IX - Convergence of Distribution Functions and Characteristic Functions

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This chapter focuses on some important topics in probability theory. It discusses the convergence of distribution functions and characteristic functions. The limit function is not uniquely determined, but the two nondecreasing functions that are identical on a dense set are identical everywhere if both functions are known to be left-continuous everywhere or if both are standardized. The chapter discusses the convergence of distribution functions to a distribution function in terms of characteristic functions. It presents a supposition that a sequence Fn(x) ∈ K. If Fn(x) converges weakly to a nondecreasing function F(x) in (—π, π), then Fn(x) (—∞ <x < ∞) is considered to converge weakly to the distribution function that is identical with F(x) in (—π, π) and is 0 for x ≦ 0; 1 for x ≧ 1.