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 < ∞}.

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

From a random sample X1, X2, …, XN there can be constructed N – K + 1 successive sample means of the form for 0 ≦ n ≦ N − K, where Erdös and Rényi (1970) studied the maximum Σ(N, K) of these N – K + 1 sample means. Under appropriate conditions, they showed that for a wide interval of λ's there exist constants C(λ), depending only on λ and the distribution from which the sample was selected, such that Σ(N, [C(λ) log N])→ λ a.s. as N→∞. In the present article, analogous results are developed for the maximum of the N – K + 1 successive sample medians and, more generally, for all sample quantiles.