Convergence in law of partial sum processes in p-variation norm

Abstract

Let X1, X2, … be a sequence of independent identically distributed real-valued random variables, Sn be the nth partial sum process Sn(t) ≔ X1 + ⋯ X⌊tn⌋, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n−1/2Sn converges in law to σW as n → ∞ in p-variation norm if and only if EX1 = 0 and σ2 = EX12 < ∞. The result is applied to test the stability of a regression model.