On the limit superior of Gaussian sequences and sample continuity of a Gaussian Markov process. (English. Russian original) Zbl 0699.60027

Solntsev, S. A.

Theory Probab. Math. Stat. 40, 129-135 (1990); translation from Teor. Veroyatn. Mat. Stat., Kiev 40, 108-114 (1989).

Summary: Suppose that \(\xi =(\xi_ n\), \(n\geq 1)\) is a centered Gaussian Markov sequence, and let \(E \xi^ 2_ n=\sigma^ 2_ n\) and \(r_{i,i+1}=E \xi_ i\xi_{i+1}/\sigma_ i\sigma_ j\). It is shown that if \(\sigma^ 2_ n\downarrow 0\) as \(n\to \infty\), then \(\overline{\lim}_{n\to \infty}\xi_ n=C_{\xi}\) almost surely, where \[ C_{\xi}=\overline{\lim}_{n\to \infty}\sigma_ n(2 \ln (1+\sum^{n-1}_{i=1}\min \{1,\ln | r^{-1}_{i,i+1}| \}))^{1/2}. \] As a corollary, a condition is presented for the sample continuity of a Gaussian Markov process on [a,b]\(\subset R\), expressed in covariance terms.

MSC:

60G15	Gaussian processes
60G17	Sample path properties
60J05	Discrete-time Markov processes on general state spaces

Keywords:

Gaussian Markov sequence; sample continuity