In this wiki article on the law of the iterated logarithm, one states that, given $M>0$, the event $$A=\left\{\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}} > M\right\}$$ has probability $0$ or $1$ by Kolmogorov Zero One Law. This leads me to believe that $A$ is a tail event.

While trying to show that $A$ is a tail event, I observed that $S_n$ is measurable with respect to $\sigma\left(\bigcup\limits_{k=1}^n \mathcal{F}_k\right)$ where $\mathcal{F_k} = \sigma(X_k)$. Even though the random variables $X_n$ are independent, the random variables $S_n$ are not. So how does one show that $A$ is a tail event?

I looked at this question, but I think that there is a jump where they say that $A$ is in $\sigma\left(\bigcup\limits_{k=n}^\infty \mathcal{F}_k\right)$.

Kindly help me in understanding how to go about this problem. Hints and tips as always are appreciated.

  • 3
    $\begingroup$ For every fixed $N$, $$\left\{\limsup_{n \to \infty} \frac{S_n}{\sqrt{n}} > M\right\}=\left\{\limsup_{n \to \infty} \frac{S_n-S_N}{\sqrt{n}} > M\right\}\in\sigma((X_n)_{n>N}).$$ $\endgroup$ – Did Apr 1 '15 at 13:55
  • $\begingroup$ Thanks Did. If you write that as an answer, I'll be able to credit you. $\endgroup$ – Gautam Shenoy Apr 1 '15 at 17:35

Credit to Did for this:

$$\left\{ \limsup_{n \to \infty}\frac{S_n}{\sqrt{n}} > M \right\}= \left\{ \limsup_{n \to \infty}\frac{S_n-S_m}{\sqrt{n}} > M \right\}$$ for every $m \geq 1$.

Now $$\left\{\frac{S_n-S_m}{\sqrt{n}} > M\right\} \in \sigma\left(\bigcup_{k=m}^n \mathcal{F_k}\right)$$ and $$\left\{\limsup_{n \to \infty}\frac{S_n-S_m}{\sqrt{n}} > M\right\} \in \sigma\left(\bigcup_{k=m}^\infty \mathcal{F_k}\right)$$ Since this is true for every $m$, we have $$\left\{ \limsup_{n \to \infty}\frac{S_n}{\sqrt{n}} > M \right\} \in \bigcap_{m=1}^\infty\sigma\left(\bigcup_{k=m}^\infty \mathcal{F_k}\right)$$

Thus it is a tail event.

Update: I'd like to point out that the source of my confusion was also from the Hewitt Savage Zero One Law article on Wikipedia which said that one could not apply Kolmogorov Zero One Law to $S_n$ as the events were not independent. However they were talking about a specific subsequence and not the $\limsup$ subsequence.


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