Maxima and sums of non-stationary random length sequences

Extremes of Sums and Maxima with Application to Random Networks

“…Remark 2. If there are in total d + 1 stationary mutually independent "column" sequences having the same tail index k 1 and extremal indices θ 1 , ..., θ d+1 , then Y * n (z, l n ) and Y n (z, l n ) have the tail index k 1 and the extremal index that is a superposition of θ 1 , ..., θ d+1 as derived in [13], see Theorem 2 in [18].…”

“…Random length sequences and distribution tails of their sums and maxima attract the interest of many researchers due to numerous applications including queues, branching processes and random networks [1], [10], [15], [16], [18], [20], [21], [22]. Let {Y n,i : n, i ≥ 1} be a doubly-indexed array of nonnegative random variables (r.v.s) in which the "row index" n corresponds to time, and the "column index" i enumerates the series.…”

“…An extremal index that is close to zero implies a kind of a strong local dependence. Let us recall Theorem 4 derived in [18]. It is assumed that the "column" sequences {Y n,i : i ≥ 1} have stationary distributions (1) in n with positive tail indices {k 1 , k 2 , ...} and extremal indices {θ 1 , θ 2 , ...} for each fixed i, where {ℓ i (x)} are restricted by the condition: for all A > 1, δ > 0 there exists x 0 (A, δ) such that for all i ≥ 1…”

“…The tail of N n does not dominate the tail of the most heavy-tailed term Y n,1 . Let , and the positive weights {z i } are assumed to be bounded as in [18]. Theorem 1.…”

“…Theorem 1. [18] Let the sets of slowly varying functions { ln (x)} n≥1 in (4) and {ℓ i (x)} i≥1 in (1) satisfy the condition (3). Suppose that k 1 < k and…”