Related papers: A sharpening of Tusn\'ady's inequality

We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\,…

Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $\sigma_{X_i}^2$, $i=1,\ldots,M$, and $\sigma_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$.…

We prove a non-asymptotic generalization of the refined continuity correction developed in Cressie (1978) for the Binomial distribution, which we then use to improve the versions of Tusn\'ady's inequality from Massart (2002) and Carter &…

In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known. This result holds for one-dimensional or multivariate continuous probability distributions with finite…

Let X_1,...., X_n be a collection of iid discrete random variables, and Y_1,..., Y_m a set of noisy observations of such variables. Assume each observation Y_a to be a random function of some a random subset of the X_i's, and consider the…

Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ \psi_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad…

An inequality for the variance of an additive function defined on random decomposable structures, called assemblies, is established. The result generalizes estimates obtained earlier in the cases of permutations and mappings of a finite set…

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 well-known Koml\'os-Major-Tusn\'ady inequalities [Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] provide sharp inequalities to partial sums of iid standard exponential random variables by a…

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple…

We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} : i\geq 1} $ are i.i.d. random variables, independent of ${Y_{i} : i…

We establish the rate of convergence of distributions of sums of independent identically distributed random variables to the Gaussian distribution in terms of truncated pseudomoments by implementing the idea of Yu. Studnyev for getting…

Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general…

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 $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X/Y$ is derived. Some basic distributional properties are also derived, including…

In this work, a generalization of Chebyshev functional is presented. New inequalities of Gruss type via Pompeiu's mean value theorem are established. Improvements of some old inequalities are proved. A generalization of pre-Gruss inequality…

Let $\{X_i,i\geq1\}$ be a sequence of negatively associated random variables, and let $\{X_i^\ast,i\geq 1\}$ be a sequence of independent random variables such that $X_i^\ast$ and $X_i$ have the same distribution for each $i$. Denote by…

Gau\ss (1823) proved a sharp upper bound on the probability that a random variable falls outside a symmetric interval around zero when its distribution is unimodal with mode at zero. For the class of all distributions with mean at zero,…

For a random variable with a unimodal distribution and finite second moment Gau\ss \, (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it is…

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the…