We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

Let $X_i$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X={\sum }_{i\ne j}{a}_{i,j}{X}_i{X}_j$, where $a_{i,j}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.

For random variables $S={\sum }_{i=1}^{\infty }{\alpha }_i{\xi }_i$, where $\left({\xi }_i\right)$ is a sequence of symmetric, independent, identically distributed random variables such that $lnP\left(\right|{\xi }_i\left|\ge t\right)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

The paper presents some connections between two tail orderings of distributions and the total time on test transform. The procedure for testing the pure-tail ordering is proposed.

Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality...

We establish the Doob inequality for martingale difference arrays and provide a sufficient condition so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Some corollaries are derived from the main results, they are more general than some well-known ones.

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.'s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

In this paper, we introduce a new stochastic order between continuous non-negative random variables called the PLR (proportional likelihood ratio) order, which is closely related to the usual likelihood ratio order. The PLR order can be used to characterize random variables whose logarithms have log-concave (log-convex) densities. Many income random variables satisfy this property and they are said to have the IPLR (increasing proportional likelihood ratio) property (DPLR property). As an application,...