C. (8%)

Feller Weak Law of Large Numbers

Suppose that $X_1, X_2, \ldots, X_n$ are iid random variables and define $S_n = \sum_{j=1}^n X_j$. If

$$\lim_{x \to \infty} x P[|X_1| > x] = 0,$$

then

$$\frac{S_n}{n} - E[X_1 1_{[|X_1| \leq n]}] \xrightarrow{p} 0 \text{ as } n \to \infty,$$

where $1_{[\cdot]}$ is an indicator function.

Let $F$ be a symmetric distribution and suppose

$$1 - F(x) = \frac{e}{2x \ln x}, \quad x \geq e.$$

Suppose that $\{X_n, n \geq 1\}$ is an iid sequence of random variables with comment distribution $F$. Show that the sample mean converges in probability to 0.