The Strong Law of Large Numbers (SLLN) is one of the basic limit theorems in probability. The SLLN is proven in the Etemadi form which only requires pairwise independence for a sequence of integrable, identically distributed random variables. A remarkable converse statement due to Kolmogorov is also included in this chapter. As applications, a problem about normal numbers and the speed of convergence of Monte Carlo methods are considered, and the solution to the classical needle problem of Buffon is given.

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Strong Law of Large Numbers

  • Hannah Geiss,
  • Stefan Geiss

摘要

The Strong Law of Large Numbers (SLLN) is one of the basic limit theorems in probability. The SLLN is proven in the Etemadi form which only requires pairwise independence for a sequence of integrable, identically distributed random variables. A remarkable converse statement due to Kolmogorov is also included in this chapter. As applications, a problem about normal numbers and the speed of convergence of Monte Carlo methods are considered, and the solution to the classical needle problem of Buffon is given.