The fourth product moment of infinitely clipped noise

This paper considers the fourth product moment, w(τ
1
, τ
2
, τ
3
) = E[x(t)x(t + τ
1
)x(t + τ
2
)x(t + τ
3
)], when x(t) is infinitely clipped noise with a mean value of zero. If the noise is Gaussian before clipping, the moment w is not obtainable in closed form. For this reason, the Gaussian assumption is withdrawn and other assumptions are employed. If the zeros of x(t) obey the Poisson distribution, a particularly simple result follows for w and for all higher moments. An alternative assumption is the following. Let unspecified events occur at times τ
0
, τ
1
, τ
2
, … according to the Poisson distribution. If alternate events, i.e., those at τ
1
, τ
3
, τ
5
, …, are designated as the zeros of x(t), both the autocorrelation function and w(τ
1
, τ
2
, τ
3
) can be derived. The results are in terms of elementary functions. A comparison is made between these models and clipped Gaussian processes.