Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications

Abstract

Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng [12] [14], we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov's strong law of larger numbers holds for independent and identically distributed random variables under a continuous sub-linear expectation if and only if the corresponding Choquet integral is finite. Keywordssub-linear expectation, capacity, Kolmogorov's inequality, Rosenthal's inequality, negative dependence, strong laws of large numbers MSC(2010) 60F15, 60F05

Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications

“…For 0 < p < 1, if C V (|X 1 | p ) < ∞, then S n /n 1/p → 0 a.s. V. And for 1 < p < 2, supposeÊ(X n ) =Ê(X n ) = µ and lim a→∞ E ((|X 1 | − a) + ) = 0, if C V (|X 1 | p ) < ∞, then (S n − nµ)/n 1/p → 0 a.s. V. Remark 3.1. If V is continuous and for some µ, (S n − nµ)/n 1/p → 0 a.s. V, then (X n − µ)/n 1/p → 0 a.s. V. With the same arguments in the proof of Theorem 3.3 (b) of Zhang [12], we can derive that…”

“…(i) For 0 < p < 1, we have lim a→∞ E (|Y n | ∧ a) = E (|Y n |) thanks to the fact that Y n is less than or equal to n 1/p . Recall the Lemma 3.9(b) in Zhang [12], we have…”

“…The main purpose of this paper is to establish three series theorem for independent random variables in the general sub-linear expectation spaces and use it to study the strong limit theorems. The Rosenthal's inequalities for the maximum partial sums of independent random variables under the sub-linear expectations established by Zhang [12] are our main tools to prove our results. Because the Rosenthal's inequalities are established for random variables with zero sub-linear expectations or sub-linear expectations near to zero and it is impossible to centralize a random variable such that both its upper expectation and lower expectation are zeros, the proof of the three series theorem is much technical.…”

“…Since |X| ∧ n 1/p 2 ≤ n 2/p , we have lim a→∞ E |X| ∧ n 1/p 2 ∧ a = E |X| ∧ n 1/p 2 . According to the Lemma 3.9(b) in Zhang [12], we have…”

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

“…Zhang [13]), the strong law of large numbers (cf. Chen [1], Chen et al [3], Hu [6], Zhang [15], Zhang and Lin [17]), and the law of the iterated logarithm (cf. Chen [2], Zhang [14]).…”