Martingale Invariance Principles

Let \(Sₙⱼ, Fₙⱼ), 1 j kₙ\ be a square-integrable martingale for each n = 1,2,3,⋯. Define Xₙⱼ = Sₙⱼ - Sn,j-1 (Sₙ₀ = 0), U²ₙⱼ = ∑ʲᵢ₌₁ X²ₙᵢ, U²ₙ = U²nkₙ, and for each z ∈ 0, 1 let ξₙ(z) = U⁻¹ₙ ∑kₙⱼ₌₁ Xₙⱼ I(U⁻²ₙ U²ₙⱼ z) and ηₙ(z) = ∑kₙⱼ₌₁ Xₙⱼ I(U⁻²ₙ U²ₙⱼ z); ξₙ and ηₙ are random elements of D 0, 1. Sufficient conditions are given for ξₙ to converge in distribution to Brownian motion and for ηₙ to converge to a mixture of Brownian motion distributions. We give several applications and examples.