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If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability.

My original approach was the following:

Let $\epsilon > 0$. Then $\lim_{n,m \rightarrow \infty}P(|Y_m-Y_n| \geq \epsilon )= \lim_{n,m \rightarrow \infty}P(|\frac{X^m}{m}- \frac{X^n}{n}| \geq \epsilon$). This is where I am stuck. I would like to write the probability out for this random variable $\frac{X^m}{m}- \frac{X^n}{n}$ but I am not sure how.

Thanks!

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1 Answer 1

Obviously, $Y_n\to0$ on $[X\leqslant1]$ and $Y_n\to\infty$ on $[X\gt1]$. Thus, if one accepts random variables that are infinite on events of positive probability, then $Y_n\to Y$ almost surely, with $Y=\infty\cdot\mathbf 1_{X\gt1}$, and it follows that $Y_n\to Y$ in probability. If one does not, $Y_n$ does not converge in probability.

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