## Autocorrelation function

For the $N$ generated pseudo-random number sequence $x_0,x_1,…,x_{N-1}$, let the shifted $k$ pieces be as follows,

$(x_0,x_k ),(x_1,x_{1+k} ),…,(x_{N-1},x_{N-1-k} )$

Autocorrelation function $r_{x y}^k$ is given by the following equation.

$r_{x y}^k=\frac{\sum _{i=0}^{N-1}\left(x_i-\overset{-}{x}\right)\left(x_{i+k}-\overset{-}{x}\right)}{\sum _{i=0}^{N-1}\left(x_i-\overset{-}{x}\right){}^2}$

where

$\overset{-}{x}=\frac{\sum _{i=0}^{N-1} x_i}{N}$

$x_{N+k}=x_{k}$

If it is assumed that the pseudo-random number sequence $x_0,x_1,…,x_{N-1}$ follows the same distribution independently, $r_{x y}^k$ with different $k$ approaches normal distribution $N\left(0,\frac{1}{N}\right)$ independently when $N\rightarrow \infty$.

References:
• K.Wakimoto, The knowledge of random numbers, Morikita Shuppan, 1970
• Statistics Section, Department of Social Sciences, College of Arts and Sciences, The University of Tokyo, Analysis of Scientific Data, University of Tokyo Press, 1992

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