Statistical Independence in Probability, Analysis and Number Theory by Mark Kac

Author:Mark Kac
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2018-08-31T16:00:00+00:00

and our theorem can also be stated as follows:

If a sequence of distribution functions σn(ω) is such that for every real ξ

then

where

An attentive reader will notice a slight logical gap. If we are simply given a sequence of distribution functions σn(ω), the last formulation follows from the preceding one only if we can exhibit a sequence of functions fn(t), 0 ≤ t ≤ 1, such that

One can circumvent this step by repeating, in essence, the argument of § 3. But the construction of the functions fn(t) is exceedingly simple. In fact, we can simply take for fn(t) the inverse of σn(ω), with the understanding that the intervals of constancy of σn(ω) are reflected in discontinuities of fn(t) and discontinuities of σn(ω) in intervals of constancy of fn(t). We leave the details to the reader. The conclusion that (4.5) implies (4.6) is a special case of an important general theorem known as the continuity theorem for Fourier-Stieltjes transforms. This theorem can be stated as follows: If σn(ω) is a sequence of distribution functions such that for every real ξ

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