Convergence and the Law of Large Numbers
Marcelo Sampaio de Alencar, Raphael T. Alencar · River Publishers eBooks · 2023
Number theory is another example, especially regarding the convergence limits of sequences. Weierstrass was the first to recognize the dichotomy between point-wise convergence and uniform convergence, and to show that the limit of the integral of a sequence of functions was not always equal to the integral of the limit of that sequence, at least for the Riemann integral. The convergence of a sequence of random variables for a given limit is conceptually important in the theory of probability, with applications in statistics and stochastic processes. Many mathematicians contributed to the refinement of the deduction, including Tchebyshev, Markov, Borel, and Kolmogorov. The research led to two forms of the law of large numbers, namely the weak law and the strong law, that define different modes to represent the convergence of probability observed for real life probability. The central limit theorem is a fundamental result of statistics, for it states that when the sample size becomes larger.