probability and stochastic process

probability and stochastic process

Let p be a probability of F and (Z_n ) be sequence of indicator r.v such
that
∀n≥1,p(Z_n=1)=1/n^2 we define
∀(n,k)∈N^*×N,A_(n,k)={∑_(i=1)^n▒〖Z_i≥k〗},A_k=⋃(n≥1)▒A(n,k)
Using Markov inequality, give an estimate of p(A_n,k) which do not depend
on n Show that for any fix k, the sequence of event (A_n,k) is increasing.
Deduce an estimate of p(A_k ) Show that the sequence of event (A_k
)(k∈N) is decreasing Deduce that the series
∑(i=1)^∞▒Z_i converges almost surely