[Paper Review] Asymptotics of randomly weighted sums without moment conditions of random weights

Qingwu Gao, Dimitrios G. Konstantinides+3 more|arXiv (Cornell University)|Feb 26, 2026

Probability and Risk Models0 citations

TL;DR

The paper derives uniform tail asymptotics for randomly weighted sums and stopped sums without moment conditions on weights, and applies results to finite-time and random-time ruin probabilities in discrete-time risk models.

ABSTRACT

In the paper, we investigate the asymptotic behaviors of the randomly weighted sums with upper tail asymptotically independent increments under new conditions without requiring moment assumptions on random weights.An application of the obtained results is established to asymptotically estimate for finite-time ruin probability in a discrete-time risk model. For the case of increments with regularly varying tails, we obtain more explicit results via an extension of Breiman's theorem.

Motivation & Objective

• Motivate the study by removing moment conditions on random weights in tail asymptotics of weighted sums.
• Establish uniform asymptotics for weighted sums with an extended convergence range.
• Introduce and utilize upper tail asymptotic independence (UTAI) and related dependence structures.
• Apply the asymptotic results to finite-time and random-time ruin probabilities in discrete-time risk models.

Proposed method

• Use upper tail asymptotic independence (UTAI) framework to analyze dependent increments.
• Derive uniform convergence results for weighted sums with weights drawn from intervals [f1(x), f2(x)].
• Develop conditions under which P(sum w_i X_i > x) ~ sum P(w_i X_i > x) uniformly in n and w_i.
• Extend Breiman-type results for regularly varying increments to obtain explicit asymptotics.
• Provide proofs that relax moment conditions on weights and show necessity of certain assumptions via examples.
• Apply the results to ruin probabilities in discrete-time risk models with finite-time and random-time horizons.

Experimental results

Research questions

• RQ1 Can the range of uniform asymptotics for weighted sums be extended beyond existing results?
• RQ2 Can the number of terms in the weighted sums be increased while preserving asymptotics?
• RQ3 Can uniform asymptotics be extended from TAI to the broader UTAI setting?
• RQ4 What happens to ruin probabilities in discrete-time risk models under randomly weighted sums without weight moment conditions?

Key findings

• For TAI increments with tails in L ∩ D, P(sum w_i X_i > x) is asymptotically equivalent to sum P(w_i X_i > x) uniformly over weights in [f1(x), f2(x)].
• The uniform asymptotic results extend to UTAI increments under additional conditions, showing robustness of the single large jump principle in broader dependence settings.
• A random-weight extension (with independent weights) yields P(sum W_i X_i > x) ~ sum P(W_i X_i > x) under suitable decay relations between G_i and F, and the results hold for finite and random stopping times.
• In the regularly varying setting, an extended Breiman-type theorem provides more explicit asymptotics for weighted sums and stopped sums.
• The paper provides examples demonstrating the necessity of conditions and showing possibilities where upper tail asymptotic independence is present without tail asymptotic independence.]
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