A note on sums of independent random variables (original) (raw)

A Comparison Inequality for Sums of Independent Random Variables

Journal of Mathematical Analysis and Applications, 2001

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 X 1 , . . . , X n be independent Banach-valued random variables. Let I be a random variable independent of X 1 , . . . , X n and uniformly distributed over {1, . . . , n}. PutX 1 = X I , and letX 2 , . . . ,X n be independent identically distributed copies ofX 1 . Then, P ( X 1 + · · · + X n ≥ λ) ≤ cP ( X 1 + · · · +X n ≥ λ/c) for all λ ≥ 0, where c is an absolute constant. , Comparisons between tail probabilities of sums of independent symmetric random variables, Ann. Inst. Poincaré Probab. Statist. 33 (1997), 651-671.

Concentration inequalities for sums of random variables, each having power bounded tails

2019

In this work we present concentration inequalities for the sum SnS_nSn of independent integer-valued not necessary indentically distributed random variables, where each variable has tail function that can be bounded by some power function with exponent −alpha-\alpha−alpha. We show that when 010 101, then the sum SnS_nSn is concentrated around its mean. Since the r.vs. that constitute the sum has tails, which can be bounded by some power function, it follows that results of this paper are applicable to a wide range of different distributions, including the exponentially decaying ones.

Upper and lower bounds for sums of random variables

Insurance Mathematics & Economics, 2000

In this contribution, the upper bounds for sums of dependent random variables Xl + X 2 + ... + Xn derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the Xi, given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of lognormal random variables is given.

Inequalities for Sums of Independent Random Variables

Proceedings of the American Mathematical Society, 1988

A moment inequality is proved for sums of independent random variables in the Lorentz spaces LPtq, thus extending an inequality of Rosenthal. The latter result is used in combination with a square function inequality to give a proof of a Banach space isomorphism theorem. Further moment inequalities are also proved.

Sums of dependent nonnegative random variables with subexponential tails

Journal of Applied Probability, 2008

In this paper we study the asymptotic tail probabilities of sums of subexponential nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.

Moment Inequalities for Sums of Certain Dependent Random Variables

Theory of Probability & Its Applications, 2003

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