org/abs/2504.17126v1 |
in our setting, a natural discontinuity arises due to the deterministic assignment of treatment based on a thresholded score function. Furthermore, the inclusion of Equation (1.4) may suggest that our method directly falls under the purview of the instrumental variable (IV) framework [Angrist et al., 1996], where the m... | https://arxiv.org/abs/2504.17126v1 |
how this method can be adapted for the practical scenario where the ηi’s are unknown. 6 Asη(i+1)−η(i)is small (typically of the order of n−1), we expect ℓ(η(i+1))−ℓ(η(i))to be negligible as long as fhas minimal smoothness (e.g., fisα-Hölder with α >1/2). Therefore, we have Y(i+1)−Y(i)≈(X(i+1)−X(i))⊤β0+ϵ(i+1)−ϵ(i). Now,... | https://arxiv.org/abs/2504.17126v1 |
We sort ˆηifor Algorithm 1 Estimation of the ATT θ0 Input: i.i.d. data {(Xi, Yi, Zi, Qi)}n i=1, threshold τ0 Output: A√n-consistent and asymptotically normal estimator of θ0 1:Partition {1,···, n}intoI1,I2,I3of roughly equal sizes. 2:Perform OLS of Qiagainst Zifori∈I1; obtain ˆγand set ˆηi=Qi−Z⊤ iˆγfor all i∈I2∪I3. 3:O... | https://arxiv.org/abs/2504.17126v1 |
characteristics X(such as age and race) and innate abilities η. We now present Algorithm 2, a slightly modified version of Algorithm 1 to estimate the ITE α0(X, η). Algorithm 2 Estimation of the ITE α0(X, η) Input: i.i.d. data {(Xi, Yi, Zi, Qi)}n i=1, threshold τ0 Output: An estimate of α0(X) 1:Follow Steps 1 to 5 of A... | https://arxiv.org/abs/2504.17126v1 |
from zero. Furthermore, for any sequence {δn}n≥1that converges to 0andb∈supp( η), we have f0,δn(b)→f0,0(b)andf1,δn(b)→f1,0(b). The compactness of XandZin Assumption 1 is made for technical convenience and can be relaxed with careful truncation arguments. However, in practical scenarios, XandZare naturally bounded or ca... | https://arxiv.org/abs/2504.17126v1 |
is the main result of this paper, which proves that the ATT can be estimated at a parametric rate despite a non-parametric correlation between the unobserved errors in Equations (1.3) and(1.4). The key steps of the proof are presented in Section 7, while the detailed proof can be found in Appendix B. Remark 5 (Cross-fi... | https://arxiv.org/abs/2504.17126v1 |
that each split is of size ˜n= 4,000. For each iteration 1≤k≤1,000, we compute ζk=√ ˜n(ˆθk−θ0), where ˆθkis the estimate of the ATT θ0using iteration kfollowing Algorithm 1. The sample mean and variance of the ζk’s are around 0.05and12.5, respectively, close to 0and11.455. Moreover, the histogram of the ζk’s (not shown... | https://arxiv.org/abs/2504.17126v1 |
chosen via 4-fold cross-validation, and quadratic interaction terms. Table 2 shows that the mean squared error (MSE) of ˆα(X)among all treated individuals tends to decrease as the sample size nincreases. Moreover, Figure 3 provides a comparison between ˆα(X)andα0(X)for(X2, X3) = (±0.7,±0.2) when n= 50,000, demonstratin... | https://arxiv.org/abs/2504.17126v1 |
of population above the age of 60 in 2000; (10) the ratio of males to females in 2000; and (11) the average household size in 2000. Using our method, we find an estimated ATT of ˆθ= 0.65. To test the null hypothesis H0: θ0= 0versus the alternative hypothesis H1:θ0̸= 0, we employ bootstrapping with B= 500 bootstrap samp... | https://arxiv.org/abs/2504.17126v1 |
work may focus on establishing the consistency of the bootstrap variance estimator and conducting inference on the ITE and CATE estimators. 7 Roadmap of theoretical proofs In this section, we outline proof sketches for Proposition 1 and Theorem 1, with full proofs provided in Appendices A and B, respectively. 7.1 Proof... | https://arxiv.org/abs/2504.17126v1 |
observe that F=1√ ˜n|IC 2|−1X i=1 ℓ(η(i+1))−ℓ(ˆη(i+1)) uˆδ˜n(i+1)−uˆδ˜n(i) −1√ ˜n|IC 2|−1X i=1 ℓ(η(i))−ℓ(ˆη(i)) uˆδ˜n(i+1)−uˆδ˜n(i) +1√ ˜n|IC 2|−1X i=1 ℓ(ˆη(i+1))−ℓ(ˆη(i)) uˆδ˜n(i+1)−uˆδ˜n(i) . The last summand can be shown to be asymptotically negligible, again due to the fact that the ordering is done on ... | https://arxiv.org/abs/2504.17126v1 |
i=2˜n+1ti(α0(Xi, ηi)−E(α0(X, η)|Q≥τ0)), (2) =√ ˜n ˜n1(β0−ˆβ˜n)⊤3˜nX i=2˜n+1 ti−(1−ti)Kˆδ˜n,i Xi, (3) =√ ˜n ˜n13˜nX i=2˜n+1 ti−(1−ti)Kˆδ˜n,i ϵi, (4) =√ ˜n ˜n13˜nX i=2˜n+1ti ℓ(ηi)−ℓ(ηc(i)) . We initially focus on examining terms (2) and (4) before returning to discuss terms (1) and (3) in Step 4. Step 2: Term (2) W... | https://arxiv.org/abs/2504.17126v1 |
need to apply the martingale central limit theorem [Billingsley, 1995] to establish normality jointly. Details are provided in Appendix B. 25 References A. Abadie and G. W. Imbens. Matching on the estimated propensity score. Econometrica , 84(2): 781–807, 2016. M. L. Anderson, C. Dobkin, and D. Gorry. The effect of inf... | https://arxiv.org/abs/2504.17126v1 |
L. Powell, and F. Vella. Nonparametric estimation of triangular simultaneous equations models. Econometrica , 67(3):565–603, 1999. J. Pinkse. Nonparametric two-step regression estimation when regressors and error are dependent. Canadian Journal of Statistics , 28(2):289–300, 2000. J. M. Robins, A. Rotnitzky, and L. P. ... | https://arxiv.org/abs/2504.17126v1 |
˜nvuut|IC 2|−1X i=1 ˆη(i+1)−ˆη(i)2vuut2|IC 2|X i=1(uk,ˆδ˜ni)2 =Op(n−1) for some Lipschitz constant ν2using the Cauchy-Schwarz inequality and the elementary inequality (a−b)2≤2(a2+b2), Assumptions 1 and 3, and the fact that the ordering is done on the ˆηi’s. Specifically, Assumption 1 implies that E(u2 k,ˆδ˜n|Q < τ 0)... | https://arxiv.org/abs/2504.17126v1 |
≥ϵ 2∩ ||ˆδ˜n||2< ξ +P(||ˆδ˜n||2≥ξ) ≤P Λ˜n,t−2P(Q < τ 0)Σu,ˆδ˜n,t ≥ϵ 2 +P(||ˆδ˜n||2≥ξ). Note that the first term goes to 0as established above, and so does the second term since ˆδ˜nP−→0. This implies Λ˜n,tP−→2P(Q < τ 0)Σu,tfor every coordinate t, which means Λ˜nP−→2P(Q < τ 0)Σu. This completes the proof. From Lemma ... | https://arxiv.org/abs/2504.17126v1 |
τ 0 →E ℓ′(η)u0(w0)⊤|Q < τ 0 , where uˆδ˜n=X−E(X|ˆη, Q < τ 0),wˆδ˜n=Z−E(Z|ˆη, Q < τ 0),u0=X−E(X|η, Q < τ 0), andw0=Z−E(Z|η, Q < τ 0). Proof. It is easy to see that the statement we want to show is equivalent to E(ℓ′(ˆη)cov(X, Z|ˆη, Q < τ 0)|Q < τ 0)→E(ℓ′(η)cov(X, Z|η, Q < τ 0)|Q < τ 0). Note that for any ˜n, we have ... | https://arxiv.org/abs/2504.17126v1 |
pair of sufficient conditions isP˜n i=1E(|ξ˜n,i|3)→0andP2˜n i=˜n+1E(|ξ˜n,i|3)→0 as˜n→ ∞ , whence the Lyapunov’s (and consequently Lindeberg’s) condition is satisfied. 36 Observe that ˜nX i=1E(|ξ˜n,i|3) =˜nX i=1E E 1√ ˜nc⊤A1 ˜Z⊤˜Z ˜n!−1 Ziηi 3 |Z1:˜n =1 ˜n3/2˜nX i=1E c⊤A1 ˜Z⊤˜Z ˜n!−1 Zi 3 E(|ηi|3|Zi) ≤ν... | https://arxiv.org/abs/2504.17126v1 |
i=2˜n+1Xi P0,i−f1,ˆδ˜n(ˆηi) f0,ˆδ˜n(ˆηi)F0,ˆδ˜n(ˆη(i+1))−F0,ˆδ˜n(ˆη(i−1)) 2! =op(1), 2˜n+˜n0X i=2˜n+1Xif1,ˆδ˜n(ˆηi) f0,ˆδ˜n(ˆηi)F0,ˆδ˜n(ˆη(i+1))−F0,ˆδ˜n(ˆη(i−1)) 2−E f1,ˆδ˜n(ˆη) f0,ˆδ˜n(ˆη)X|Q < τ 0! =op(1). From Lemma 4, conditional on ˆδ˜n, we have 1 ˜n3˜nX i=2˜n+1(1−ti)Kˆδ˜n,iXi−P(Q≥τ0)E f1,ˆδ˜n(ˆη) f0,ˆδ˜n(ˆη)X|Q <... | https://arxiv.org/abs/2504.17126v1 |
η1:2˜n, X1:2˜n, ϵ˜n+1:2˜n), F˜n,2˜n+1=σ(Z1:2˜n, η1:2˜n+1, X1:2˜n+1, ϵ˜n+1:2˜n),···,F˜n,3˜n=σ(Z1:2˜n, η1:3˜n, X1:3˜n, ϵ˜n+1:2˜n), F˜n,3˜n+1=σ(Z1:3˜n, η1:3˜n, X1:3˜n, ϵ˜n+1:2˜n+1),···,F˜n,4˜n=σ(Z1:3˜n, η1:3˜n, X1:3˜n, ϵ˜n+1:3˜n). For each ˜n, it is easy to see that(iX j=1ξ˜n,j,F˜n,i,1≤i≤4˜n) is a martingale. We now use B... | https://arxiv.org/abs/2504.17126v1 |
arXiv:2504.17175v1 [math.PR] 24 Apr 2025Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: The correlated case. Soukaina Douissi∗, Philip Ernst†, Frederi Viens‡ Friday 25thApril, 2025 Abstract We study the continuous-time version of the empirical corre lation coefficient between the paths of two pos... | https://arxiv.org/abs/2504.17175v1 |
comes an application of Jensen’s inequality where equality does not hold because the paths are not constant. All other details, including the definition of ρin discrete time, are omitted, since many references includ ing several cited below, such as [11, 2], cover this topic. The topic of independence testing for contin... | https://arxiv.org/abs/2504.17175v1 |
case r= 0was posted in 2022 as [10], which predates the publication of the paper [2]. In t he paper [2], the case of r= 0was studied in detail, and a speed of convergence in this CLT was establishe d, at the so-called Berry-Esséen rate 1/√ T, using tools from the so-called Wiener chaos analysis. That p aper also studie... | https://arxiv.org/abs/2504.17175v1 |
Wiener chaos. Another major difference between [7] and the current p aper is that the latter is in continuous time and the former is in discrete time; this is perhaps a superfici al distinction in terms of results, since both papers concentrate on increasing-horizon asymptotics. Ho wever, in terms of proofs, whether the ... | https://arxiv.org/abs/2504.17175v1 |
independence for solutions of the stochastic heat equation. But first, in Section 2, we begin wi th some preliminary information on analysis on Wiener space, to help make this paper essentially self-co ntained beyond the construction of basic objects like the Wiener process. 2 Elements of the analysis on Wiener space Th... | https://arxiv.org/abs/2504.17175v1 |
Xq∈Hqfor every q/greaterorequalslant1. This is summarized in the direct-orthogonal-sum notation L2(Ω) = ⊕∞ q=0Hq. HereH0denotes the constants. •Relation with Hermite polynomials. Multiple Wiener integr als. The mapping Iq(h⊗q) : = q!Hq(W(h))is a linear isometry between the symmetric tensor product H⊙qspace of functions... | https://arxiv.org/abs/2504.17175v1 |
domain can be expressed explicitly for any Xas in (4): X∈D1,2if and only if/summationtext qqq!/bardblfq/bardbl2 H⊗q<∞. •Generator Lof the Ornstein-Uhlenbeck semigroup . The linear operator Lis defined as being diagonal under the Wiener chaos expansion of L2(Ω):Hqis the eigenspace of Lwith eigenvalue−q, i.e. for any X∈Hq... | https://arxiv.org/abs/2504.17175v1 |
n, such that cmax/braceleftbig E/bracketleftbig X4 n/bracketrightbig −3,/vextendsingle/vextendsingleE/bracketleftbig X3 n/bracketrightbig/vextendsingle/vextendsingle/bracerightbig /lessorequalslantdTV(Xn,N)/lessorequalslantCmax/braceleftbig E/bracketleftbig X4 n/bracketrightbig −3,/vextendsingle/vextendsingleE/bracketl... | https://arxiv.org/abs/2504.17175v1 |
Y12(T)√ T−r√ T 2θ/parenrightBigg L−→N/parenleftbigg 0,1 2θ3/parenleftbigg1 2+r2 2/parenrightbigg/parenrightbigg asT→+∞. Proof. We will first prove a CLT for the second- Wiener chaos term Ar(T), indeed we can write Ar(T) :=Ar,1(T)+Ar,2(T) (15) We claim that as T→+∞ Ar,1(T) :=c1(r)√ T/integraltextT 0IU1 2(f⊗2 u)duL−→... | https://arxiv.org/abs/2504.17175v1 |
0f⊗2 tdt/parenrightBigg2 = 2/bardbl1√ T/integraldisplayT 0f⊗2 tdt/bardblL2[0,T]2 =2 T/integraldisplayT 0/integraldisplayT 0/a\}bracketle{tf⊗2 t,f⊗2 s/a\}bracketri}htdtds =2 T/integraldisplayT 0/integraldisplayT 0(/a\}bracketle{tft,fs/a\}bracketri}ht)2dtds =2 T/integraldisplayT 0/integraldisplayT 0/parenleftbigg/integ... | https://arxiv.org/abs/2504.17175v1 |
E[|Yθ(T)|]/lessorequalslant1√ T1 θ2(r+/radicalbig 1−r2). (26) which finishes the proof of Theorem 1. Proposition 6 Letp/greaterorequalslant1, then there exists a constant depending only on θandp, such that E/bracketleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2θ/radicalbigg Y11(T) T×Y22(... | https://arxiv.org/abs/2504.17175v1 |
>2, then we can write using the hypercontractivity property of multiple Wiener integrals (7) : +∞/summationdisplay n=1P/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleAr(n)√n/vextendsingle/vextendsingle/vextendsingle/vextendsingle> ε/parenrightbigg =+∞/summationdisplay n=1P/parenleftbigg/vextendsi... | https://arxiv.org/abs/2504.17175v1 |
E[¯Y−4 11(T)1/braceleftBig |¯Y11(T)−1|<1√ T/bracerightBig],we have 1−1√ T<¯Y11(T)<1+1√ Ta.s. we can say that for all T > T 0:= 1,¯Y11(T)>0a.s. thus it’s bounded away from 0. Hence sup T/greaterorequalslantT0E[¯Y−4 11(T)1/braceleftBig |¯Y11(T)−1|<1√ T/bracerightBig]< +∞.For the first expectation E[¯Y−4 11(T)1/braceleftBi... | https://arxiv.org/abs/2504.17175v1 |
, cαa threshold depending on α, that we will determine. On the other hand, by definition of type I error and 40, we can wr ite that for Tlarge, we have α≈PH0/parenleftBig√ T|ρ(T)|> cα/parenrightBig =PH0/parenleftBig√ T√ θ|ρ(T)|>√ θcα/parenrightBig We infer that a natural rejection regions Rαare of the form : Rα:=/bracel... | https://arxiv.org/abs/2504.17175v1 |
statist ic of our test and therefore to reject independence if/braceleftBig/vextendsingle/vextendsingle/vextendsingleY12(T)√ T/vextendsingle/vextendsingle/vextendsingle> cα/bracerightBig ,whereα≈PH0/parenleftBig/vextendsingle/vextendsingle/vextendsingleY12(T)√ T/vextendsingle/vextendsingle/vextendsingle> cα/parenrightB... | https://arxiv.org/abs/2504.17175v1 |
To control the first term1 σr,θIW 2(hT), we will use the following proposition. 26 Proposition 16 There exists T0/greaterorequalslant0, such that for all T/greaterorequalslantT0,∀x∈R, /vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbigg1 σr,θIW 2(hT)< x/parenrightbigg −φ(x)/vextendsingle/vextendsingle... | https://arxiv.org/abs/2504.17175v1 |
following constants : c1(T,r,θ) =K(θ,r)√ T,C′=C∨2, K(θ,r) =/parenleftBig θ2 8|r|∧1 2c(θ)√ 1−r2∧2θ2 √ r2+1/parenrightBig σr,θ c2(T,r,θ) =|r| 4θ21 σr,θ√ T Then, it’s easy to check that the function ε/ma√sto→g(ε) :=C′e−c1(T,r,θ)(ε 2−c2(T,r,θ))+εis convex on (0,+∞)and thatarg inf ε>0g(ε) =ε∗(T) =/parenleft... | https://arxiv.org/abs/2504.17175v1 |
in the non-explicit nature of the Wiener chaos kernels n eeded to represent the solution of AR( p) as a Gaussian time series, and its functionals that go into c omputing the Pearson correlation of a pair of AR( p) processes. This could be technically challenging, though not conceptually so. The case of time series with... | https://arxiv.org/abs/2504.17175v1 |
a family of independent standard Brownian motions and {hk,k/greaterorequalslant1}are the eigenfunctions of ∆, given by : hk(x) =/radicalbigg 2 πsin(kx), k/greaterorequalslant1, (56) •{hk,k/greaterorequalslant1}forms a complete orthonormal system in L2([0,π]). In this case, using the diagnalization afforded by the eigen-... | https://arxiv.org/abs/2504.17175v1 |
are other possible options, such as using differe nt numbers of Fourier modes for each copy, different time horizons (which could also occur if N= 1), and different correlations rkfor everyk. We may also study other spatial noise structures for white n oise in (54). For a spatial covariance operator QforWin /(55) is co-di... | https://arxiv.org/abs/2504.17175v1 |
same arguments as above, combined with Propositio n 13, shows that, if instead, we define our test using the numerator Y12,kof the empirical correlation ρkofuk,1anduk,2instead of the full ρkitself, as definded in Section 4.2, then the Type-II error βis bounded above as β/lessorequalslant(lnT)N TN/2N/productdisplay k=1C(k... | https://arxiv.org/abs/2504.17175v1 |
2, where Tis fixed and Ntends to infinity, which is a more complex endeavor, since the main propositions in this paper are not tailored to asympotics for fixed time horizon. However, the observation frequency discussi on above regarding Scenario 3 is an indication that asymptotics for Ntending to infinity and Tfixed probabl... | https://arxiv.org/abs/2504.17175v1 |
vi=ui−u1,i/greaterorequalslant2, then : kp(˜FT) =2p−1(cp 1(r)+cp 2(r))×(p−1)! (√ T√ 2/bardblhT/bardbl)p/integraldisplayT 0/integraldisplayT−u1 −u1.../integraldisplayT−u1 −u1δ(vp)δ(vp−vp−1)×...×δ(v3−v2)dv2dv3...dvpdu1 On the other hand, by dominated convergence theorem, we have 1 T/integraldisplayT−u1 −u1.../integraldis... | https://arxiv.org/abs/2504.17175v1 |
Ornstein-Uhlenbeck processes of general Hurst parameter. Statistical Inferen ce for Stochastic Processes, 22(1), 111-142. [15] Hu, Y. and Song, J. (2013). Parameter estimation for fra ctional Ornstein-Uhlenbeck processes with discrete observations. F. Viens et al (eds), Malliavin Calc ulus and Stochastic Analysis: A Fe... | https://arxiv.org/abs/2504.17175v1 |
arXiv:2504.17202v1 [math.ST] 24 Apr 2025Graph Quasirandomness for Hypothesis Testing of Stochastic Block Models Kiril Bangachev∗Guy Bresler† April 25, 2025 Abstract The celebrated theorem of Chung, Graham, and Wilson on quasirand om graphs implies that if the 4-cycle and edge counts in a graph Gare both close to their ... | https://arxiv.org/abs/2504.17202v1 |
. . . . . . . . . . . . . . . . . . . . . . . 11 2 Preliminaries 14 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Testing via Low Degree Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Simple Facts about Fourier Coefficients of Stochasti... | https://arxiv.org/abs/2504.17202v1 |
a sample from G(n,1/2) or, in their terminology, quasirandom . To describe the result, we denote by CH(G) the number of labeled copies of Hin a graph G(so, for example, CH(Kn) = n(n−1)···(n−|V(H)|+1) for any H) and by C∗ H(G) the number of induced labeled copies. Theorem 1.1 ([CGW88]).The following conditions are equiv... | https://arxiv.org/abs/2504.17202v1 |
increasingly popular framework of low-degree polynomial t ests (see Section 2.2 for definitions), this would give evidence for computational indistinguishabili ty. The reason this argument fails is that Theorem 1.1 is quantitatively too weak in the o(1) dependence on the number of vertices. For example, consider Pn=G(n,... | https://arxiv.org/abs/2504.17202v1 |
Gu,vis distributed as Bern(1/2),due to the randomness of the label of v. Our Conjecture and Results. We make the following quasirandomness conjecture for hypot h- esis testing of a stochastic block model ( Definition 1 below) against Erd˝ os-R´ enyi. Conjecture 1. Suppose that Pnis a stochastic block model (whose parame... | https://arxiv.org/abs/2504.17202v1 |
there exists a low-degree distinguisher implementable in nearly-quadratic time or there does not ex ist any constant-degree polynomial distinguisher. On Finding A Distinguisher. The practical distinguisher is easy to find as noted in [ YZZ24] – it is a signed star, triangle, or 4-cycle count. Hence, a stat istician in p... | https://arxiv.org/abs/2504.17202v1 |
one can further observe that IE G∼PnSCH(G) = IE G∼Pn/productdisplay (ij)∈E(H)(2Gij−1)×/parenleftBigg/summationdisplay H1∼H1/parenrightBigg = Θ/parenleftBigg n|V(H)|×IE G∼Pn/bracketleftBigg/productdisplay (ij)∈E(H)(2Gij−1)/bracketrightBigg/parenrightBigg . The quantity ΦPn(H):= IE G∼Pn/bracketleftBigg/productdisplay (ij... | https://arxiv.org/abs/2504.17202v1 |
(2Gi,j−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(xi)1≤i≤h/bracketrightBigg/bracketrightBigg = IE/bracketleftBigg/productdisplay (i,j)∈E(H)IE/bracketleftBigg (21+Qxi,xj 2−1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(xi)1≤i≤h/bracketrightBigg/bracketrightBigg = I... | https://arxiv.org/abs/2504.17202v1 |
the case of SBMs with non-vanishing community frac- tions and 2-SBMs (as demonstrated in Theorem A.8 ). Instead, we rely on many-to-one compar- isons. For example in the case of 2-SBMs, we prove that for any Hand anySBM(p,Q),there exists someKSBM(p,Q)∈ {stars on at most Dedges}∪{4-cycles}such that ΨSBM(p,Q)(H)/lessorsi... | https://arxiv.org/abs/2504.17202v1 |
. 1.5 Prior Techniques and Barriers Our two main results are parts 3 and 4 of Theorem 1.4 . We begin with an overview of previous techniques appearing in the recent work on planted graph mod els [YZZ24] and the classical quasir- andomness theory [ CGW88] and outline two barriers which prevent those techniques fr om wor... | https://arxiv.org/abs/2504.17202v1 |
for when the subgraph counts of graphs resemble that of Erd˝ os-R´ enyi. One way to r ewrite their results to make them more similar to our setting is as follows. Let Gbe any graph on nvertices and let GbeGafter a uniformly random vertex permutation. Then, the condition |CH(G)|= (1+o(1))n|V(H)|2−|E(H)| can be equivalen... | https://arxiv.org/abs/2504.17202v1 |
too weak as we obtain |ΦSBM(p,Q)(T)|1 |E(T)|≤ |ΦSBM(p,Q)(Cyc4)|1 |E(Cyc4)|and|E(T)|=|V(T)|−1. 64 325 1 GraphHLeaf-Isolation Triangle Inequality =⇒4 321/unionsqtext 675 GraphH′Star2 Figure 1: Illustration of the leaf-isolation inequality ov er the graph Hwith leaves 5 ,6.It effectively compares |ΦSBM(p,Q)(H)|to|ΦSBM(p,Q)... | https://arxiv.org/abs/2504.17202v1 |
following two types of cancellations occurs: 1.Within-community cancellations: when there is a cancellation within community 1, corre- sponding to |p1Q1,1+p2Q1,2|=o(p1|Q1,1|+p2|Q1,2|) or, similarly in community 2, |p1Q1,2+ p2Q2,2|=o(p1|Q1,2|+p2|Q2,2|) . 2.Between-community cancellations: when there are no in-community ... | https://arxiv.org/abs/2504.17202v1 |
G∼Pnf(G)−IE G∼G(n,1/2)f(G)/vextendsingle/vextendsingle/vextendsingle=ω/parenleftBigg max/parenleftBig Var G∼G(n,1/2)[f(G)]1/2,Var G∼Pn[f(G)]1/2/parenrightBig/parenrightBigg . Conversely, if a polynomial fof degree at most Dsatisfies /vextendsingle/vextendsingle/vextendsingleIE G∼Pnf(G)−IE G∼G(n,1/2)f(G)/vextendsingle/ve... | https://arxiv.org/abs/2504.17202v1 |
of graphs without isolated vertices and at most Dedges), there exists some Hsuch that/summationtext H1⊆Kn:H∼H1c2 H1/\e}a⊔io\slash= 0 and |ΦPn(H)|×/parenleftBig/summationtext H1⊆Kn:H∼H1c2 H1/parenrightBig1/2 n|V(H)|/2=ω/parenleftBigg/parenleftBig/summationdisplay H1⊆Kn:H∼H1c2 H1/parenrightBig1/2/parenrightBigg . the cla... | https://arxiv.org/abs/2504.17202v1 |
Proofs of Main Results 3.1 Diagonal SBMs In this section, we study the family of SBM(p,Q) models where all off-diagonal entries of Qare non- zero. This is equivalent to saying that whenever two vertice s have distinct labels, the probability they are adjacent is exactly 1 /2. Theorem 3.1 (Maximizing Partition Functions ... | https://arxiv.org/abs/2504.17202v1 |
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