I have to contruct a sequence of random variables that follow Weak Law of Large Numbers, but don't follow Strong Law of Large Numbers. Can canyone give me any hint please? Basically i need to choose such a sequence of integrable r.v-s that:
1) $$(S_n-\mathbb{E}S_n)/n \rightarrow 0 \qquad \mbox{in probability}$$
and
2) $$\mathbb{P}(\{\omega:(S_n-\mathbb{E}S_n)/n \rightarrow 0 \})<1$$
$S_n:=X_1+\ldots+X_n$
The only thing I know is that if $X_1,X_2,\ldots$ were identically distributed and independent, then its impossible.
A counterexample to the question exactly as stated is very simple. Let $A_n:[0,1]\to\mathbb C$ be any sequence of functions that tends to $0$ in measure but not pointwise. Ok, also say $\int_0^1 A_n=0$. Let $S_n=nA_n$, then let $X_1=S_1$ and $X_n=S_n-S_{n-1}$ for $n>1$.