Related papers: An Exponential Inequality for Symmetric Random Var…

Let $X_1, X_2,\ldots, X_n$ be $n$ independent and identically distributed random variables, here $n \geq 2.$ Let $X_{(1)}, X_{(2)}, \ldots, X_{(n)}$ be the order statistics of $X_1, X_2,..., X_n.$ In this note we proved that: (I) If $X_1,…

We present a new exponential inequality as a generalization of that of Sung \textit{et al.} \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^{x}…

The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots,…

For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of random variables satisfying $\mathbb{E} \lvert X_{n} \rvert < \infty$ for all $n \geqslant 1$, a maximal inequality is established, and used to obtain strong law of large numbers for…

Let X1, ..., Xn be arbitrary non-negative independent random variables with respective expected values $\mu_{i}$ at most one. We sketch but do not prove an equivalent conjecture to Feige's Conjecture $\mathbb{P} \left( \sum_{i=1}^{n} X_{i}…

We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand.

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence…

For a bivariate random vector (X,Y), symmetry conditions are presented that yield stochastic orderings among |X|, |Y|, |max(X,Y)|, and | min(X, Y)|. Partial extensions of these results for multivariate random vectors (X1,...,Xn) are also…

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not…

Let $\BS_1,...,\BS_n$ be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter $p\in(0,1)$. Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0<p\le 1/2$ and $m_*(p):=1$ if…

The sum of independent, but not necessary identically distributed, exponential random variables follows hypoexponential distribution. We focus on a particular case when all, but one rate parameters of the exponential variables are…

Given a sequence $(X_n)$ of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series $\sum_{n=1}^\infty X_n$ is almost surely convergent. For…

Let X_1,X_2,... be a sequence of [0,1]-valued i.i.d. random variables, let c\geq 0 be a sampling cost for each observation and let Y_i=X_i-ic, i=1,2,.... For n=1,2,..., let M(Y_1,...,Y_n)=E(max_{1\leq i\leq n}Y_i) and…

We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let…

Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

Let $X_1,X_2, \ldots $ be a sequence of $i.i.d$ real (complex) $d \times d $ invertible random matrices with common distribution $\mu$ and $\sigma_1(n), \sigma_2(n), \ldots , \sigma_d(n)$ be the singular values, $\lambda_1(n), \lambda_2(n),…