The exponential rate of convergence of the distribution of the maximum of a random walk

Abstract

Let G n ( x ) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G ( x ) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of G n ( x ) — G ( x ) is asymptotically equal to c.H(x)n −3/2 γ n as n → ∞ where c and γ are known constants and H ( x ) is a function solely depending on x .