2017/03/22 by Deli Li, Li, Deli, Han‐Ying Liang +3
Decision Sciences · #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.1703.07868
Let (B, ‖⋅‖) be a real separable Banach space. Let φ(⋅) and ψ(⋅) be two continuous and increasing functions defined on [0, ∞) such that φ(0) = ψ(0) = 0, limt → ∞ φ(t) = ∞, and (ψ(⋅))/(φ(⋅)) is a nondecreasing function on [0, ∞). Let \Vn;~n ≥ 1 \ be a sequence of independent and symmetric \bf B-valued random variables. In this note, we establish a probability inequality for sums of independent \bf B-valued random variables by showing that for every n ≥ 1 and all t ≥ 0, ℙ(‖∑i=1n Vi ‖ gt; t bn ) ≤ 4 ℙ (‖∑i=1n φ(ψ-1(‖Vi‖)) \fracVi‖Vi‖ ‖ gt; t an ) + ∑i=1nℙ(‖Vi‖ gt; bn ), where an = φ(n) and bn = ψ(n), n ≥ 1. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed \bf B-valued random variables.