Limit of cumulative distribution functions is a cumulative distribution function

Question

Le $X_n$ be a sequence of random variables with a cumulative distribution function $F_{X_n}$. Now of the sequence of function $F_{X_n}$ converge pointwise to a function $F$. Is $F$ itself a cumulative distribution function of some random variable?

Answer 1

No. Consider $$F_{X_n}=\begin{cases} 1-\frac{1}{x-n}, & x \geq n+1\\ 0, & \text{otherwise} \end{cases} $$ The function converges pointwise to zero, which is not a cumulative distribution function.

Answer 2

CDFs have some properties, e.g. if $F$ is a CDF then

• $F(x) \to 0$ as $x \to -\infty$
• $F(x) \to 1$ as $x \to +\infty$
• $F$ is non-decreasing
• $F$ is right-continuous

Can you check if all of them hold in the limit?

HINT the second one can fail, so can the first one, if the support decreases correctly

UPDATE as suggested in the comments, the last one can fail as well