Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is 0,2 ^k/2/k!] or −2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .