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doi: 10.3934/mfc.2022017
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New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach

1. 

School of Statistic and Data Science, Qufu Normal University, Qufu 273165, Shandong, China

2. 

Department of Mathematics, Jining University, Qufu 273199, Shandong, China

*Corresponding author: Feng Hu

Received  September 2021 Revised  February 2022 Early access June 2022

Fund Project: The third author is supported by National Science Foundation of China (No.11801307) and Natural Science Foundation of Shandong Province (No. ZR2021MA009)

Both Khinchin's law of large numbers (Khinchin's LLN) and Lindeberg's central limit theorem (Lindeberg's CLT) are fundamental results in probability theory. In this paper, we give the new proofs of these two theorems. A law of large numbers and a central limit theorem are proved for independent and non-identical distributed random variables. Indeed, these results include the Khinchin's LLN and Lindeberg's CLT. Our main tool is the viscosity solution theory of partial differential equation (PDE).

Citation: Xue Meng, Miaomiao Gao, Feng Hu. New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022017
References:
[1]

J. Bernoulli, ARS Conjectandi, Basileae: Thurnisiorum, Swiss, 1713.

[2]

P. Billingsley, Probability and Measure, Third edition, Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995.

[3]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[4]

R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.

[6]

W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc., 51 (1945), 800-832.  doi: 10.1090/S0002-9904-1945-08448-1.

[7]

H. Fischer, A History of the Central Limit Theorem From Classical to Modern Probability Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011. doi: 10.1007/978-0-387-87857-7.

[8]

S. T. Ho and L. H. Y. Chen, An ${L^p}$ bound for the remainder in a combinatorial central limit theorem, The Annals of Probability, 6 (1978), 231-249.  doi: 10.1214/aop/1176995570.

[9]

A. Y. Khintchine, Sur la lois des grands nombres, C.R. Acad. Sci., Paris, 188 (1929), 477-479. 

[10]

A. Y. Khintchine, Su una legge dei grandi numeri generalizzata, Giorn. Ist. Ital. Attuari, 7 (1936), 365-377. 

[11]

J. W. Lindeberg, Über das exponentialgesetz in der wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae, 16 (1920), 1-23. 

[12]

J. W. Lindeberg, Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, 15 (1922), 211-225.  doi: 10.1007/BF01494395.

[13]

J. W. Lindeberg, Über das Gauss'sche fehlergesetz, Skandinavisk Aktuarietid-skrift, 5 (1922), 217-234. 

[14]

M. Loeve, Probability Theory. I, Fourth edition, Graduate Texts in Mathematics, Vol. 45. Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9464-8.

[15]

S. G. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 4 (2019), Paper No. 4, 8 pp, arXiv: math/0702358. doi: 10.1186/s41546-019-0038-2.

[16]

S. G. Peng, A new central limit theorem under sublinear expectations, preprint, arXiv: 0803.2656.

[17]

S. G. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Probability Theory and Stochastic Modelling, 95. Springer, Berlin, 2019. doi: 10.1007/978-3-662-59903-7.

[18]

S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.

[19]

L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  doi: 10.1002/cpa.3160450202.

show all references

References:
[1]

J. Bernoulli, ARS Conjectandi, Basileae: Thurnisiorum, Swiss, 1713.

[2]

P. Billingsley, Probability and Measure, Third edition, Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995.

[3]

M. G. CrandallH. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[4]

R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.

[6]

W. Feller, The fundamental limit theorems in probability, Bull. Amer. Math. Soc., 51 (1945), 800-832.  doi: 10.1090/S0002-9904-1945-08448-1.

[7]

H. Fischer, A History of the Central Limit Theorem From Classical to Modern Probability Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, New York, 2011. doi: 10.1007/978-0-387-87857-7.

[8]

S. T. Ho and L. H. Y. Chen, An ${L^p}$ bound for the remainder in a combinatorial central limit theorem, The Annals of Probability, 6 (1978), 231-249.  doi: 10.1214/aop/1176995570.

[9]

A. Y. Khintchine, Sur la lois des grands nombres, C.R. Acad. Sci., Paris, 188 (1929), 477-479. 

[10]

A. Y. Khintchine, Su una legge dei grandi numeri generalizzata, Giorn. Ist. Ital. Attuari, 7 (1936), 365-377. 

[11]

J. W. Lindeberg, Über das exponentialgesetz in der wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae, 16 (1920), 1-23. 

[12]

J. W. Lindeberg, Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, 15 (1922), 211-225.  doi: 10.1007/BF01494395.

[13]

J. W. Lindeberg, Über das Gauss'sche fehlergesetz, Skandinavisk Aktuarietid-skrift, 5 (1922), 217-234. 

[14]

M. Loeve, Probability Theory. I, Fourth edition, Graduate Texts in Mathematics, Vol. 45. Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4684-9464-8.

[15]

S. G. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 4 (2019), Paper No. 4, 8 pp, arXiv: math/0702358. doi: 10.1186/s41546-019-0038-2.

[16]

S. G. Peng, A new central limit theorem under sublinear expectations, preprint, arXiv: 0803.2656.

[17]

S. G. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Probability Theory and Stochastic Modelling, 95. Springer, Berlin, 2019. doi: 10.1007/978-3-662-59903-7.

[18]

S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411.  doi: 10.1007/s11425-009-0121-8.

[19]

L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  doi: 10.1002/cpa.3160450202.

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