Electronic Journal of Probability

Functional Erdős-Rényi law of large numbers for nonconventional sums under weak dependence

Yuri Kifer

Full-text: Open access


We obtain a functional Erdős–Rényi law of large numbers for “nonconventional” sums of the form $\Sigma _n=\sum _{m=1}^n F(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of exponentially fast $\psi $-mixing random vectors and $F$ is a Borel vector function extending in several directions [18] where only i.i.d. random variables $X_1,X_2,...$ were considered.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 23, 17 pp.

Received: 5 August 2016
Accepted: 16 February 2017
First available in Project Euclid: 1 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60F10, 60F17, 37D20

laws of large numbers large deviations nonconventional sums hyperbolic diffeomorphisms Markov chains

Creative Commons Attribution 4.0 International License.


Kifer, Yuri. Functional Erdős-Rényi law of large numbers for nonconventional sums under weak dependence. Electron. J. Probab. 22 (2017), paper no. 23, 17 pp. doi:10.1214/17-EJP39. https://projecteuclid.org/euclid.ejp/1488337348

Export citation


  • [1] K.A. Borovkov, A functional form of the Erdős-Rényi law of large numbers, Theory Probab. Appl. 35 (1991), 762–766.
  • [2] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer–Verlag, Berlin, 1975.
  • [3] R.C. Bradley, On the $\varphi $-mixing condition for stationary random sequences, Duke Math. J. 47 (1980), 421–433.
  • [4] R.C. Bradley, Introduction to Strong Mixing Conditions, Kendrick Press, Heber City, 2007.
  • [5] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin, 2010.
  • [6] M. Denker and Z. Kabluchko, An Erdös-Rényi law for mixing processes, Probab. Math. Stat. 27 (2007), 139–149.
  • [7] M.D. Donsker and S.R.S. Varadhan, Asymptotic evaluation of certain Markov processes expectations for large time. I, Comm. Pure Appl. Math. 28 (1975), 1–47.
  • [8] P. Erdös and A. Rényi, On a new law of large numbers, J. Anal. Math. 23 (1970), 103–111.
  • [9] M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, 3d ed., (2012), Springer–Verlag, New York.
  • [10] H.Furstenberg, Nonconventional ergodic averages, Proc. Symp. Pure Math. 50 (1990), 43–56.
  • [11] Y. Guivarc’h and J. Hardy, Théorm̀es limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H.Poincaré (Prob. Statist.) 24 (1988), 73–98.
  • [12] I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters–Noordhoff, Groningen (1971).
  • [13] Yu. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc., 321 (1990), 505–524.
  • [14] Yu. Kifer, Averaging in dynamical systems and large deviations, Invent. Math., 110 (1992), 337–370.
  • [15] Yu. Kifer, Optimal stopping and strong approximation theorems, Stochastics 79 (2007), 253–273.
  • [16] Yu. Kifer, Large Deviations and Adiabatic Transitions for Dynamical Systems and Markov Processes in Fully Coupled Averaging, Memoirs of AMS 944, AMS, Providence R.I. (2009).
  • [17] Yu. Kifer, Nonconventional limit theorems, Probab. Th. Rel. Fields 148 (2010), 71–106.
  • [18] Yu. Kifer, An Erdős-Rényi law for nonconventional sums, Electron. Commun. Probab. 20 (2015), no.83; Erratum: 21 (2016), no.33.
  • [19] Yu.Kifer and S.R.S Varadhan, Nonconventional limit theorems in discrete and continuous time via martingales, Ann. Probab. 42 (2014), 649–688.
  • [20] Yu.Kifer and S.R.S Varadhan, Nonconventional large deviations theorem, Probab. Th. Rel. Fields, 158 (2014), 197–224.
  • [21] S.V. Nagaev, Some limit theorems for stationary Markov chains, Theory Probab. Appl. 2 (1957), 378–406.
  • [22] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188 (1990).