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

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