Give an example of a sequence of independent and identically distributed r.v.'s $\{Xn\}$ with mean $0$ and variance $1$ and a sequence of positive integer-valued r.v.'s $v_n$ tending to $\infty$ a.e. such that $S_{v_n}/s_{v_n}$ does not converge in distribution.
My solution
Let $\{X_n\}$ i.i.d random variables; $P(X_1 = 1) = P(X_1 = −1) = \frac{1}{2}$. Let Sn, n ≥ 1, be the partial sums and let $v(n),\,\,\, n ≥ 1$, be the index of $S_n$ at the time of the $nth$ visit to $0$. $$P(|S_{n}|>\epsilon)\leq \frac{E(S_n)}{\epsilon}=0, \,\,\,since \,\,\, E(X_n)=0$$ $$\sum^{\infty}_{n=1}P(|S_{n}|>\epsilon)<\infty,\,\,\, thus \,\,\,(for, B.C)\,\,\,\,\,P(|S_{n}|>\epsilon\,\, i.o)=0,\,\,\, $$ then $P(S_n = 0 \,\,\,i.o) = 1,$ so that $v(n)\to ∞\,\,\,\,\,$ $a.s$ $\,\,\,\,\,n → ∞.$
However $$\frac{S_{v(n)}}{s_{v(n)}}=\frac{S_{v(n)}}{\sqrt{v(n)}}=0,\,\,\,\,\,\, n\to \infty\,\,\,\,\,\,(*)?$$
but as $$\frac{S_{n}}{\sqrt{n}}{\to}^{d} \,\,\,N(0,1)$$
this is a contradiction? .........................
I think this is an example for the solution, but I'm not sure any help.