A Comparison Inequality for Sums of Independent Random Variables∗
نویسندگان
چکیده
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 X1, . . . , Xn be independent Banach-valued random variables. Let I be a random variable independent of X1, . . . , Xn and uniformly distributed over {1, . . . , n}. Put X̃1 = XI , and let X̃2, . . . , X̃n be independent identically distributed copies of X̃1. Then, P (‖X1 + · · · + Xn‖ ≥ λ) ≤ cP (‖X̃1 + · · · + X̃n‖ ≥ λ/c) for all λ ≥ 0, where c is an absolute constant. The independent Banach-valued random variables X1, . . . , Xn are said to regularly cover (the distribution of) a random variable Y provided that
منابع مشابه
Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables
Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...
متن کاملSOME PROBABILISTIC INEQUALITIES FOR FUZZY RANDOM VARIABLES
In this paper, the concepts of positive dependence and linearlypositive quadrant dependence are introduced for fuzzy random variables. Also,an inequality is obtained for partial sums of linearly positive quadrant depen-dent fuzzy random variables. Moreover, a weak law of large numbers is estab-lished for linearly positive quadrant dependent fuzzy random variables. Weextend some well known inequ...
متن کاملComparison of Sums of Independent Identically Distributed Random Variables
Let Sk be the k-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of ‖Sk‖ with that of ‖Sj‖, and deduce some tail distribution maximal inequalities. The main result of this paper was inspired by the inequality from [dP–M] that says that Pr(‖X1‖ > t) ≤ 5 Pr(‖X1 +X2‖ > t/2) whenever X1 and X2 are independent...
متن کاملXXXX A Note on Sums of Independent Random
In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Lataa la on bounds on moments of such sums. We also give a new proof of Lataa la's result for nonnegative random variables, and improve one of the constants in his inequality.
متن کاملComplete Convergence forWeighted Sums of Negatively Superadditive Dependent Random Variables
Abstract. Let {Xn,n≥1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {ank,1≤ k≤n,n≥1} be an array of real numbers. Under some suitable conditions, we present some results on complete convergence for weighted sums ∑k=1ankXk of NSD random variables by using the Rosenthal type inequality. The results obtained in the paper generalize some corresponding one...
متن کامل