Boundary Crossing Problems for Sample Means

Motivated by several classical sequential decision problems, we study herein the following type of boundary crossing problems for certain nonlinear functions of sample means. Let X₁, X₂,… be i.i.d. random vectors whose common density belongs to the k-dimensional exponential family h_θ(x) = exp\θ'x - ψ(θ)\ with respect to some nondegenerate measure ν. Let X̄ₙ = (X₁ + ⋯ + Xₙ)/n, θₙ = (∇ψ)⁻¹(X̄ₙ), and let I(θ, λ) = E_θlog_θ(X₁)/h_λ(X₁)\ ( = Kullback-Leibler information number). Consider stopping times of the form Tc(λ) = inf: I(θₙ, λ) ≥ n⁻¹g(cn)\, c > 0, where g is a positive function such that g(t) ~ α log t⁻¹ as t → 0. We obtain asymptotic approximations to the moments E_θ Tʳc(λ) as c → 0 that are uniform in θ and λ with |λ - θ|²/c → ∞. We also study the probability that X̄Tc(λ) lies in certain cones with vertex ∇ψ (λ). In particular, in the one-dimensional case with λ > θ, we consider boundary crossing probabilities of the form P_θ\θₙ ≥ λ and I(θₙ, λ) ≥ n⁻¹ g(cn) for some n\. Asymptotic approximations (as c → 0) to these boundary crossing probabilities are obtained that are uniform in θ and λ with |λ - θ|²/c → ∞.