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Let $\\varphi(\\cdot)$ and $\\psi(\\cdot)$ be two continuous and increasing functions defined on $[0, \\infty)$ such that $\\varphi(0) = \\psi(0) = 0$, $\\lim_{t \\rightarrow \\infty} \\varphi(t) = \\infty$, and $\\frac{\\psi(\\cdot)}{\\varphi(\\cdot)}$ is a nondecreasing function on $[0, \\infty)$. Let $\\{V_{n};~n \\geq 1 \\}$ be a sequence of independent and symmetric {\\bf B}-valued random variables. 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