The law of large numbers is a theorem in probability theory that describes the average of a large number of random variables. The law states that the average of a large number of random variables converges to the expected value in some sense under certain conditions. In this chapter, we introduce some laws of large numbers.

16.1 Basic Concepts and Facts

Definition 16.1 (Almost Surely Convergence). Let X, X1, X2,… be random variables on a probability space (Ω, , P). The sequence {Xn}n≥1 is said to converge to X almost surely if and only if it converges to X almost everywhere; that is, there is a set A such that XnX on A and P(Ac) = P(Ω\A) = 0.

Definition 16.2 (Convergence in Probability). Let X1, X2,…, be a sequence of random variables on (Ω, , P). The sequence Xi is said to be convergent to a random variable X on (Ω, , P) if for every > 0, we have

Definition ...

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