Iterated random functions and regularly varying tails

References

• G. Alsmeyer, On the stationary tail index of iterated random Lipschitz functions, Stoch. Process. Appl. 126 (2016), pp. 209–233. doi: 10.1016/j.spa.2015.08.004
   Web of Science ®Google Scholar
• L. Arnold and H. Crauel, Iterated function systems and multiplicative ergodic theory, in Diffusion Theory and Related Problems in Analysis II, M. A. Pinsky and V. Wihstatz, eds., Birkhauser, Boston, 1992, pp. 283–305.
   Google Scholar
• N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation, Vol. 27, Cambridge University Press, Cambridge, 1989.
   Google Scholar
• S. Brofferio and D. Buraczewski, On unbounded invariant measures of stochastic dynamical systems, Ann. Probab. 43 (2015), pp. 1456–1492. doi: 10.1214/13-AOP903
   Web of Science ®Google Scholar
• D. Buraczewski and E. Damek, A simple proof of heavy tail estimates for affine type Lipschitz recursions, Stoch. Process. Appl.127 (2016), pp. 657–668 doi: 10.1016/j.spa.2016.06.022
   Web of Science ®Google Scholar
• D. Buraczewski, E. Damek and T. Mikosch, Stochastic Models with Power-Law Tails, Springer, New York, 2016.
   Google Scholar
• V. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching random processes, Theory Probab. Appl. 9(4) (1964), pp. 640–648. doi: 10.1137/1109088
   Google Scholar
• J. Chover, P. Ney and S. Wainger, Degeneracy properties of subcritical branching processes, Ann. Probab. 1 (1973), pp. 663–673. doi: 10.1214/aop/1176996893
   Web of Science ®Google Scholar
• J. Chover, P. Ney and S. Wainger, Functions of probability measures, J. Anal. Math. 26(1) (1973), pp. 255–302. doi: 10.1007/BF02790433
   Web of Science ®Google Scholar
• D.B.H. Cline, Convolution tails, product tails and domains of attraction, Probab. Theory Relat. Fields 72(4) (1986), pp. 529–557. doi: 10.1007/BF00344720
   Web of Science ®Google Scholar
• E. Damek and B. Kołodziejek, Stochastic recursions: between Kesten's and Grey's assumptions. Available at arXiv preprint arXiv:1701.02625, 2017.
   Google Scholar
• P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 41(1) (1999), pp. 45–76. doi: 10.1137/S0036144598338446
   Web of Science ®Google Scholar
• M. Duflo, Random Iterative Models, Springer-Verlag, Berlin, 1997.
   Google Scholar
• P. Dyszewski, Iterated random functions and slowly varying tails, Stoch. Process. Appl. 126(2) (2016), pp. 392–413. doi: 10.1016/j.spa.2015.09.005
   Web of Science ®Google Scholar
• J.H. Elton, A multiplicative ergodic theorem for Lipschitz maps, Stoch. Process. Appl. 34(1) (1990), pp. 39–47. doi: 10.1016/0304-4149(90)90055-W
   Web of Science ®Google Scholar
• P. Embrechts, A property of the generalized inverse Gaussian distribution with some applications, J. Appl. Probab. (201983), pp. 537–544. doi: 10.2307/3213890
   Web of Science ®Google Scholar
• P. Embrechts and C.M. Goldie, On convolution tails, Stoch. Process. Appl. 13(3) (1982), pp. 263–278. doi: 10.1016/0304-4149(82)90013-8
   Google Scholar
• S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Vol. 6, Springer-Verlag, New York, 2011.
   Google Scholar
• C.M. Goldie, Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1(1) (1991), pp. 126–166. doi: 10.1214/aoap/1177005985
   Google Scholar
• D.R. Grey, Regular variation in the tail behaviour of solutions of random difference equations, Ann. Appl. Probab. 4(1) (1994), pp. 169–183. doi: 10.1214/aoap/1177005205
   Web of Science ®Google Scholar
• A.K. Grincevićius, One limit distribution for a random walk on the line, Lith. Math. J. 15(4) (1975), pp. 580–589. doi: 10.1007/BF00969789
   Google Scholar
• F. Guillemin, P. Robert and B. Zwart, Aimd algorithms and exponential functionals, Ann. Appl. Probab. 14(1) (2004), pp. 90–117. doi: 10.1214/aoap/1075828048
   Web of Science ®Google Scholar
• H. Hennion and L. Hervé, Central limit theorems for iterated random Lipschitz mappings, Ann. Probab. 32(3A) (2004), pp. 1934–1984. doi: 10.1214/009117904000000469
   Web of Science ®Google Scholar
• A. Iksanov, Renewal Theory for Perturbed Random Walks and Similar Processes, Birkhäuser, Basel, 2016.
   Google Scholar
• H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131(1) (1973), pp. 207–248. doi: 10.1007/BF02392040
   Web of Science ®Google Scholar
• P. Kevei, A note on the Kesten–Grincevičius–Goldie theorem, Electron. Commun. Probab. 21 (2016), pp. 1–12. doi: 10.1214/16-ECP9
   Web of Science ®Google Scholar
• C. Klüppelberg, Subexponential distributions and characterizations of related classes, Probab. Theory Relat. Fields 82(2) (1989), pp. 259–269. doi: 10.1007/BF00354763
   Web of Science ®Google Scholar
• C. Klüppelberg, A. Lindner and R. Maller, Continuous time volatility modelling: Cogarch versus Ornstein–Uhlenbeck models. in From Stochastic Calculus to Mathematical Finance, Springer-Verlag, Berlin, 2006, pp. 393–419.
   Google Scholar
• J. Li and Q. Tang, Interplay of insurance and financial risks in a discrete-time model with strongly regular variation, Bernoulli 21(3) (2015), pp. 1800–1823. doi: 10.3150/14-BEJ625
   Web of Science ®Google Scholar
• M. Mirek, Heavy tail phenomenon and convergence to stable laws for iterated lipschitz maps, Probab. Theory Relat. Fields 151(3–4) (2011), pp. 705–734. doi: 10.1007/s00440-010-0312-9
   Web of Science ®Google Scholar
• V. Rivero, Tail asymptotics for exponential functionals of Lévy processes: the convolution equivalent case. in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 48, Institut Henri Poincaré, 2012, pp. 1081–1102.
   Google Scholar