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The Emergence of the Normal Distribution in Deterministic Chaotic Maps (2404.03808v1)
Published 4 Apr 2024 in physics.data-an, cond-mat.stat-mech, math-ph, math.MP, math.ST, and stat.TH
Abstract: The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The condition of independence, however, only holds in real systems as an approximation. To extend the theorem to more general situations, previous studies have derived a version of the Central Limit Theorem that also holds for variables that are not independent. Here, we present numerical results that characterize how convergence is attained when the variables being summed are deterministically related to one another by the recurrent application of an ergodic mapping. In all the explored cases, the convergence to the limit distribution is slower than for random sampling. Yet, the speed at which convergence is attained varies substantially from system to system, and these variations imply differences in the way information about the deterministic nature of the dynamics is progressively lost as the number of summands increases. Some of the identified factors in shaping the convergence process are the strength of mixing induced by the mapping and the shape of the marginal distribution of each variable, most particularly, the presence of divergences or fat tails.
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Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Araujo, A., Giné, E. The central limit theorem for real and Banach valued random variables. Wiley: New York, 1980. [7] Billingsley, P. Probability and Measure. Wiley: New York, 1995. [8] Burton, R.; Denker, M. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc. 1987, 302, 715. [9] Denker, M. The central limit theorem for dynamical systems. Banach Center Publ. 1989, 1, 33. [10] Liverani, C. Central limit theorem for deterministic systems. Pitman Res. Notes Math. Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Billingsley, P. Probability and Measure. Wiley: New York, 1995. [8] Burton, R.; Denker, M. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc. 1987, 302, 715. [9] Denker, M. The central limit theorem for dynamical systems. Banach Center Publ. 1989, 1, 33. [10] Liverani, C. Central limit theorem for deterministic systems. Pitman Res. Notes Math. Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Burton, R.; Denker, M. On the central limit theorem for dynamical systems. Trans. Amer. Math. Soc. 1987, 302, 715. [9] Denker, M. The central limit theorem for dynamical systems. Banach Center Publ. 1989, 1, 33. [10] Liverani, C. Central limit theorem for deterministic systems. Pitman Res. Notes Math. Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Denker, M. The central limit theorem for dynamical systems. Banach Center Publ. 1989, 1, 33. [10] Liverani, C. Central limit theorem for deterministic systems. Pitman Res. Notes Math. Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Liverani, C. Central limit theorem for deterministic systems. Pitman Res. Notes Math. Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. [25] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Theor. Prob. Appl. 2010, 54, 638. [26] Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997. Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. 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Ser. 1996, 362, 56. [11] Nicol, M; Török, A; Vaienti, S. Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Theor. Dyn. Syst. 2018, 38, 1127. [12] Kosloff, Z.; Volný, D. Local limit theorem in deterministic systems. Ann. Inst. Henri Poincaré Probab. Stat. 2022 58, 548. [13] Kosloff, Z.; Volný, D. Stable CLT for deterministic systems. arXiv:2211.03448 [math.DS] 2023. [14] Buzzi, J. Chaos and Ergodic Theory. In Meyers, R. (ed.) Encyclopedia of Complexity and Systems Science; Springer: New York, NY, 2009. [15] Wouters, J. Deviations from Gaussianity in deterministic discrete time dynamical systems. Chaos 2020, 30, 023117. [16] Cover, T. M.; Thomas, J. A. Elements of Information Theory. Wiley-Interscience: Hoboken, NJ, 2006. [17] Irwin, J. O. On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. 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On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s type II. Biometrika 1927, 19, 225. [18] Hall, P. The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 1927, 19, 240. [19] May, R. M. Simple mathematical models with very complicated dynamics. Nature 1976, 261, 459. [20] Tsuchiya, T., Yamagishi, D. The complete bifurcation diagram for the logistic map. Z. Naturforsch. 1997 52a, 513. [21] Jakobson, M. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 1981 81, 39. [22] Schroder, E. Über iterirte Functionen. Math. Ann. 1870, 3, 296. [23] Maritz, M. F. A note on exact solutions of the logistic map. Chaos 2020, 30, 033136. [24] Hazewinkel, M., ed. Encyclopaedia of Mathematics. Reidel: Dordrecht, 1988. 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- Gardiner, C. W. Handbook of Stochastic Methods. Springer: Berlin, 1997.