The Strong Law of Large Numbers

Summary

15.1 This section gives some fundamental definitions in the theory of probability, such as the definitions of a probability space and a random variable.

15.2 In this section the fundamental concept of independence is developed.

15.3 We give an example of singular function, which arises naturally from probability theory.

15.4 Given Ω, set Ω _n = Ω for all n ∈ N and Ω^N = Π _n Ω _n . The one-sided shift transformation v on Ω^N is defined by (x _n ) _{n ≥1} ↦ (x _{n +i}) _{n ≥i}. The probability of a v-invariant event is either 0 or 1 (Proposition 15.4.2) and v is μ-ergodic (Proposition 15.4.3)—see Section 11.4. Then, using Birkhoff’s ergodic theorem, we prove the strong law of large numbers (Theorem 15.4.1).

15.5 A number x ≠ m/bⁿ is said to be completely normal if, for every integer k and every k-tuple (u₁,…, u _k ) of base-b digits, the k-tuple appears in the base-b expansion of x with asymptotic relative frequency 1/b^k. Almost every number (for Lebesgue measure) in [0, 1] is completely normal (Proposition 15.5.2).