Functional Laws of Large Numbers in Hölder Spaces

Alfredas Račkauskas (1) , Charles Suquet (2)

1 IMI - VU Institute of Mathematics and Informatics
2 LPP - Laboratoire Paul Painlevé - UMR 8524

Let $S_n=X_1+\dots+X_n$, $n\ge 1$, where $(X_i)_{i\ge 1}$ are random variables. Let $\mu$ be a constant and $I$ be the identity function on $[0,1]$. We study the almost sure convergence to $\mu I$ of the two polygonal line partial sums processes $\zeta_n$ and $\zeta_n^{\mathrm{ad}}$ with respective vertices $(k/n,S_k)$ and $(\tau_k,S_k)$, $0\le k\le n$, where $\tau_k=T_k/T_n$ and $T_k=\abs{X_1}+\dots+\abs{X_k}$. These convergences are considered in the space $C[0,1]$ or in the Hölder spaces $\mathcal{H}_\alpha^o[0,1]$, $0\le \alpha<1$. In $C[0,1]$, any strong law of large numbers satisfied by $S_n$ is inherited by $\zeta_n$. In $\mathcal{H}_\alpha^o[0,1]$, assuming moreover that the $X_i$'s are i.i.d., $n^{-1}\zeta_n$ converges almost surely to $\mu I$ if and only if $\E\abs{X_1}^{1/(1-\alpha)}<\infty$ and $\mu=\E X_1$. In contrast, the same convergence for $\zeta_n^{\mathrm{ad}}$ is equivalent to $\E\abs{X_1}<\infty$ and $\mu=\E X_1$.

Mots clés: adaptive partial sums process; Hölder space; Marcinkiewicz-Zygmund strong law of large numbers; partial sums process; uniform law of large numbers; weighted increments

Domaines: Probabilités [math.PR]

Alfredas Račkauskas, Charles Suquet. Functional Laws of Large Numbers in Hölder Spaces. 2012. ⟨hal-00836589⟩