Expectation: Is this statment true or is there a counter example

Question

Let $(X_i)_{i\in (1,...,N)}$ be a sequence of i.i.d random variables with $0<\mathbb{E}(X_i^2)<0$. Is

$$\sup_i|X_i|<\infty$$

a consequence of the strong law of large numbers, or is there a counterexample?

Answer 1

I assume you are asking about an infinite sequence $X_1,X_2,\dots$ of iid random variables, and you want to know whether $\sup_i |X_i|<\infty$ holds almost surely.

If $X_1$ is bounded, meaning $P(|X_1|\le M)=1$ , then trivially we will have $P(\sup |X_i|\le M)=1$.

If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $\{|X_i|>M\}$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $\sup_i |X_i|\ge M$ for all positive integers $M$, so $\sup_i |X_i| = \infty$ with probability $1$.