Let $X$ be a random variable that is positive and integer-valued.

Let $X_1, X_2, ...$ also be random variables that are positive and integer-valued.

Prove that $X_n$ converges in distribution to $X$ if and only if

$$\lim_{n\to\infty} P(X_n = k) = P(X = k)$$

for every integer $k$.

My attempt:

I know that convergence in distribution for this problem means that

$$\lim_{n\to\infty} P(X_n \leq k) = P(X \leq k)$$

I don't know how this leads to the desired result. I also don't know how to prove the reverse implication.

  • 1
    $\begingroup$ Hint: $P(X_n=k) = P(X_n\le k) - P(X_n\le k-1).$ $\endgroup$ – whuber Dec 1 '18 at 22:01
  • $\begingroup$ Thanks @whuber. How do I apply that as $n$ goes to infinity? $\endgroup$ – MSE Dec 1 '18 at 22:29
  • 1
    $\begingroup$ The limit of the difference of two sequences is the difference of their limits (QED). $\endgroup$ – whuber Dec 1 '18 at 23:31

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