Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions

Abstract

Let $S_n$ denote the $n$th partial sum of i.i.d. nonconstant mean zero random variables. Given an approximation $K(n)$ of $E|S_n|$, tight bounds are obtained for the ratio $E|S_n|/K(n)$. These bounds are best possible as $n$ tends to infinity. Implications of this result relate to the law of the iterated logarithm for mean zero variables, Chebyshev's inequality and Markov's inequality. Asymptotically exact lower-bounds are obtained for expectations of functions of row-sums of triangular arrays of independent but not necessarily identically distributed random variables. Expectations of "Poissonized random sums" are also treated.