Monotonicity of an Integral of M. Klass

Abstract

For each value of $\beta, 0 < \beta < 2$, the integral $$\int^\infty_{-\infty} \{1 - \exp(-x^{-2}\sin^2tx)\}|t|^{-1-\beta}dt$$ decreases monotonically as a function of $x, x > 0$. This result is useful in approximating the absolute $\beta$th moment of the sum of zero mean i.i.d. random variables.