Abstract
We consider the random difference equations S = d (X + S)Y and T = d X + TY, where = d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.
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Alsmeyer G, Mentemeier S. Tail behaviour of stationary solutions of random difference equations: The case of regular matrices. J Difference Equ Appl, 2012, 18: 1305–1332
Asimit A V, Badescu A L. Extremes on the discounted aggregate claims in a time dependent risk model. Scand Actuar J, 2010, 2010: 93–104
Asimit A V, Furman E, Tang Q, et al. Asymptotics for risk capital allocations based on conditional tail expectation. Insurance Math Econom, 2011, 49: 310–324
Bingham N H, Goldie C M, Teugels J L. Regular Variation. Cambridge: Cambridge University Press, 1987
Buraczewski D, Damek E, Mikosch T. Stochastic Models with Power-Law Tails. New York: Springer, 2016
Buraczewski D, Damek E, Mikosch T, et al. Large deviations for solutions to stochastic recurrence equations under Kesten’s condition. Ann Probab, 2013, 41: 2755–2790
Buraczewski D, Damek E, Mirek M. Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems. Stochastic Process Appl, 2012, 122: 42–67
Chamayou J, Letac G. Explicit stationary distributions for compositions of random functions and products of random matrices. J Theoret Probab, 1991, 4: 3–36
Chen Y, Yang Y. Ruin probabilities with insurance and financial risks having an FGM dependence structure. Sci China Math, 2014, 57: 1071–1082
Chen Y, Yuen K C. Precise large deviations of aggregate claims in a size-dependent renewal risk model. Insurance Math Econom, 2012, 51: 457–461
Cline D B H. Convolution tails, product tails and domains of attraction. Probab Theory Related Fields, 1986, 72: 529–557
Collamore J F. Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann Appl Probab, 2009, 19: 1404–1458
Denisov D, Zwart B. On a theorem of Breiman and a class of random difference equations. J Appl Probab, 2007, 44: 1031–1046
Dufresne D. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand Actuar J, 1990, 1990: 39–79
Dufresne D. On the stochastic equation L(X) = L[B(X + C)] and a property of gamma distributions. Bernoulli, 1996, 2: 287–291
Dyszewski P. Iterated random functions and slowly varying tails. Stochastic Process Appl, 2016, 126: 392–413
Embrechts P, Goldie C M. Perpetuities and random equations. In: Contribution to Asymptotic Statistics. Berlin-Heidelberg: Springer-Verlag, 1994, 75–86
Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag, 1997
Foss S, Korshunov D, Zachary S. An Introduction to Heavy-Tailed and Subexponential Distribution, 2nd ed. New York: Springer, 2013
Goldie C M. Implicit renewal theory and tails of solutions of random equations. Ann Appl Probab, 1991, 1: 126–166
Goldie C M, Grübel R. Perpetuities with thin tails. Adv in Appl Probab, 1996, 28: 463–480
Grey D R. Regular variation in the tail behaviour of solutions of random difference equations. Ann Appl Probab, 1994, 4: 169–183
Hitczenko P, Wesołowski J. Perpetuities with thin tails revisited. Ann Appl Probab, 2009, 19: 2080–2101
Hitczenko P, Wesołowski J. Renorming divergent perpetuities. Bernoulli, 2011, 17: 880–894
Joe H. Dependence Modeling with Copulas. Boca Raton: CRC Press, 2015
Kesten H. Random difference equations and renewal theory for products of random matrices. Acta Math, 1973, 131: 207–248
Kevei P. A note on the Kesten–Grinceviˇcius–Goldie theorem. Electron Comm Probab, 2016, 21: 1–12
Konstantinides D G, Mikosch T. Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann Probab, 2005, 39: 1992–2035
Lehtomaa J. Asymptotic behaviour of ruin probabilities in a general discrete risk model using moment indices. J Theoret Probab, 2015, 28: 1380–1405
Li J, Tang Q. Interplay of insurance and financial risks in a discrete-time model with strongly regular variation. Bernoulli, 2015, 21: 1800–1823
Li J, Tang Q, Wu R. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model. Adv in Appl Probab, 2010, 42: 1126–1146
Milevsky M. The present value of a stochastic perpetuity and the Gamma distribution. Insurance Math Econom, 1997, 20: 243–250
Nelsen R B. An Introduction to Copulas, 2nd ed. New York: Springer, 2006
Takahashi H, Kanagawa S, Yoshihara K I. Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors. Stochastic Anal Appl, 2015, 33: 740–755
Tang Q. The subexponentiality of products revisited. Extremes, 2006, 9: 231–241
Tang Q, Tsitsiashvili G. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process Appl, 2003, 108: 299–325
Tang Q, Yuan Z. Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes, 2014, 17: 467–493
Vervaat W. On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv in Appl Probab, 1979, 11: 750–783
Yang Y, Konstantinides D G. Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks. Scand Actuar J, 2015, 2015: 641–659
Yang Y, Wang Y. Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes, 2013, 16: 55–74
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In memory of Professor Xiru Chen (1934–2005)
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Tang, Q., Yuan, Z. Random difference equations with subexponential innovations. Sci. China Math. 59, 2411–2426 (2016). https://doi.org/10.1007/s11425-016-0146-0
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DOI: https://doi.org/10.1007/s11425-016-0146-0
Keywords
- asymptotics
- Karamata index
- long tail
- random difference equation
- subexponentiality
- tail probability
- uniformity