An intuitive proof of Prokhorov’s theorem I just want to understand why tightness condition implies that the measure sequence would admit a converging subsequence. I wonder if someone can explain to me how Prokhorov's theorem is proved intuitively without too much analysis formalism. Why not being able to escape to infinity warrants convergence?
 A: I always thought the example given in Billingsley 's book was illustrative as to why tightness is needed for this result. Consider the sequence $F_n$ of distribution functions
(or the associated probability measures on $\mathbb{R}$) corresponding to $Unif(-n,n)$ random variables. In this case every subsequence converges to the limiting "extended distribution function" $F_\infty(x)=1/2$. Notice that this CDF does not correspond to a probability measure on $\mathbb{R}$, and intuitively why this is occurring is that the mass of the distributions of $F_n$ is escaping to infinity. In general it can be shown that every sequence of CDF's must have a subsequence converging to some sort of "extended distribution function" like the example above (this is usually called Helly's selection theorem). Tightness it turns out is a necessary and sufficient condition to imply that this extended distribution function corresponds to a probability measure. See the below slides regarding Helly's theorem.
https://faculty.math.illinois.edu/~psdey/MATH561SP19/week10.pdf
