Electronic Communications in Probability

Functional limit theorems for divergent perpetuities in the contractive case

Dariusz Buraczewski and Alexander Iksanov

Full-text: Open access

Abstract

Let $\big(M_k, Q_k\big)_{k\in\mathbb{N}}$ be independent copies of an $\mathbb{R}^2$-valued random vector. It is known that if $Y_n:=Q_1+M_1Q_2+\ldots+M_1\cdot\ldots\cdot M_{n-1}Q_n$ converges a.s. to a random variable $Y$, then the law of $Y$ satisfies the stochastic fixed-point equation $Y \overset{d}{=} Q_1+M_1Y$, where $(Q_1, M_1)$ is independent of $Y$. In the present paper we consider the situation when $|Y_n|$ diverges to $\infty$ in probability because $|Q_1|$ takes large values with high probability, whereas the multiplicative random walk with steps $M_k$'s tends to zero a.s. Under a regular variation assumption we show that $\log |Y_n|$, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the $J_1$-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the $J_1$-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.

Article information

Source
Electron. Commun. Probab. Volume 20 (2015), paper no. 10, 14 pp.

Dates
Accepted: 31 January 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320937

Digital Object Identifier
doi:10.1214/ECP.v20-3915

Mathematical Reviews number (MathSciNet)
MR3314645

Zentralblatt MATH identifier
1307.60026

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
extremal process functional limit theorem perpetuity random difference equation

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Buraczewski, Dariusz; Iksanov, Alexander. Functional limit theorems for divergent perpetuities in the contractive case. Electron. Commun. Probab. 20 (2015), paper no. 10, 14 pp. doi:10.1214/ECP.v20-3915. https://projecteuclid.org/euclid.ecp/1465320937


Export citation

References

  • Alsmeyer, Gerold; Iksanov, Alex; Roesler, Uwe. On distributional properties of perpetuities. J. Theoret. Probab. 22 (2009), no. 3, 666–682.
  • Babillot, Martine; Bougerol, Philippe; Elie, Laure. The random difference equation $X_ n=A_ nX_ {n-1}+B_ n$ in the critical case. Ann. Probab. 25 (1997), no. 1, 478–493.
  • Basu, Ranojoy; Roitershtein, Alexander. Divergent perpetuities modulated by regime switches. Stoch. Models 29 (2013), no. 2, 129–148.
  • Brofferio, Sara. How a centred random walk on the affine group goes to infinity. Ann. Inst. H. Poincare Probab. Statist. 39 (2003), no. 3, 371–384.
  • Brofferio, S. and Buraczewski D.: On unbounded invariant measures of stochastic dynamical systems. Ann. Probab. (2015+), to appear.
  • Buraczewski, Dariusz. On invariant measures of stochastic recursions in a critical case. Ann. Appl. Probab. 17 (2007), no. 4, 1245–1272.
  • Glynn, Peter W.; Whitt, Ward. Ordinary CLT and WLLN versions of $L=\lambda W$. Math. Oper. Res. 13 (1988), no. 4, 674–692.
  • Goldie, Charles M.; Maller, Ross A. Stability of perpetuities. Ann. Probab. 28 (2000), no. 3, 1195–1218.
  • Grincevicjus, A. K.: Limit theorems for products of random linear transformations on the line. Litovsk. Mat. Sb. 15, (1975), 61–77, 241.
  • Hitczenko, Paweł; Wesołowski, Jacek. Renorming divergent perpetuities. Bernoulli 17 (2011), no. 3, 880–894.
  • Iksanov, Alexander; Pilipenko, Andrey. On the maximum of a perturbed random walk. Statist. Probab. Lett. 92 (2014), 168–172.
  • Kellerer, H. G.: Ergodic behaviour of affine recursions I: criteria for recurrence and transience. Technical report, (1992), University of Munich, Germany. Available at http://www.mathematik.uni-muenchen.de
  • Pakes, Anthony G. Some properties of a random linear difference equation. Austral. J. Statist. 25 (1983), no. 2, 345–357.
  • Pruitt, William E. General one-sided laws of the iterated logarithm. Ann. Probab. 9 (1981), no. 1, 1–48.
  • Rachev, Svetlozar T.; Samorodnitsky, Gennady. Limit laws for a stochastic process and random recursion arising in probabilistic modelling. Adv. in Appl. Probab. 27 (1995), no. 1, 185–202.
  • Resnick, Sidney I. Extreme values, regular variation, and point processes. Applied Probability. A Series of the Applied Probability Trust, 4. Springer-Verlag, New York, 1987. xii+320 pp. ISBN: 0-387-96481-9
  • Resnick, Sidney I. Heavy-tail phenomena. Probabilistic and statistical modeling. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2007. xx+404 pp. ISBN: 978-0-387-24272-9; 0-387-24272-4
  • Vervaat, Wim. On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 (1979), no. 4, 750–783.
  • Zeevi, Assaf; Glynn, Peter W. Recurrence properties of autoregressive processes with super-heavy-tailed innovations. J. Appl. Probab. 41 (2004), no. 3, 639–653.