Limit Laws of Erdos-Renyi-Shepp Type

Let Sₙ = X₁ + ⋯ + Xₙ be the nth partial sum of an i.i.d. sequence of random variables. We describe the limiting behavior of {equation*}{split}T_n = max1≤ i≤ n(Si+κ(i) - S_i), \\ U_n = max0≤ i≤ n-k(Sᵢ₊ₖ - S_i), \\ W_n = max0≤ i≤ n-k max1≤ j≤ k(Sᵢ₊ⱼ - S_i) \\ {split}{equation*} and Vₙ = max0≤ i≤ n-k min1≤ j≤ k(k/j)(Sᵢ₊ⱼ - Sᵢ), for k = κ(n) = c log n, and where $c > 0$ is a given constant. We assume that the random variables Xᵢ are centered and have a finite moment generating function in a right neighborhood of zero, and obtain among other results the full form of the Erdos-Renyi (1970) and Shepp (1964) theorems. Our conditions extend those of Deheuvels, Devroye and Lynch (1986) to cover a larger class of distributions.