TES: A class of methods for generating autocorrelated uniform variates.

*(English)*Zbl 0764.65002This paper introduces a new class of simple generation methods for Markovian sequences of random variables with uniform marginals, called transform-expand-sample (TES). A basic TES method is a nonlinear autoregressive scheme with modulo-1 arithmetic; each is determined by a transformation and two parameters \(\alpha\in[0,1]\) and \(\varphi\in[- 1,1]\).

The first autocorrelation of TES sequences is analytically computed and it is shown that, for two fundamental TES methods, the resulting lag-1 autocorrelation \(\rho_ 1\) as a function of \(\alpha\) and \(\varphi\) spans every values in \((-1,1)\) and is monotonic quadratic in both \(\alpha\) and \(\varphi\).

On the other hand, higher autocorrelations are investigated empirically by simulation. The sample paths of TES are cyclical and exhibit discontinuity in the neighborhood of point 0 due to wraparound. So, transformations of TES methods are presented to make the sample paths more continuous-looking while preserving their marginal uniformity.

The first autocorrelation of TES sequences is analytically computed and it is shown that, for two fundamental TES methods, the resulting lag-1 autocorrelation \(\rho_ 1\) as a function of \(\alpha\) and \(\varphi\) spans every values in \((-1,1)\) and is monotonic quadratic in both \(\alpha\) and \(\varphi\).

On the other hand, higher autocorrelations are investigated empirically by simulation. The sample paths of TES are cyclical and exhibit discontinuity in the neighborhood of point 0 due to wraparound. So, transformations of TES methods are presented to make the sample paths more continuous-looking while preserving their marginal uniformity.

Reviewer: K.Uosaki (Tottori)

##### MSC:

65C10 | Random number generation in numerical analysis |