# Martingale Approach to Derive Lundberg-Type Inequalities

*Tautvydas Kuras* (),
*Jonas Sprindys* () and
*Jonas Šiaulys* ()

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Tautvydas Kuras: Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Jonas Sprindys: Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Jonas Šiaulys: Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

*Mathematics*, 2020, vol. 8, issue 10, 1-18

**Abstract:**
In this paper, we find the upper bound for the tail probability P sup n ⩾ 0 ∑ I = 1 n ξ I > x with random summands ξ 1 , ξ 2 , … having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ 1 exp { − ϱ 2 x } with some positive constants ϱ 1 and ϱ 2 . For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

**Keywords:** exponential estimate; supremum of sums; tail probability; risk model; inhomogeneity; ruin probability; Lundberg’s inequality (search for similar items in EconPapers)

**JEL-codes:** C (search for similar items in EconPapers)

**Date:** 2020

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