ASYMPTOTICS FOR WEIGHTED RANDOM SUMS

作者：Olvera Cravioto Mariana^*

来源：Advances in Applied Probability, 2012, 44(4): 1142-1172.

• Randomly weighted sum
• randomly stopped sum
• heavy tail
• intermediate regular variation
• regular variation
• Breiman's theorem

Let {X-i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail (F) over bar. Let (N, C-1, C-2, ... ) be a nonnegative random vector independent of the {X-i) with N epsilon NU {infinity}. We study the weighted random sum S-N =Sigma(N)(i=1) CiXi, and its maximum, M-N = sup(1 <= k<N+1) Sigma(k)(i=1) CiXi. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M-N > x) similar to P(S-N > x) similar to E[Sigma(N)(i=1) (F) over bar (x/C-i)] as x -> infinity. When E[X-1] > 0 and the distribution of Z(N) =Sigma(N)(i=1) C-i is also intermediate regularly varying, we obtain the asymptotics P(M-N > x) similar to P(S-N > x) similar to E[Sigma(N)(i=1) (F) over bar (x/C-i)] + P (Z(N) > x/E[X-1]). For completeness, when the distribution of Z(N) is intermediate regularly varying and heavier than (F) over bar, we also obtain conditions under which the asymptotic relations P(M-N > x) similar to P(S-N > x) similar to P(Z(N) > x/E[X-1]) hold.

• 出版日期2012-12