Probability Theory (Convergence of Random Variables: The Strong Law of Large Numbers)

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2. Recurrent events. Catastrophes occur at the times 71 , T2, . . . where Ti = X1+ X2+. . .+ X; and the X ; are independent identically distributed positive random variables. Let N(t) = max {n : Tn S t} be the number of catastrophes which have occurred by time t. Prove that if E(X ] ) co, almost surely