Find $X_n$ such that $Var(X_n)\to \infty$ but $X_n \to 0$ in probability. I have found a similar example where the expectation runs to infinity but the actual random variable converges to $0$. However, I can't think of an example where my question is satisfied.
Define $X_n = Z_n X_{n-1}$ where the $Z_n's$ are IID and have probability mass function: \begin{align} Z_n = \begin{cases} 1/3 & z = 3\\ 2/3 & z = 1/3 \end{cases} \end{align}
Assume that $X_0 = 1$. Then, $\mathbb{E}[X_n] = \mathbb{E}[X_{n-1}]\mathbb{E}[Z_n]$
But if I work it out then $\mathbb{E}[X_{n-1}]$ is similarly defined as $\mathbb{E}[X_n]$ and we end up iterating the identity $n$ times but the expectation of $Z_n$ remains the same until we reach $\mathbb{E}[X_0] = 100$. The expectation of $Z_n$ is: \begin{align} 1/3\times 3 + 2/3 \times 1/3 = 11/9 \end{align} Thus, \begin{align} \mathbb{E}[X_n] = \frac{11}{9}^n \end{align} and this goes to $\infty$ as $n \to \infty$ since we are raising a number larger than $1$ to an infinite power.
However, we can prove the random variable $X_n \to 0$ in probability.