Consider a sequence U1, U2, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un+1, · ··, Un+K, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X1, X2, · ·· with d.f. F satisfying a strict monotonicity condition.

Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

Consider a sequence of observations Yk = Xk + ek, where {Xk : 1 ≦ k < ∞} are i.i.d. random variables having distribution function F and {ek : 1 ≦ k < ∞} are arbitrary random errors of observation. The stochastic geyser problem asks for conditions under which F can be uniquely determined from a knowledge of the sequence of Yk's. The objective of the present article is to show that, if F is continuous and strictly increasing and the sample quantiles of successive blocks {Xn+ 1Xn+ 2, …, Xn+ K} of particular lengths K can be a.s. estimated to within an error of size o(1) as K →∞, then we can almost surely determine F from a single realization of {Yk : 1 ≦ k < ∞}.