We establish preservation results for the stochastic comparison of multivariate random sums of stationary, not necessarily independent, sequences of nonnegative random variables. We consider convex-type orderings, i.e. convex, coordinatewise convex, upper orthant convex and directionally convex orderings. Our theorems generalize the well-known results for the stochastic ordering of random sums of independent random variables.
Recently the order preserving property of estimators has been intensively studied, e.g. by Gan and Balakrishnan and collaborators. In this paper we prove the stochastic monotonicity of moment estimators of gamma distribution parameters using the standard coupling method and majorization theory. We also give some properties of the moment estimator of the shape parameter and derive an approximate confidence interval for this parameter.
We study how iterated convolutions of probability measures compare under stochastic domination. We give necessary and sufficient conditions for the existence of an integer n such that μ*n is stochastically dominated by ν*n for two given probability measures μ and ν. As a consequence we obtain a similar theorem on the majorization order for vectors in Rd. In particular we prove results about catalysis in quantum information theory.
Let be a sequence of independent and identically distributed random variables with continuous distribution function F(x). Denote by X(1,k),X(2,k),... the kth record values corresponding to We obtain some stochastic comparison results involving the random kth record values X(N,k), where N is a positive integer-valued random variable which is independent of the .
Let X be the branching particle diffusion corresponding to the operator Lu+β(u2−u) on D⊆ℝd (where β≥0 and β≢0). Let λc denote the generalized principal eigenvalue for the operator L+β on D and assume that it is finite. When λc>0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology as t tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from...
Nello spazio delle variabili aleatorie subgaussiane definite su si studia l'equivalenza tra la norma subgaussiana e la norma di Fernique, dando valutazioni numeriche delle costanti di equivalenza. A tale scopo si fa uso di una nuova caratterizzazione della norma subgaussiana delle variabili aleatorie simmetriche.
We formulate and discuss a conjecture concerning lower bounds for norms of log-concave vectors, which generalizes the classical Sudakov minoration principle for Gaussian vectors. We show that the conjecture holds for some special classes of log-concave measures and some weaker forms of it are satisfied in the general case. We also present some applications based on chaining techniques.
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 be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable , where 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 , where is a sequence of symmetric, independent, identically distributed random variables such that 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.