Two counterexamples, addressing questions raised in cite{AD} and cite{PZ}, are provided. Both counterexamples are related to chaoses. Let $F_n=Y_n+Z_n$, where the random variables $Y_n$ and $Z_n$ belong to different chaoses of uniformly bounded degree. It may be that $F_noverset{a.s.}longrightarrow 0$, $F_noverset{L_{2+delta}}longrightarrow 0$ and $Eigl{sup_n,abs{F_n}^deltaigr}0$, and yet $Y_n$ fails to converge to 0 a.s.
Pratelli, L., Rigo, P. (2021). On the almost sure convergence of sums. STATISTICS & PROBABILITY LETTERS, 172, 1-5 [10.1016/j.spl.2021.109045].
On the almost sure convergence of sums
Rigo Pietro
2021
Abstract
Two counterexamples, addressing questions raised in cite{AD} and cite{PZ}, are provided. Both counterexamples are related to chaoses. Let $F_n=Y_n+Z_n$, where the random variables $Y_n$ and $Z_n$ belong to different chaoses of uniformly bounded degree. It may be that $F_noverset{a.s.}longrightarrow 0$, $F_noverset{L_{2+delta}}longrightarrow 0$ and $Eigl{sup_n,abs{F_n}^deltaigr}0$, and yet $Y_n$ fails to converge to 0 a.s.File in questo prodotto:
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