Functional Laws of Large Numbers in Hölder Spaces

Let Sn = X1 + · · · +Xn, n ≥ 1, where (Xi)i≥1 are random variables. Let μ be a constant and I be the identity function on [0, 1]. We study the almost sure convergence to μI of the two polygonal line partial sums processes ζn and ζ ad n with respective vertices (k/n, Sk) and (τk, Sk), 0 ≤ k ≤ n, where τk = Tk/Tn and Tk = |X1|+ · · ·+ |Xk|. These convergences are considered in the space C[0, 1] or in the Hölder spaces H α[0, 1], 0 ≤ α < 1. In C[0, 1], any strong law of large numbers satisfied by Sn is inherited by ζn. In H α[0, 1], assuming moreover that the Xi’s are i.i.d., nζn converges almost surely to μI if and only if E |X1| < ∞ and μ = EX1. In contrast, the same convergence for ζ ad n is equivalent to E |X1| < ∞ and μ = EX1.