Infinite expectation implies infinite random variable?

Question

Let $X$ be a nonnegative integer-valued random variable depending on a parameter $L$.

Assume that $E[X] \rightarrow \infty$ as $L\rightarrow \infty$. Does this imply that for any $k$, $P(X > k)$ is going to 1 with $L$?

Answer 1

Define $X_n$ to be such that $X_n$ is $0$ with probability $1-\frac{1}{n}$ and $n^2$ with probability $\frac{1}{n}$.

It is the case that $E[X_n]=n \to \infty$. But for any positive $k$ we have $\mathbb{P}(X_n > k) = \frac{1}{n} \to 0$. Thus showing a counter example.

Answer 2

Let $X_L$ be $L$ times a Bernoulli random variable (that is, it takes values $0$ and $L$ with equal probability). Its expectation is $L/2$ which tends to infinity with $L$. However, $P(X_L>0)=1/2$ for all $L$.