Properties of Optimal Extended-Valued Stopping Rules for Sₙ/n¹

Let X₁, X₂, ⋯ be i.i.d. nondegenerate mean zero random variables with common distribution function F such that EX₁⁺ log⁺ X₁ < ∞. Let Sₙ = X₁ + ⋯ + Xₙ and T_∞ be the collection of randomized extended-valued stopping rules. For t∈ T_∞, define E(Sₜ/t) = E(Sₜ/t)1_ < ∞\. There exists τ∈ T_∞ such that E(S_τ/τ) = t∈ T_∞ E(Sₜ/t) < ∞. We wish to determine whether or not P(τ = ∞) > 0. Let b⁻¹(x) = 1/(∫^∞ₓ (y/x)log (y/x) dF(y)). We find that P(Xₙ b(n) i.o.) = 1 ⇒ P(τ < ∞) = 1. As a corollary, if E(X₁⁺)^α < ∞ for some α > 1, P(τ < ∞) = 1. Conversely, if P(|Xₙ| b(n) i.o.) = 0, then P(τ = ∞) > 0. Moreover, if g(n) and ∑^∞ₙ₌₁ g(2ⁿ)⁻¹ < ∞ then P(|Xₙ| b(n)g(n)i.o.) = 1 ⇒ P(τ < ∞) = 1. Examples satisfying these latter conditions are given. An outgrowth of this work is that for any i.i.d. sequence ₙ\ of mean zero random variables and c < 1/4, P(Sₙ cE|Sₙ|i.o.) = 1. The importance of this result stems from the fact that we may also have Sₙ(log n)^ε/n → -∞ in probability (see [15]). In order to be completely rigorous, a section was included which provides a useful characterization of the general form of the class of optimal rules for sequential decision problems.