Limit Theorems

Abstract

Suppose X ₁,X ₂,… are independent, identically distributed ℝ^d-valued random variables and consider, as usual, the random walk k ↦ S _k = X ₁ + ⋯ + X _k . If 피[X ₁] = 0 and 피[X ²₁ ] < + ∞, the classical central limit theory on ℝ^d implies that as n ↦ ∞, the random vector n ^-1/2 S _n converges in distribution to an ℝ^d-valued Gaussian random vector. It turns out that much more is true; namely, as n → ∞, the distribution of the process t ↦ n ^-1/2 S _⌊nt⌋ starts to approximate that of a suitable Gaussian process. This approximation is good enough to show that various functional of the random walk path converge in distribution to those of the limiting Gaussian process.