# LAW OF LARGE NUMBERS

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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