A characterization of the infinitely divisible squared Gaussian processes. (English) Zbl 1102.60031

The authors deal with the infinite divisibility of the square of a Gaussian process. The main statement is as follows. Let \(\eta \) be a centered Gaussian vector indexed by a finite set \(E\) with a positive definite covariance function \(G(x,y)\), \((x,y)\in E\times E.\) The vector \(\eta ^{2}\) is infinitely divisible if and only if there exists a real-valued function \(d \) on \(E\) such that for any \(x,y\in E\):
\[ G(x,y)=d(x)g(x,y)d(y) \]
where the function \(g\) is the Green function of a transient symmetric Markov process. Under an additional joint continuity assumption on the covariance function a similar statement holds for a Gaussian process indexed by \( R^{1}.\) At last it is proved that the square of the Brownian sheet is not infinitely divisible.


60G15 Gaussian processes
60E07 Infinitely divisible distributions; stable distributions
60J25 Continuous-time Markov processes on general state spaces
60J55 Local time and additive functionals
Full Text: DOI arXiv


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