Bounds for the stop-loss distance of an independent random sum via Stein's method

Punyapat Kammoo; Kritsana Neammanee; Kittipong Laipaporn; Punyapat Kammoo; Kritsana Neammanee; Kittipong Laipaporn

doi:10.3934/math.2025587

AIMS Mathematics

2025, Volume 10, Issue 6: 13082-13103. doi: 10.3934/math.2025587

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Research article

Bounds for the stop-loss distance of an independent random sum via Stein's method

1.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2.
Centre of Excellence in Mathematics, Ministry of Higher Education, Science, Research and Innovation, National University of Sciences, Bangkok 10400, Thailand
3.
Department of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand
4.
Center of Excellence for Ecoinformatics, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand

Received: 23 January 2025 Revised: 19 May 2025 Accepted: 22 May 2025 Published: 06 June 2025
MSC : 60F05

Let $ W = X_1+X_2+\cdots + X_N $ be a random sum and $ Z $ be the standard normal random variable. In this paper, we investigated uniform and non-uniform bounds of the stop-loss distance, which measures the difference between two random variables, $ W $ and $ Z $, using the expression $ |Eh_k(W) - Eh_k(Z)| $, where $ h_k(x) = (x-k)^+ $ is a call function. In particular, we focused on the case that $ X_1, X_2, \ldots $ are independent random variables, and $ N $ is a non-negative, integer-valued random variable independent of the $ X_j $'s. Our methods were Stein's method and the concentration inequality approach. The value $ Eh_k(W) = E(W-k)^+ $ represents the excess over a threshold and is relevant to applications in collateralized debt obligations (CDOs) and the collective risk model.
- random sums,
- Stein's method,
- stop-loss distance,
- concentration inequality,
- normal approximation,
- non-uniform bounds
Citation: Punyapat Kammoo, Kritsana Neammanee, Kittipong Laipaporn. Bounds for the stop-loss distance of an independent random sum via Stein's method[J]. AIMS Mathematics, 2025, 10(6): 13082-13103. doi: 10.3934/math.2025587

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Abstract

Let $ W = X_1+X_2+\cdots + X_N $ be a random sum and $ Z $ be the standard normal random variable. In this paper, we investigated uniform and non-uniform bounds of the stop-loss distance, which measures the difference between two random variables, $ W $ and $ Z $, using the expression $ |Eh_k(W) - Eh_k(Z)| $, where $ h_k(x) = (x-k)^+ $ is a call function. In particular, we focused on the case that $ X_1, X_2, \ldots $ are independent random variables, and $ N $ is a non-negative, integer-valued random variable independent of the $ X_j $'s. Our methods were Stein's method and the concentration inequality approach. The value $ Eh_k(W) = E(W-k)^+ $ represents the excess over a threshold and is relevant to applications in collateralized debt obligations (CDOs) and the collective risk model.

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