Weakly Convergent Subsequence: Iff-Condition? Helly's theorem states that we can extract a convergent subsequence from any sequence of distribution functions. Moreover, one can show that the limit of this subsequence is a distribution function (and hence we have weak convergence), if the distribution functions have a tight distribution. 
In the lecture it was mentioned, that we would have an iff-condition for being able to extract a convergent subsequence to a distribution function. Does this mean that if the measure is not tight, then the resulting function is not a distribution function, hence there is no weak convergence?
 A: One can try a typical example of non-tight sequence of probability measures on the real line $\mu_n:=\delta_n$. The distribution function of $\mu_n$ is $F_n\colon t\mapsto \mathbf 1_{[n,+\infty)}(t)$. The sequence $(F_n(\cdot))_{n\geqslant 1}$ (as well as all its subsequences) converges pointwise to the null function, which is not a distribution function. 
In the general case, if a sequence of probability measures on the real line converges in distribution to a probability measure, then the considered sequence is tight.
A: I am providing the proof of the last statement used by @Davide in his answer about which OP asked in the comments.
Let $\{F_n\}_{n\geq 1}$ be a sequence of d.f. that converges weakly to some limit d.f. $F$(say).
Now, suppose assume $\{F_n\}_{n\geq 1}$ is not tight
$\Rightarrow$ $\exists \ \ \epsilon_0>0$ and for each $M>0,\exists\ n_M$(a subsequence), satisfying,
$F_{n_M}(M) - F_{n_M}(-M) \leq 1-\epsilon_0$
Now, we can take limit on both sides for $n_M\longrightarrow\infty$, which gives us (by using that fact from weak convergence of $\{F_n\}_{n\geq1}$) that $F_{n_M}(M)\longrightarrow F(M),\ \&\ F_{n_M}(-M)\longrightarrow F(-M)$.
Now, we get,
$F(M) - F(-M)\leq 1-\epsilon_0$.
Taking limit $M\longrightarrow\infty$ on both sides give $1-0\leq1-\epsilon_0\Rightarrow \epsilon_0\leq 0(\Rightarrow\Leftarrow)$.
Hence, a sequence of d.f. that converges weakly is considered to be a tight sequence.
