Normal comparison lemma and Slepian's inequality are essential tools in the study of Gaussian processes. In this paper we extend normal comparison lemma and derive various related comparison inequalities including Slepian's inequality for order statistics of two Gaussian arrays.The derived results can be applied in numerous problems related to the study of the supremum of order statistics of Gaussian processes. In order to illustrate the range of possible applications, we analyze the lower tail behaviour of order statistics of self-similar Gaussian processes and derive mixed Gumbel limit theorems.
We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest.
Study of time series data often involves measuring the strength of temporal dependence, on which statistical properties like consistency and central limit theorem are built. Historically, various dependence measures have been proposed. In this note, we first survey some of the most well-used dependence measures as well as various probability and moment inequalities built upon them under a high-dimensional triangular array time series setting. We then argue that this triangular array setting will pose substantially new challenges to the verification of some dependence conditions. In particular,…
The Hoeffding-type-inequalities are obtained for the distribution tails of canonical (degenerate) U- and V-statistics of an arbitrary order based on samples from a stationary sequence of observations satisfying $ρ$-mixing.
Hoeffding-type exponential inequalities are obtained for the distribution tails of canonical von Mises' statistics of arbitrary order based on samples from a stationary sequence of random variables satisfying the φ-mixing condition. The present paper weakens the restrictions on the coefficient φ() which are contained in the paper "Exponential inequalities for the distributions of canonical U- and V-statistics of dependent observations" by Borisov and Volodko (2009). At the same time, we correct the corresponding proof in this paper.
Let $(ξ_i)_{i=1,...,n}$ be a sequence of independent and symmetric random variables. We consider the upper bounds on tail probabilities of self-normalized deviations $$ \mathbf{P} \Big( \max_{1\leq k \leq n} \sum_{i=1}^{k} |ξ_i|\big/ \big(\sum_{i=1}^{n} |ξ_i|^β\big)^{1/β} \geq x \Big) $$ for $x>0$ and $β>1.$ Our bound is the best that can be obtained from the Bernstein inequality under the present assumption. An application to Student's $t$-statistics is also given.
We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.
This note is concerned with concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds. Some statistical illustrations complete the exposition.
Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts…
We establish exponential inequalities for a class of V-statistics under strong mixing conditions. Our theory is developed via a novel kernel expansion based on random Fourier features and the use of a probabilistic method. This type of expansion is new and useful for handling many notorious classes of kernels.
There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña \cite{d} establishes a nice exponential inequality for discrete time locally square integrable martingale . In this paper, we obtain de la Peña's inequalities for stochastic integral of multivariate point processes. The proof is primarily based on Doléans-Dade exponential formula and the optional stopping theorem. As application, we obtain an exponential inequality for block counting process in $Λ-$coalescents.
We consider the three dimensional array $\mathcal{A} = \{a_{i,j,k}\}_{1\le i,j,k \le n}$, with $a_{i,j,k} \in [0,1]$, and the two random statistics $T_{1}:= \sum_{i=1}^n \sum_{j=1}^n a_{i,j,σ(i)}$ and $T_{2}:= \sum_{i=1}^{n} a_{i,σ(i),π(i)}$, where $σ$ and $π$ are chosen independently from the set of permutations of $\{1,2,\ldots,n \}.$ These can be viewed as natural three dimensional generalizations of the statistic $T_{3}=\sum_{i=1}^{n} a_{i,σ(i)}$, considered by Hoeffding \cite{Hoe51}. Here we give Bernstein type concentration inequalities for $T_{1}$ and $T_{2}$ by extending the argument f…
We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. A possible application -- a CLT for the hitting time for random walk on random graphs -- will be presented in \cite{LoTe20b}
We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentr…
In this paper, we obtain the upper and lower bounds for two inequalities related to the range statistics. The first one is concerning the one-variable case and the second one is about the bivariate case.
We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetr…
