Law of Large Numbers – Definition

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Law of Large Numbers Definition

This is a concept of probability denoting that the prevalence of events with a similar chance of occurrence eventually level out over a number of enough trials. As the number of instances continues to increase, the actual ratio of outcomes intersects at the theoretical or expected ratio of outcomes.

This law outlines that the larger the sample size, the closer the mean is to the average of the total population. It was developed to address the fact a rapidly growing large entity can’t keep the stride over a long time.

A Little More on the Law of Large Numbers

For example, assume a coin is tossed a million times, it is safe to say that the approximately half the tosses will be heads and half will be tails. Therefore, this ratio will be almost 1:1. When the coin is tossed twenty times, the ratio will be differing and might be 3:7 or any other. This law is sometimes misapplied to situations with very few experiments, and it results to an error in logic referred to as the gambler’s fallacy.

The law of large numbers, also known as the law of averages, is a theory used to explain the results arising from conducting a similar experiment many times. It states that the statistical probability of a sample with a specific value edges closer to the statistical probability of a collection of samples in the universe as the sample increases. Political polls use this method, and that’s why they become more accurate the larger their sample size gets.

Back in July 2015, Wal-Mart Stores Inc. generated an income of $485.5 billion, and Amazon made $95.8 billion. For them to increase their income by 50% based on this numbers, Walmart would require a total of $242.8 billion, and Amazon would need just $47.9. According to the law of large numbers, this increase would be more difficult for Walmart than for Amazon. This law assures stable long-term results for the averages of various random events and is therefore very essential.

References for Law of Large Numbers

Academic Research on Law of Large Numbers

Complete convergence and the law of large numbers, Hsu, P. L., & Robbins, H. (1947). Proceedings of the National Academy of Sciences of the United States of America, 33(2), 25. This paper defines and distinguishes the standard terms in the theory of probability which include the probability space, a real-valued P-measurable function X = X among others.

Convergence rates in the law of large numbers, Baum, L. E., & Katz, M. (1965). Transactions of the American Mathematical Society, 120(1), 108-123. This paper gives attention to the sequences of independent and similarly distributed random variables and presents various propositions and examples for the case of independent but differently distributed random variables.

A law of large numbers for large economies, Uhlig, H. (1996). Economic Theory, 8(1), 41-50. This article presents the law of large numbers by interpreting the integral as a Pettis-integral. It also provides proof using the calculation of variances and shows that the measurability problem identified by Judd in 1985 is avoidable through convergence in the mean square instead of convergence almost everywhere.

The strong law of large numbers for u-statistics., Hoeffding, W. (1961). North Carolina State University. Dept. of Statistics. This paper examines the strong law of large numbers for the class of U-statistics under the moment condition leading to a generalization of this law.

Consistency in nonlinear econometric models: A generic uniform law of large numbers, Andrews, D. W. (1987). Econometrica: Journal of the Econometric Society, 1465-1471.  This article aims at providing a universal uniform law of large numbers which is essentially general and includes the majority of applications of the uniform law of large numbers in the literature of nonlinear econometrics.

Local convergence of martingales and the law of large numbers. Chow, Y. S. (1965). The Annals of Mathematical Statistics, 36(2), 552-558. This paper generalizes Neveu’s results from when he proved the existence of a new submartingale convergence theorem.

A note on the strong law of large numbers for positively dependent random variables, Birkel, T. (1988). Statistics & Probability Letters, 7(1), 17-20. This article presents strong laws of large numbers for sequences of random variables that are associated or pairwise positive quadrant dependent.

The exact law of large numbers via Fubini extension and characterization of insurable risks, Sun, Y. (2006). Journal of Economic Theory, 126(1), 31-69. This study introduces simple measure-theoretic methods to this framework in order to get several versions of this law and their converses stochastic processes or a continuum of random variables while also proposing a Fubini extension as a probability space which expands the probability space while retaining the Fubini property.

A general approach to the strong law of large numbers, Fazekas, I., & Klesov, O. (2001). Theory of Probability & Its Applications, 45(3), 436-449. This paper studies a general method based on abstract Hájek–Rényi type maximal inequalities to obtain strong laws of large numbers.

A strong law of large numbers for capacities, Maccheroni, F., & Marinacci, M. (2005). The Annals of Probability, 33(3), 1171-1178. In this article, a wholly monotone capacity on a polish space and a sequence of bounded p.i.i.d random variables are considered.

A law of large numbers: bidding and compulsory competitive tendering for refuse collection contracts, Gomez-Lobo, A., & Szymanski, S. (2001). Review of Industrial Organization, 18(1), 105-113. This is an investigation of the relationship existing between costs and the number of bidders for the refuse collection contracts of UK local authorities.

A law of large numbers in the theory of consumer’s choice under uncertainty, Yaari, M. E. (1976). Journal of Economic theory, 12(2), 202-217. This paper examines a hypothesis that states that under certainty, consumption is dependent on income just to the extent that income affects lifetime wealth.

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