Check if the weak law of large numbers holds true for the following sequence of random variables

Question

Suppose we have $n$ independent discrete random variables, whose distribution is as follows:

$X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values:

$-\sqrt{k}$ with a probability of $1/k$

$0$ with a probability of $1 - 2/k$

$\sqrt{k}$ with a probability of $1/k$

Does the sequence $X(k)$ adhere to the weak law of large numbers?

Answer 1

Yes. \begin{align} \text{Var} \left [ \frac{\sum_{k=2}^{n} X(k)}{n - 1} \right ] &= \frac{1}{(n - 1)^2} \sum_{k=2}^{n} \text{E}[X(k)^2] \\ &= \frac{2}{n - 1} \end{align} since $\text{E}[X(k)^2] = 2$ for every $k$. Hence $\sum_{k=2}^{n} X(k) / (n - 1) \stackrel{\text{p}}{\to} 0$ by Chevyshev's inequality.