Electronic Journal of Probability

On the rate of convergence in the Kesten renewal theorem

Dariusz Buraczewski, Ewa Damek, and Tomasz Przebinda

Full-text: Open access


We consider the stochastic recursion $X_{n+1} = M_{n+1}X_n + Q_{n+1}$ on $\mathbb{R}^d$,

where ($M_n, Q_n$) are i.i.d. random variables such that $Q_n$ are translations, $M_n$ are similarities of the Euclidean space $\mathbb{R}^d$. Under some standard assumptions the sequence $X_n$ converges to a random variable $R$ and the law $\nu$ of $R$ is the unique stationary measure of the process. Moreover,

the weak limit of properly dilated measure $\nu$ exists, defining thus a homogeneous tail measure $\Lambda$. In this paper we study the rate of convergence of dilations of $\nu$ to $\Lambda$


In particular in the one dimensional setting, when $(M_n,Q_n) \in \mathbb{R}^+\times \mathbb{R}$, $\mathbb{E} M_n^{\alpha }=1$ and $X_n\in \mathbb{R}$, the Kesten renewal theorem says that $t^\alpha\mathbb{P}[|R|>t]$ converges to some strictly positive constant $C_+$. Our main result says that $$\big|t^\alpha\mathbb{P}[|R|>t]-C_+\big|\le C (\log t)^{-\sigma},$$ for some $\sigma>0$ and large $t$. It generalizes the previous one by Goldie.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 22, 35 pp.

Accepted: 3 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}

stochastic recursions random difference equation stationary measure rate of convergence renewal theorem spectral gap

This work is licensed under aCreative Commons Attribution 3.0 License.


Buraczewski, Dariusz; Damek, Ewa; Przebinda, Tomasz. On the rate of convergence in the Kesten renewal theorem. Electron. J. Probab. 20 (2015), paper no. 22, 35 pp. doi:10.1214/EJP.v20-3708. https://projecteuclid.org/euclid.ejp/1465067128

Export citation


  • Doob, J. L. Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statistics 20, (1949). 393–403.
  • Gnedenko, B. V.; Kolmogorov, A. N. Limit distributions for sums of independent random variables. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. ix+264 pp.
  • Itô, Kiyosi. Multiple Wiener integral. J. Math. Soc. Japan 3, (1951). 157–169.
  • Lévy, Paul. Sur certains processus stochastiques homogènes. (French) Compositio Math. 7, (1939). 283–339.
  • Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, ARXIVmath.DG/0211159
  • Smirnov, S. and Schramm, O.: On the scaling limits of planar percolation, ARXIV1101.5820
  • Alsmeyer, Gerold. On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theoret. Probab. 16 (2003), no. 1, 217–247.
  • Alsmeyer, G: On the stationary tail index of iterated random Lipschitz functions, ARXIV1409.2663
  • Alsmeyer, Gerold; Iksanov, Alex; Rösler, Uwe. On distributional properties of perpetuities. J. Theoret. Probab. 22 (2009), no. 3, 666–682.
  • Alsmeyer, Gerold; Mentemeier, Sebastian. Tail behaviour of stationary solutions of random difference equations: the case of regular matrices. J. Difference Equ. Appl. 18 (2012), no. 8, 1305–1332.
  • Arnold, Ludwig; Crauel, Hans. Iterated function systems and multiplicative ergodic theory. Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990), 283–305, Progr. Probab., 27, Birkhäuser Boston, Boston, MA, 1992.
  • 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.
  • Bartkiewicz, Katarzyna; Jakubowski, Adam; Mikosch, Thomas; Wintenberger, Olivier. Stable limits for sums of dependent infinite variance random variables. Probab. Theory Related Fields 150 (2011), no. 3-4, 337–372.
  • Bougerol, Philippe; Picard, Nico. Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), no. 4, 1714–1730.
  • Bourgain, Jean; Gamburd, Alex. On the spectral gap for finitely-generated subgroups of $\rm SU(2)$. Invent. Math. 171 (2008), no. 1, 83–121.
  • Bourgain, J.; Gamburd, A. A spectral gap theorem in ${\rm SU}(d)$. J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1455–1511.
  • Brandt, Andreas. The stochastic equation $Y_ {n+1}=A_ nY_ n+B_ n$ with stationary coefficients. Adv. in Appl. Probab. 18 (1986), no. 1, 211–220.
  • Buraczewski, Dariusz; Damek, Ewa; Guivarc'h, Yves. Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Related Fields 148 (2010), no. 3-4, 333–402.
  • Buraczewski, Dariusz; Damek, Ewa; Guivarc'h, Yves; Hulanicki, Andrzej; Urban, Roman. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Related Fields 145 (2009), no. 3-4, 385–420.
  • Buraczewski, Dariusz; Damek, Ewa; Mirek, Mariusz. Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems. Stochastic Process. Appl. 122 (2012), no. 1, 42–67.
  • Carlsson, Hasse. Remainder term estimates of the renewal function. Ann. Probab. 11 (1983), no. 1, 143–157.
  • Collamore, Jeffrey F.; Vidyashankar, Anand N. Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stochastic Process. Appl. 123 (2013), no. 9, 3378–3429.
  • de Saporta, Benoéte; Guivarc'h, Yves; Le Page, Emile. On the multidimensional stochastic equation $Y_ {n+1}=A_ nY_ n+B_ n$. C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 499–502.
  • Diaconis, Persi; Freedman, David. Iterated random functions. SIAM Rev. 41 (1999), no. 1, 45–76.
  • Diaconis, Persi; Saloff-Coste, Laurent. Bounds for Kac's master equation. Comm. Math. Phys. 209 (2000), no. 3, 729–755.
  • Dixmier, Jacques. $C^*$-algebras. Translated from the French by Francis Jellett. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xiii+492 pp. ISBN: 0-7204-0762-1
  • Duflo, Marie. Random iterative models. Translated from the 1990 French original by Stephen S. Wilson and revised by the author. Applications of Mathematics (New York), 34. Springer-Verlag, Berlin, 1997. xviii+385 pp. ISBN: 3-540-57100-0
  • Elton, John H. A multiplicative ergodic theorem for Lipschitz maps. Stochastic Process. Appl. 34 (1990), no. 1, 39–47.
  • Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier. A probabilistic representation of constants in Kesten's renewal theorem. Probab. Theory Related Fields 144 (2009), no. 3-4, 581–613.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • Frennemo, Lennart. On general Tauberian remainder theorems. Math. Scand. 17 1965 77–88.
  • Furstenberg, H.; Kesten, H. Products of random matrices. Ann. Math. Statist. 31 1960 457–469.
  • Goldie, Charles M. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 (1991), no. 1, 126–166.
  • Goldie, Charles M.; Maller, Ross A. Stability of perpetuities. Ann. Probab. 28 (2000), no. 3, 1195–1218.
  • Guivarc'h, Yves. Heavy tail properties of stationary solutions of multidimensional stochastic recursions. Dynamics & stochastics, 85–99, IMS Lecture Notes Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, 2006.
  • Guivarc'h, Y. and Le Page, E.: Spectral gap properties and asymptotics of stationary measures for affine random walks, ARXIV1204.6004
  • Guivarc'h, Y. and Le Page, E.: On the homogeneity at infinity of the stationary probability for an affine random walk, HAL archives-ouvertes ID: hal-00868944.
  • Helgason, Sigurdur. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xv+628 pp. ISBN: 0-12-338460-5
  • Hennion, Hubert; Hervé, Loíc. Central limit theorems for iterated random Lipschitz mappings. Ann. Probab. 32 (2004), no. 3A, 1934–1984.
  • Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis. Springer Verlag, (1963).
  • Janvresse, Elise. Spectral gap for Kac's model of Boltzmann equation. Ann. Probab. 29 (2001), no. 1, 288–304.
  • Kesten, Harry. Random difference equations and renewal theory for products of random matrices. Acta Math. 131 (1973), 207–248.
  • Klöppelberg, Claudia; Pergamenchtchikov, Serguei. The tail of the stationary distribution of a random coefficient ${\rm AR}(q)$ model. Ann. Appl. Probab. 14 (2004), no. 2, 971–1005.
  • Le Page, É. Théorèmes de renouvellement pour les produits de matrices aléatoires. équations aux différences aléatoires. (French) [Renewal theorems for products of random matrices. Random difference equations] Séminaires de probabilités Rennes 1983, 116 pp., Publ. Sém. Math., Univ. Rennes I, Rennes, 1983.
  • Lubotzky, A.; Phillips, R.; Sarnak, P. Hecke operators and distributing points on the sphere. I. Frontiers of the mathematical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149–S186.
  • Lubotzky, A.; Phillips, R.; Sarnak, P. Hecke operators and distributing points on $S^ 2$. II. Comm. Pure Appl. Math. 40 (1987), no. 4, 401–420.
  • Mirek, Mariusz. Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Related Fields 151 (2011), no. 3-4, 705–734.
  • Oh, Hee. The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group. Probability in mathematics. Israel J. Math. 149 (2005), 301–316.
  • 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.
  • Rosenthal, Jeffrey S. Random rotations: characters and random walks on ${\rm SO}(N)$. Ann. Probab. 22 (1994), no. 1, 398–423.
  • Rudin, Walter. Principles of mathematical analysis. Second edition McGraw-Hill Book Co., New York 1964 ix+270 pp.
  • Stone, Charles. On absolutely continuous components and renewal theory. Ann. Math. Statist. 37 1966 271–275.
  • Porod, Ursula. The cut-off phenomenon for random reflections. Ann. Probab. 24 (1996), no. 1, 74–96.
  • Porod, U. The cut-off phenomenon for random reflections. II. Complex and quaternionic cases. Probab. Theory Related Fields 104 (1996), no. 2, 181–209.
  • 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.
  • Zolotarev, V. M. Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces. (Russian) Teor. Verojatnost. i Primenen. 21 (1976), no. 4, 741–758.
  • Zolotarev, V. M. Ideal metrics in the problem of approximating the distributions of sums of independent random variables. (Russian) Teor. Verojatnost. i Primenen. 22 (1977), no. 3, 449–465.