Stephen Montgomery-Smith, Comparison of Sums of independent Identically Distributed Random Variables.
Prob. and Math. Stat. 14, (1993), 281-285.
Let \(S_k\) be the \(k\)-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of \(\|S_k\|\) with that of \(\|S_j\|\), and deduce some tail distribution maximal inequalities.
Theorem: There is universal constant \(c\) such that for \(j \lt k\) we have \(\Pr(\|S_j\| > t) \le c \Pr (\|S_k\| > t/c)\).
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