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In this work, we study convergence in probability and almost sure convergence for weighted partial sums of random variables that are related to the class of generalized Oppenheim expansions. It is worth noting that the random variables…
Consider the matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices. Under the optimal third moment condition, we first establish a…
Understanding the distributions of spectral estimators in low-rank random matrix models, also known as signal-plus-noise matrix models, is fundamentally important in various statistical learning problems, including network analysis, matrix…
This paper is mainly concerned with asymptotic studies of weighted bootstrap for u- and v-statistics. We derive the consistency of the weighted bootstrap u- and v-statistics, based on i.i.d. and non i.i.d. observations, from some more…
A family of explicit modified Euler methods (MEMs) is constructed for long-time approximations of super-linear SODEs driven by multiplicative noise. The proposed schemes can preserve the same Lyapunov structure as the continuous problems.…
This study develops a non-asymptotic Gaussian approximation theory for distributions of M-estimators, which are defined as maximizers of empirical criterion functions. In existing mathematical statistics literature, numerous studies have…
We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder…
Diffusion models for continuous state spaces based on Gaussian noising processes are now relatively well understood from both practical and theoretical perspectives. In contrast, results for diffusion models on discrete state spaces remain…
Testing for the equality of two high-dimensional distributions is a challenging problem, and this becomes even more challenging when the sample size is small. Over the last few decades, several graph-based two-sample tests have been…
This paper derives non-asymptotic error bounds for nonlinear stochastic approximation algorithms in the Wasserstein-$p$ distance. To obtain explicit finite-sample guarantees for the last iterate, we develop a coupling argument that compares…
We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are…
Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this…
Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for…
This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs…
Consider the sum $Z = \sum_{n=1}^\infty \lambda_n (\eta_n - \mathbb{E}\eta_n)$, where $\eta_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $\lambda_n$'s are predetermined weights. We study the asymptotic behavior…
Several new methods have been proposed for performing valid inference after model selection. An older method is sampling splitting: use part of the data for model selection and part for inference. In this paper we revisit sample splitting…
The "typical" asymptotic behavior of the weighted sums of independent, identically distibuted random vectors in k-dimensional space is considered. It is shown that under finitnes of fifth absolute moment of an individual term the rate of…
We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard errors. Our results hold in the increasing dimension setting and…
This paper is concerned with finite sample approximations to the supremum of a non-degenerate $U$-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the…
In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,\dots,X_n$ over the class of rectangles in the case when the covariance…