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A comparison inequality for sums of independent random variables
TLDR
In this paper, the authors 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.Abstract:
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}. Put Z_1 = X_I, and let Z_2,...,Z_n be independent identically distributed copies of Z_1. Then, P(||X_1+...+X_n|| > t) t/c), for all t>0, where c is an absolute constant.read more
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Ranking of Z-Numbers and Its Application in Decision Making
TL;DR: This paper suggests a human-like fundamental approach for ranking of Z-numbers which is based on two main ideas to compute optimality degrees of Z -numbers and to adjust the obtained degrees by using a human being’s opinion formalized by a degree of pessimism.
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Convergence rates for the law of large numbers for arrays
Víctor Hernández,Henar Urmeneta +1 more
TL;DR: In this paper, Baum and Katz type moment conditions for the convergence rate of large numbers for rowwise arrays are presented, where each row regularly covers a random variable in the array.
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Probability inequalities and tail estimates for metric semigroups
TL;DR: In this article, the authors study probability inequalities leading to tail estimates in a general semigroup with a translation-invariant metric and obtain approximate bounds for sums of independent semigroup-valued random variables.
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The Khinchin-Kahane inequality and Banach space embeddings for metric groups
Apoorva Khare,Bala Rajaratnam +1 more
TL;DR: In this paper, the authors extend the Khinchin-Kahane inequality to an arbitrary abelian metric group and prove a refinement which is sharp and which extends the sharp version for Banach spaces.
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Local characteristics and tangency of vector-valued martingales
TL;DR: In this paper, the definition of tangency through local characteristics, basic $Lp$- and φ$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent Martingales and local characteristics in infinite dimensions are presented.
References
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Book
A course in probability theory
TL;DR: This edition of A Course in Probability Theory includes an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.
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On vector measures
Joe Diestel,B. Faires +1 more
TL;DR: In this paper, the Radon-Nikodym theorem is generalized to the case of strongly bounded vector measures, which is a generalization of a result due to E. Leonard and K. Sundaresan.
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Complete Convergence and the Law of Large Numbers
P. L. Hsu,Herbert Robbins +1 more
TL;DR: The set of all ω to such that the relation within the braces holds, is the distribution function of X, and the random variables of a sequence X1, X2 ... are independent if, for every sequence x1, x2, … of real numbers, the sets are independent.