Uniform Lower Bounds for Randomly Stopped Banach Space Valued Random Sums

Abstract

Let $\{X_n\}$ be a sequence of independent Banach space valued random elements with partial sums $S_n = X_1 + \cdots + X_n$. Then let $T$ be any possibly randomized stopping time based on $\{X_n\}$. Fix any $\alpha > 0$ and let $\Phi(\cdot)$ be any nondecreasing continuous function on $\lbrack 0, \infty)$ with $\Phi(0) = 0$ such that $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x \geq 0, c \geq 2$. Put $S^\ast_n = \max_{1 \leq k \leq n}\|S_k\|$ and $a^\ast_n = E\Phi(S^\ast_n)$. It is proved that there exists a universal constant $c^\ast_\alpha < \infty$ depending only on $\alpha$ [and otherwise independent of $(B, \|\cdot\|), \{X_n\}, T \text{and} \Phi(\cdot)$] such that $Ea^\ast_T \leq c^\ast_\alpha E\Phi(S^\ast_T)$. As a consequence, $E\Phi(S^\ast_{T_c}) = \infty$ whenever $P(T_c < \infty) = 1$ and $c \geq c^\ast_\alpha$, where \begin{equation*}T_c = \begin{cases} \text{first} n: c^\Phi(S^\ast_n) < a^\ast_n,\\ \infty, \text{if no such} n \text{exists}.\end{cases}\end{equation*} In fact, $Ea^\ast_{T_c} = \infty$, too. An upper bound for $c^\ast_\alpha$ is constructed.