Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Abstract

Let \{X_i={(X_{1,i}, . . .,X_{m,i})}^{\mathrm{T}}, i\ge 1\}be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X_1 are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence

\{X_i, i\ge 1\}

.Under several mild assumptions, precise large deviations for

S_n={\displaystyle {\sum }_{i=1}^n{X}_i}\mathrm{ }

and S_{N(t)}={\displaystyle {\sum }_{i=1}^{N(t)}{X}_i}\mathrm{ }are investigated. Meanwhile, some simulation examples are also given to illustrate the results.

Keywords

Precise large deviations / multi-dimensional / consistently varying distributions / random sums

References

[1] BaltrūnasA, LeipusR, ŠiaulysJ. Precise large deviation results for the total claim amount under subexponential claim sizes. Statist Probab Lett, 2008, 78: 1206–1214

[2] BinghamN H, GoldieC M, TeugelsJ L. Regular Variation. Cambridge: Cambridge Univ Press, 1987

[3] ClineD B H, SamorodnitskyG. Subexponentiality of the product of independent random variables. Stochastic Process Appl, 1994, 49: 75–98

[4] EmbrechtsP, KlüppelbergC, MikoschT. Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag, 1997

[5] KaasR, TangQ. A large deviation result for aggregate claims with dependent claim occurrences. Insurance Math Econom, 2005, 36: 251–259

[6] KlüppelbergC, MikoschT. Large deviations of heavy-tailed random sums with applications in insurance and finance. J Appl Probab, 1997, 34: 293–308

[7] LuD. Lower bounds of large deviation for sums of long-tailed claims in a multi-risk model. Statist Probab Lett, 2012, 82: 1242–1250

[8] NelsenR B. An Introduction to Copulas. New York: Springer, 2006

[9] NgK W, TangQ, YanJ, YangH. Precise large deviations for the prospective-loss process. J Appl Probab, 2003, 40: 391–400

[10] NgK W, TangQ, YanJ, YangH. Precise large deviations for sums of random variables with consistently varying tails. J Appl Probab, 2004, 41: 93–107

[11] ShenX, TianH. Precise large deviations for sums of two-dimensional random vectors with dependent components heavy tails. Comm Statist Theory Methods, 2016, 45(21): 6357–6368

[12] TangQ, SuC, JiangT, ZhangJ. Large deviations for heavy-tailed random sums in compound renewal model. Statist Probab Lett, 2001, 52: 91–100

[13] WangS, WangW. Precise large deviations for sums of random variables with consistently varying tails in multi-risk models. J Appl Prob, 2007, 44: 889–900

[14] WangS,WangW. Precise large deviations for sums of random variables with consistent variation in dependent multi-risk models. Comm Statist Theory Methods, 2013, 42: 4444–4459

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