np linalg cholesky sigma randn ( ,nplt scatter ([ [ ,:],[ [ ,:]alpha = figure realizations of bivariate normal distributions with means zerovariances and correlation coefficients (leftand (rightin some casesthe covariance matrix has special structure which can be exploited to create even faster generation algorithmsas illustrated in the following example example (simulating normal vectors in ( timesuppose that the random vector [ xn ]represents the values at times kdk of zeromean gaussian process ( ( ) that is weakly stationarymeaning that cov( ( ) ( )depends only on - then clearly the covariance matrix of xsay an is symmetric toeplitz matrix suppose for simplicity that var ( then the covariance matrix is in fact correlation matrixand will have the following structurea an- an- - an : an- an- an- using the levinson-durbin algorithm we can compute lower diagonal matrix ln and diagonal matrix dn in ( time such that ln an > dn see theorem if we simulate ( in )then the solution of the linear systemln / zn has the desired distribution ( an the linear system is solved in ( time via forward substitution