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May  2020, 40(5): 2827-2873. doi: 10.3934/dcds.2020151

A functional CLT for nonconventional polynomial arrays

Yeor Hafouta ,

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: Yeor Hafouta

Received  July 2019 Revised  December 2019 Published  March 2020

In this paper we will prove a functional central limit theorem (CLT) for random functions of the form
$ {\mathcal S}_N(t) = N^{-\frac12}\sum\limits_{n = 1}^{[Nt]} F(\xi_{q_1(n, N)}, \xi_{q_2(n, N)}, ..., \xi_{q_\ell(n, N)}) $
where the
$ q_i $
's are certain type of bivariate polynomials,
$ F = F(x_1, ..., x_\ell) $
 is a locally Hölder continuous function and the sequence of random variables
$ \{\xi_n\} $
 satisfies some mixing and moment conditions. This paper continues the line of research started in [15] and [17], and it is a generalization of the results in [9] and Chapter 3 of [11]. We will also prove a strong law of large numbers (SLLN) for the averages
$ N^{-\frac12} {\mathcal S}_N(1) $
 which extends the results from the beginning of Chapter 3 of [11] to general bivariate polynomial functions
$ q_i $
. Our results hold true for sequences
$ \{\xi_n\} $
 generated by a wide class of Markov chains and dynamical systems. As an application we obtain functional CLT's for expressions of the form
$ N^{-\frac12}M([Nt]) $
, where
$ M(N) $
 counts the number of multiple recurrence of the sequence
$ \{\xi_n\} $
 to certain sets
$ A_1, ..., A_\ell $
 which occur at the times
$ q_1(n, N), ..., q_\ell(n, N) $
, as well as SLLN's for these
$ M(N) $
's. One of the simplest examples is when
$ \xi_n $
 is
$ n $
-the digit of a random
$ m $
-base or continued fraction expansion, and each
$ A_i $
 is singleton (i.e. it represent one possible value of a digit).
Keywords: Functional central limit theorems, nonconventional setup, ergodic theorems, Stein's method, Markov chains, dynamical systems.
Mathematics Subject Classification: Primary: 60F17, 60F05, 60J05; Secondary: 37D99.
Citation: Yeor Hafouta. A functional CLT for nonconventional polynomial arrays. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2827-2873. doi: 10.3934/dcds.2020151
References:
[1]

A. D. Barbour, Stein's Method for diffusion approximations, Probab. Th. Rel. Fields, 84 (1990), 297-322.  doi: 10.1007/BF01197887.  Google Scholar

[2]

A. D. Barbour and S. Janson, A functional combinatorial central limit theorem, Electron. J. Probab., 14 (2009), 2352-2370.  doi: 10.1214/EJP.v14-709.  Google Scholar

[3]

V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.  Google Scholar

[5] R. C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007.   Google Scholar
[6]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[7]

H. Furstenberg, Nonconventional ergodic averages, The Legacy of John von Neumann, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 50 (1990), 43-56.  doi: 10.1090/pspum/050/1067751.  Google Scholar

[8]

Y. Hafouta and Y. Kifer, Berry-Esseen type estimates for nonconventional sums, Stoch. Proc. Appl., 126 (2016), 2430-2464.  doi: 10.1016/j.spa.2016.02.006.  Google Scholar

[9]

Y. Hafouta and Y. Kifer, Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591.  doi: 10.1080/17442508.2016.1267181.  Google Scholar

[10]

Y. Hafouta, Stein's method for nonconventional sums, Electron. Commun. Probab., 23 (2018), 14 pp. doi: 10.1214/18-ECP142.  Google Scholar

[11]

Y. Hafouta and Y. Kifer, Nonconventional Limit Theorems and Random Dynamics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10849.  Google Scholar

[12]

Y. Hafouta, Nonconventional moderate deviations theorems and exponential concentration inequalities, Ann. Inst. H. Poincaré Probab. Statist., 56 (2020), 428–448, arXiv: 1805.00849. doi: 10.1214/19-AIHP967.  Google Scholar

[13] P. Hall and C. C. Hyde, Martingale Central Limit Theory and Its Application, Academic Press, Inc., New York-London, 1980.   Google Scholar
[14]

N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations, Nonlinearity, 27 (2014), 1323-1349.  doi: 10.1088/0951-7715/27/6/1323.  Google Scholar

[15]

Y. Kifer, Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106.  doi: 10.1007/s00440-009-0223-9.  Google Scholar

[16]

Y. Kifer, A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets, Stoch. Dyn., 12 (2012), 1150023, 21 pp. doi: 10.1142/S0219493711500237.  Google Scholar

[17]

Y. Kifer and S. R. S. Varadhan, Nonconventional limit theorems in discrete and continuous time via martingales, Ann. Probab., 42 (2014), 649-688.  doi: 10.1214/12-AOP796.  Google Scholar

[18]

Y. Kifer, Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem, Discrete Contin. Dyn. Syst., 38 (2018), 2687-2716.  doi: 10.3934/dcds.2018113.  Google Scholar

[19]

V. Maume-Deschamps, Projective metrics and mixing properties on towers, Trans. Amer. Math. Soc., 353 (2001), 3371-3389.  doi: 10.1090/S0002-9947-01-02786-6.  Google Scholar

[20]

I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146.  doi: 10.1007/s00220-005-1407-5.  Google Scholar

[21]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64-94.  doi: 10.1215/ijm/1255631807.  Google Scholar

[22]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[23]

L.-S. Young, Recurrence time and rate of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

show all references

References:
[1]

A. D. Barbour, Stein's Method for diffusion approximations, Probab. Th. Rel. Fields, 84 (1990), 297-322.  doi: 10.1007/BF01197887.  Google Scholar

[2]

A. D. Barbour and S. Janson, A functional combinatorial central limit theorem, Electron. J. Probab., 14 (2009), 2352-2370.  doi: 10.1214/EJP.v14-709.  Google Scholar

[3]

V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.  doi: 10.1017/S0143385700004090.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.  Google Scholar

[5] R. C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007.   Google Scholar
[6]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[7]

H. Furstenberg, Nonconventional ergodic averages, The Legacy of John von Neumann, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 50 (1990), 43-56.  doi: 10.1090/pspum/050/1067751.  Google Scholar

[8]

Y. Hafouta and Y. Kifer, Berry-Esseen type estimates for nonconventional sums, Stoch. Proc. Appl., 126 (2016), 2430-2464.  doi: 10.1016/j.spa.2016.02.006.  Google Scholar

[9]

Y. Hafouta and Y. Kifer, Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591.  doi: 10.1080/17442508.2016.1267181.  Google Scholar

[10]

Y. Hafouta, Stein's method for nonconventional sums, Electron. Commun. Probab., 23 (2018), 14 pp. doi: 10.1214/18-ECP142.  Google Scholar

[11]

Y. Hafouta and Y. Kifer, Nonconventional Limit Theorems and Random Dynamics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. doi: 10.1142/10849.  Google Scholar

[12]

Y. Hafouta, Nonconventional moderate deviations theorems and exponential concentration inequalities, Ann. Inst. H. Poincaré Probab. Statist., 56 (2020), 428–448, arXiv: 1805.00849. doi: 10.1214/19-AIHP967.  Google Scholar

[13] P. Hall and C. C. Hyde, Martingale Central Limit Theory and Its Application, Academic Press, Inc., New York-London, 1980.   Google Scholar
[14]

N. T. A. Haydn and Y. Psiloyenis, Return times distribution for Markov towers with decay of correlations, Nonlinearity, 27 (2014), 1323-1349.  doi: 10.1088/0951-7715/27/6/1323.  Google Scholar

[15]

Y. Kifer, Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106.  doi: 10.1007/s00440-009-0223-9.  Google Scholar

[16]

Y. Kifer, A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets, Stoch. Dyn., 12 (2012), 1150023, 21 pp. doi: 10.1142/S0219493711500237.  Google Scholar

[17]

Y. Kifer and S. R. S. Varadhan, Nonconventional limit theorems in discrete and continuous time via martingales, Ann. Probab., 42 (2014), 649-688.  doi: 10.1214/12-AOP796.  Google Scholar

[18]

Y. Kifer, Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem, Discrete Contin. Dyn. Syst., 38 (2018), 2687-2716.  doi: 10.3934/dcds.2018113.  Google Scholar

[19]

V. Maume-Deschamps, Projective metrics and mixing properties on towers, Trans. Amer. Math. Soc., 353 (2001), 3371-3389.  doi: 10.1090/S0002-9947-01-02786-6.  Google Scholar

[20]

I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146.  doi: 10.1007/s00220-005-1407-5.  Google Scholar

[21]

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64-94.  doi: 10.1215/ijm/1255631807.  Google Scholar

[22]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

[23]

L.-S. Young, Recurrence time and rate of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

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