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$S_n$ is a symmetric random walk. How do I show that the event $A_M=[\limsup\limits_{n\to\infty}S_n\leqslant M]$ is in the tail sigma algebra $\cap_{m>n}\sigma(X_m,X_{m+1},...)$? I would be really grateful for any help.

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For a sequence $(S_n)_{n \in \mathbb{N}}$ of random variables given by $$S_n := \sum_{j=1}^n X_j,$$ it does (in general) not hold that

$$A_M := \left\{ \limsup_{n \to \infty} S_n \leq M \right\}$$

is a tail event with respect to $(X_n)_{n \in \mathbb{N}}$ - instead, it is a tail event with respect to $(S_n)_{n \in \mathbb{N}}$. Since the $\sigma$-algebras $\sigma(S_n)$, $n \in \mathbb{N}$, are (in general) not independent, this means in particular that Kolmogorov's 0-1 law is not applicable.

Remark: It is not difficult to see that $A_M$ is an element of the exchangable $\sigma$-algebra and therefore

$$\mathbb{P}(A_M) \in \{0,1\}$$

by Hewitt-Savage's 0-1-law.

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