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A functional CLT for nonconventional polynomial arrays
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
$ {\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)}) $ |
$ q_i $ |
$ F = F(x_1, ..., x_\ell) $ |
$ \{\xi_n\} $ |
$ N^{-\frac12} {\mathcal S}_N(1) $ |
$ q_i $ |
$ \{\xi_n\} $ |
$ N^{-\frac12}M([Nt]) $ |
$ M(N) $ |
$ \{\xi_n\} $ |
$ A_1, ..., A_\ell $ |
$ q_1(n, N), ..., q_\ell(n, N) $ |
$ M(N) $ |
$ \xi_n $ |
$ n $ |
$ m $ |
$ A_i $ |
References:
[1] |
A. D. Barbour,
Stein's Method for diffusion approximations, Probab. Th. Rel. Fields, 84 (1990), 297-322.
doi: 10.1007/BF01197887. |
[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. |
[3] |
V. Bergelson,
Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.
doi: 10.1017/S0143385700004090. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. |
[5] |
R. C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007.
![]() |
[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. |
[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. |
[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. |
[9] |
Y. Hafouta and Y. Kifer,
Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591.
doi: 10.1080/17442508.2016.1267181. |
[10] |
Y. Hafouta, Stein's method for nonconventional sums, Electron. Commun. Probab., 23 (2018), 14 pp.
doi: 10.1214/18-ECP142. |
[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. |
[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. |
[13] |
P. Hall and C. C. Hyde, Martingale Central Limit Theory and Its Application, Academic Press, Inc., New York-London, 1980.
![]() ![]() |
[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. |
[15] |
Y. Kifer,
Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106.
doi: 10.1007/s00440-009-0223-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
L.-S. Young,
Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.
doi: 10.2307/120960. |
[23] |
L.-S. Young,
Recurrence time and rate of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
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. |
[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. |
[3] |
V. Bergelson,
Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349.
doi: 10.1017/S0143385700004090. |
[4] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. |
[5] |
R. C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007.
![]() |
[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. |
[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. |
[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. |
[9] |
Y. Hafouta and Y. Kifer,
Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591.
doi: 10.1080/17442508.2016.1267181. |
[10] |
Y. Hafouta, Stein's method for nonconventional sums, Electron. Commun. Probab., 23 (2018), 14 pp.
doi: 10.1214/18-ECP142. |
[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. |
[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. |
[13] |
P. Hall and C. C. Hyde, Martingale Central Limit Theory and Its Application, Academic Press, Inc., New York-London, 1980.
![]() ![]() |
[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. |
[15] |
Y. Kifer,
Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106.
doi: 10.1007/s00440-009-0223-9. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
L.-S. Young,
Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.
doi: 10.2307/120960. |
[23] |
L.-S. Young,
Recurrence time and rate of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
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