Inequalities for variances of some functions of random variables
Some inequalities for variances and other measures of deviation of functions of random variables are considered. The following inequality is typical. Let \(f\) be a real-valued function, defined on \((0, \infty)\). If there exists a positive function \(g_f\) on \([0, \infty)\) with the property that \[ \bigl |f(x + h) - f(x) \bigr |\leq |h |/g_f (x) \] for all \(x > 0\) and \(h > - x\), then \[ D^2 \bigl( f(Y) \bigr) \leq D^2Y/ \bigl( g_f (EY) \bigr)^2 \] holds for any nonnegative random variables \(Y\) with \(EY > 0\) and \(EY^2 < \infty\). Based on the inequalities from the paper the author derives some corollaries concerning the variance of powers of random variables and of sums of independent random variables. For example, the following inequality \[ D^2 (X_1 + \cdots + X_n)^\alpha \leq D^2 X_1/n^{1 - 2 \alpha} (EX_1)^{2 - 2 \alpha} \] holds for any independent identically distributed random variables \(X_1, \ldots, X_n\) and \(0 < \alpha \leq 1\).
scientific article; zbMATH DE number 769862