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Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk
Volume 5, Issue 2 (2018), pp. 129–143
Dominyka Kievinaitė   Jonas Šiaulys 1  

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https://doi.org/10.15559/18-VMSTA99
Pub. online: 15 March 2018      Type: Research Article      Open accessOpen Access

1 The second author was supported by grant No S-MIP-17-72 from the Research Council of Lithuania.

Received
13 October 2017
Revised
10 February 2018
Accepted
13 February 2018
Published
15 March 2018

Abstract

Let $\{{\xi _{1}},{\xi _{2}},\dots \}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\,{\sup _{n\geqslant 0}}\,{\sum _{i=1}^{n}}{\xi _{i}}>x)$ can be bounded above by ${\varrho _{1}}\exp \{-{\varrho _{2}}x\}$ with some positive constants ${\varrho _{1}}$ and ${\varrho _{2}}$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

References

[1] 
Albrecher H., Ivanovs J., Zhou X.: Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22(3), 1364–1382 (2016). MR3474819
[2] 
Ambagaspitiya R.S.: Ultimate ruin probability in the Sparre-Andersen model with dependent claim sizes and claim occurrence times. Insur. Math. Econ. 44(3), 464–472 (2009). MR2519090
[3] 
Andrulytė I.M., Bernackaitė E., Kievinaitė D., Šiaulys J.: A Ludberg-type inequality for an inhomogeneous renewal risk model. Mod. Stoch.: Theory Appl. 2, 173–184 (2015). MR3389589
[4] 
Asmussen S., Albrecher H.: Ruin Probabilities. Word Scientific Publishing (2010). MR2766220
[5] 
Bernackaitė E., J. Šiaulys J.: The exponential moment tail of inhomogeneous renewal process. Stat. Probab. Lett. 97, 9–15 (2015). MR3299745
[6] 
Bernackaitė E., J. Šiaulys J.: The finite-time ruin probability for an inhomogeneous renewal risk model. J. Ind. Manag. Optim. 13(1), 207–222 (2017). MR3576051
[7] 
Castañer A., Claramunt M.M., Gathy M., Lefèvre Cl., Mármol M.: Ruin problems for a discrete time risk model with non-homogeneous conditions. Scand. Actuar. J. 2013(2), 83–102 (2013). MR3041119
[8] 
Chernoff H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493–507 (1952). MR0057518
[9] 
Cojocaru I.: Ruin probabilities in multivariate risk models with periodic common shock. Scand. Actuar. J. 2017(2), 159–174 (2017). MR3590439
[10] 
Constantinescu C., Kortschak D., Maume-Deschamps V.: Ruin probabilities in models with a Markov chain dependence structure. Scand. Actuar. J. 2013(6), 453–476 (2013). MR3176014
[11] 
Czarna I., Palmowski Z.: Ruin probability with Poisson delay for a spectrally negative Lévy process. J. Appl. Probab. 48(4), 984–1002 (2011). MR2896663
[12] 
Damarackas J., Šiaulys J.: Bi-seasonal discrete time risk model. Appl. Math. Comput. 247, 30–940 (2014). MR3270895
[13] 
Danilenko S., Markevičiūtė J., Šiaulys J.: Randomly stopped sums with exponential-type distributions. Nonlinear Anal. Model. Control 22(6), 793–807 (2017). MR3724621
[14] 
Embrechts P., Klüppelberg C., Mikosch T.: Modeling Extremal Events for Insurance and Finance. Springer (1997). MR1458613
[15] 
Embrechts P., Veraverbeke V.: Estimates for probability of ruin with special emphasis of the possibility of large claims. Insur. Math. Econ. 1, 55–72 (1982). MR0652832
[16] 
Gerber H.: Martingales in risk theory. Bull. Swiss Assoc. Actuar. 1973, 205–216 (1973).
[17] 
Grigutis A., Korvel A., Šiaulys J.: Ruin probabilities at a discrete-time multi risk model. Inf. Technol. Valdym. 44(4), 367–379 (2015).
[18] 
Grigutis A., Korvel A., Šiaulys J.: Ruin probability in the three-seasonal discrete-time risk model. Mod. Stoch.: Theory Appl. 2, 421–441 (2015). MR3456147
[19] 
Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963). MR0144363
[20] 
Jordanova P., Stehlík M.: Mixed Poisson process with Pareto mixing variable and its risk applications. Lith. Math. J. 56(2), 189–206 (2016). MR3504255
[21] 
Kiefer J., Wolfowitz J.: On the characteristics of the general queuing process with applications to random walk. Ann. Math. Stat. 27(1), 147–161 (1956). MR0077019
[22] 
Leipus R., Šiaulys J.: Asymptotic behavior of the finite-time ruin probability under subexponential claim sizes. Insur. Math. Econ. 40, 498–508 (2007). MR2311546
[23] 
Leipus R., Šiaulys J.: Asymptotic behavior of the finite-time ruin probability in the renewal risks models. Appl. Stoch. Models Bus. Ind. 25, 309–321 (2009). MR2541109
[24] 
Mishura Y., Ragulina O., Stroyev O.: Practical approaches to the estimation of the ruin probability in a risk model with additional funds. Mod. Stoch.: Theory Appl. 1(2), 167–180 (2014). MR3316485
[25] 
Mikosch T.: Non-life Insurance Mathematics. Springer (2009). MR2503328
[26] 
Răducan A.M., Vernic R., Zbăganu G.: Recursive calculation of ruin probabilities at or before claim instants for non-identically distributed claims. ASTIN Bull. 45(2), 421–443 (2015). MR3394025
[27] 
Răducan A.M., Vernic R., Zbăganu G.: On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues. J. Comput. Appl. Math. 290, 319–333 (2015). MR3370412
[28] 
Răducan A.M., Vernic R., Zbăganu G.: Upper and lower bounds for a finite-type ruin probability in a nonhomogeneous risks process. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 17(4), 287–292 (2016). MR3582626
[29] 
Ragulina O.: The risk model with stochastic premiums, dependence and a threshold dividend strategy. Mod. Stoch.: Theory Appl. 4(4), 315–351 (2017).
[30] 
Sgibnev M.S.: Submultiplicative moments of the supremum of a random walk with negative drift. Stat. Probab. Lett. 32, 377–383 (1997). MR1602211
[31] 
Tang Q.: Asymptotics for the finite time ruin probability in the renewal risk model with consistent variation. Stoch. Models 20(3), 281–297 (2004). MR2082126
[32] 
Wang Y., Cui Z., Wang K., Ma X.: Uniform asymptotics of the finite-time ruin probability for all times. J. Math. Anal. Appl. 390, 208–223 (2012). MR2885767
[33] 
Zhang T., Fang X.N., Liu J., Yang Y.: Asymptotics for the partial sum and its maximum of dependent random variables. Lith. Math. J. 57(1), 142–153 (2017). MR3621877
[34] 
Zhang Z., Cheng E.C.K., Yang H.: Lévy insurance risk process with Poissonian taxation. Scand. Actuar. J. 2017(1), 51–87 (2017). MR3592957

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Keywords
Exponential bound supremum of sums tail probability risk model inhomogeneity ruin probability Lundberg’s inequality

MSC2010
62E20 60E15 60G50 91B30

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