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Cesaro -> Cesàro
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LSpice
LSpice
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Extracting a subsequence CesaroCesàro converging to the limsup of the CesaroCesàro sums

Let $X_n$ be a sequence of uniformly bounded random variables -— that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.

Write $\bar X_N := \frac{1}{N} \sum_{n = 1}^N X_n$, and $Y := \limsup_{N \to \infty} \bar X_N$.

Question: Does there exist some subsequence $X_{n_k}$ such that $\frac{1}{N} \sum_{k=1}^{N} X_{n_k}$ converges to $Y$ almost surely as $N \to \infty$?

Extracting a subsequence Cesaro converging to the limsup of the Cesaro sums

Let $X_n$ be a sequence of uniformly bounded random variables - that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.

Write $\bar X_N := \frac{1}{N} \sum_{n = 1}^N X_n$, and $Y := \limsup_{N \to \infty} \bar X_N$.

Question: Does there exist some subsequence $X_{n_k}$ such that $\frac{1}{N} \sum_{k=1}^{N} X_{n_k}$ converges to $Y$ almost surely as $N \to \infty$?

Extracting a subsequence Cesàro converging to the limsup of the Cesàro sums

Let $X_n$ be a sequence of uniformly bounded random variables — that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.

Write $\bar X_N := \frac{1}{N} \sum_{n = 1}^N X_n$, and $Y := \limsup_{N \to \infty} \bar X_N$.

Question: Does there exist some subsequence $X_{n_k}$ such that $\frac{1}{N} \sum_{k=1}^{N} X_{n_k}$ converges to $Y$ almost surely as $N \to \infty$?

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Nate River
Nate River
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Extracting a subsequence Cesaro converging to the limsup of the Cesaro sums

Let $X_n$ be a sequence of uniformly bounded random variables - that is, there exists some $K > 0$ such that $|X_n| \leq K$ almost surely for all $n \in \mathbb N$.

Write $\bar X_N := \frac{1}{N} \sum_{n = 1}^N X_n$, and $Y := \limsup_{N \to \infty} \bar X_N$.

Question: Does there exist some subsequence $X_{n_k}$ such that $\frac{1}{N} \sum_{k=1}^{N} X_{n_k}$ converges to $Y$ almost surely as $N \to \infty$?