I am currently trying to collect applications of Cesaro's lemma to various fields.
I suspect that there are (elementary) applications of this lemma to probability theory but couldn't find any.
Suggestions and references would be highly appreciated.
2026-02-23 19:39:12.1771875552
Applications of Cesaro's lemma to probability theory
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A consequence of Cesàro's lemma is Kronecker's, which can be used to prove laws of large numbers.
I can think of two related (negative) results:
A sequence of i.i.d. random variables cannot converge almost surely, unless it is a deterministic sequence.
If a sequence of random variables converges in probability, then it is not necessary true for its Cesàro mean.