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11

Part of 2014 Miklós Schweitzer

Problems(1)

Limit distribution of a sum of sines

Source: Miklós Schweitzer 2014, problem 11

12/23/2014
Let UU be a random variable that is uniformly distributed on the interval [0,1][0,1], and let Sn=2∑k=1nsin⁡(2kUπ).S_n= 2\sum_{k=1}^n \sin(2kU\pi). Show that, as n→∞n\to \infty, the limit distribution of SnS_n is the Cauchy distribution with density function f(x)=1π(1+x2)f(x)=\frac1{\pi(1+x^2)}.
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