Related papers: Large deviation inequalities for sums of indicator…

Let $X_1,X_2,...$ be a sequence of independent and identically distributed random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the positive sequence $\tau_n$ and the positive increasing sequence $a_n$, we give necessary…

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables…

Let $(\xi_i)_{i=1,...,n}$ 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} |\xi_i|\big/…

The probability that the sum of independent, centered, identically distributed, heavy-tailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We…

This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Levy property. We then give a connection…

In this paper we present a tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that are not necessarily bounded. The class considered includes geometrically and subgeometrically strongly mixing…

We present a formalization of the well-known thesis that, in the case of independent identically distributed random variables $X_1,\dots,X_n$ with power-like tails of index $\alpha\in(0,2)$, large deviations of the sum $X_1+\dots+X_n$ are…

The Chernoff bound is a well-known tool for obtaining a high probability bound on the expectation of a Bernoulli random variable in terms of its sample average. This bound is commonly used in statistical learning theory to upper bound the…

We provide bounds on the tail probabilities for simple procedures that generate random samples _without replacement_, when the probabilities of being selected need not be equal.

Let X_1,X_2,... be a sequence of independent and identically distributed random variables, and put S_n=X_1+...+X_n. Under some conditions on the positive sequence tau_n and the positive increasing sequence a_n, we give necessary and…

Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of…

This paper develops an optimal Chernoff type bound for the probabilities of large deviations of sums $\sum_{k=1}^n f (X_k)$ where $f$ is a real-valued function and $(X_k)_{k \in \mathbb{Z}_{\ge 0}}$ is a finite state Markov chain with an…

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are…

The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail…

The asymptotic efficiency of a generalized likelihood ratio test proposed by Cox is studied under the large deviations framework for error probabilities developed by Chernoff. In particular, two separate parametric families of hypotheses…

We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy…

We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms…

Large deviations for sums of i.i.d.\ random variables with stretched-exponential tails (also called Weibull or semi-exponential tails) have been well understood since the 60's, going back to Nagaev's seminal work. Many extensions in the…

Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.

The sums and maxima of non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraint is that there exists a unique series in a…