MathDB

Limit distribution of a sum of sines

Source: Miklós Schweitzer 2014, problem 11

December 23, 2014

trigonometryfunctionprobability and stats

Problem Statement

Let

U

be a random variable that is uniformly distributed on the interval

[0,1]

, and let

S_n= 2\sum_{k=1}^n \sin(2kU\pi).

Show that, as

n\to \infty

, the limit distribution of

S_n

is the Cauchy distribution with density function

f(x)=\frac1{\pi(1+x^2)}