A comparison inequality for sums of independent random variables (Q5927712)

Consider \(n\) independent Banach-space-valued random variables \(X_i\), not identically distributed. Define a mixture variable \(Y\) whose distribution is given by the average distribution of the \(X_i\). Let \(Y_i\) be independent random variables with common distribution equal to that of \(Y\). The authors extend some work of \textit{S. J. Montgomery-Smith} [Probab. Math. Stat. 14, No. 2, 281-285 (1993; Zbl 0827.60005)] from the symmetric case to establish NEWLINE\[NEWLINE P[\|X_1+\cdots+X_n\|\geq \lambda] \leq c P[\|Y_1+\cdots+Y_n\|\geq \lambda/c], NEWLINE\]NEWLINE where \(c\) is an absolute constant and \(\lambda\geq 0\).