The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance

Abstract

In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary real-valued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as S w m , represents the stochastic present value of aggregate net losses during the first m periods. Suppose that the primary random variables are independent of each other with long-tailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of max 1≤m≤n S w m -the maximum of the first n weighted sums-is asymptotically equivalent to that of S w n -the last weighted sum.

Uniform asymptotics for the tail probability of weighted sums with heavy tails

“…The first main result generalizes Lemma 4.1 of Chen et al [3] with different approach in two ways. First, it increases the upper bound of the weights and decreases the lower bound of the weights.…”

“…First, it increases the upper bound of the weights and decreases the lower bound of the weights. Second, the fixed shift term A in Lemma 4.1 of Chen et al [3] is enlarged to some unbounded function, which is irrespective of the upper bound of the weights.…”

“…Proof of Theorem 3.2. The idea is from the proof of Theorem 2.1 of Chen et al [3]. Let Ω K = {X i ≥ 0 for all i ∈ K, X j < 0 for all j ∈ {1, · · · , n}\K}, K ⊆ {1, · · · , n} be a finite partition of the whole space Ω. Obviously,…”

“…holds uniformly for a ≤ c i ≤ b, 1 ≤ i ≤ n, in the sense that lim x→∞ sup a≤ci≤b,1≤i≤n Recently, Liu et al [16] and Li [14] established the same asymptotic relation for some dependent X i 's. Chen et al [3] showed that for any fixed 0 < a ≤ b < ∞ it holds that uniformly for a ≤ c i ≤ b,…”

On random coefficient INAR(1) processes

“…See for instance [23,25], more recent articles [12,13,27,38,47,58,59], and references therein. It is a well-known paradigm that such an assumption yields a rich probabilistic structure of the stationary solution and allows for a great flexibility in the modeling of its asymptotic behavior [1,39,51,52,54].…”