Can a non integrable random variable satisfy a strong law of large numbers principle?

Question

Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$.

Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but

$$\lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n X_i = 0$$

almost surely?

Note: Here $X^+ := \max(X, 0)$, $X^- := -\min(X, 0)$ denote the positive and negative parts of $X$ respectively.

Answer 1

No.

If $\mathbb E[X^+]=\infty$ then $$\infty = \mathbb E[ \lfloor X^+\rfloor ] =\sum_{n=1}^{\infty} \mathbb P(X \geq n) = \sum_{n=1}^{\infty} \mathbb P(X_n\geq n)$$ and the events $X_n \geq n$ are independent so because their probabilities sum to $\infty$, almost surely infinitely many of these events occur.

However, if $X_n \geq n$ then either $\sum_{i=1}^{n-1} X_i \leq - \frac{n}{2}$ or $\sum_{i=1}^n X_i \geq \frac{n}{2}$ and in the first case $\frac{1}{n-1} \sum_{i=1}^{n-1} X_i \leq -\frac{1}{2}$ and in the second case $\frac{1}{n} \sum_{i=1}^n X_i \geq \frac{1}{2}$.

If one of these two events occurs for infinitely many $n$, that contradicts $\lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n X_i=0$.

Answer 2

This is a slight modification of Will Sawin's answer. (Of course, the desired conclusion is well known; see e.g. this answer.

Instead of the condition $EX_+=EX_-=\infty$ in the OP, it is enough to assume that $E|X|=\infty$.

Indeed, suppose that $E|X|=\infty$, whereas $$\bar X_n:=\frac1n\,\sum_{i=1}^n X_i\to0 \tag{10}\label{10}$$ (as $n\to\infty$) almost surely (a.s.).

Then \begin{align*} \sum_{n=1}^\infty P(|X_n|\ge n)&= \sum_{n=1}^\infty P(|X|\ge n) \\ &=E\sum_{n=1}^\infty 1(|X|\ge n) \\ &=E\sum_{n=1}^{\lfloor|X|\rfloor} 1 \\ &=E\lfloor|X|\rfloor \ge E|X|-1=\infty. \end{align*} So, by the second Borel–Cantelli lemma, infinitely many events of the form $\{|X_n|\ge n\}$ occur a.s.

On the other hand, condition \eqref{10} implies that $$X_n=n\bar X_n-(n-1)\bar X_{n-1}=o(n)$$ a.s., which contradicts the conclusion that infinitely many events of the form $\{|X_n|\ge n\}$ occur a.s. $\quad\Box$

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One can also replace \eqref{10} by the more general condition $$\bar X_n\to a$$ for any real $a$ (then using $X_i-a$ and $X-a$ in place of $X_i$ and $X$).

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Given that $E|X|=\infty$, Corollary 1 in the paper The Strong Law of Large Numbers When the Mean is Undefined by Erickson (1973, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 185) provides a necessary and sufficient condition in terms of the distribution of $X$ for each of the following: (i) $\bar X_n\to\infty$ a.s.; (ii) $\bar X_n\to-\infty$ a.s.; (iii) $\limsup\bar X_n=\infty$ a.s. and $\liminf\bar X_n=-\infty$ a.s.