I am facing a big chalenge to formalize the answer. Thinking all day. Any help? Hint?
Consider $X_{1}$,$X_{2}$,$X_{3}$,$X_{4}$,$X_{5}$,... i.i.d Bernoulli(i.e $P(X_{i}=1)=p),P(X_{i}=0)=1-p)$:
i) Does $Y_{n} = \prod_{i=1}^{n} X_{i}$. $\overline{Y_{n}} = \sum \frac{Y_{n}}{n}$ converges to a constant? Which one? Almost surely or just in probability?
ii) Does $W_{n} = \sum_{i=1}^{n} X_{i}$. $\overline{W_{n}} = \sum \frac{W_{n}}{n}$ converges to a constant? Which one? Almost surely or just in probability?
iii) Does $Z_{n} =\prod_{i=n-1}^{n} X_{i}$. $\overline{Z_{n}} = \sum \frac{Z_{n}}{n}$ converges to a constant? Which one? Almost surely or just in probability?
Partial answer: I will provide some answers and leave the rest to you. Since $P(X_n=1 \forall n)=0$ (assuming $0<p<1$) we see that $Y_n=0$ for all $n$ sufficiently large, with probability $1$. This implies that $\sum \frac {Y_n} n$ converges almost surely. The sum is not a constant.
By Strong Law $\frac {W_n} n \to p$ almost surely. A series cannot converge unless its general term tends to $0$. SO the second series neither converges almost surely nor does it converge in probability.