On the quadratic variation of a class of processes of Gladyshev type. (English. Russian original) Zbl 0626.60029

Shklyar, G. A.

Theory Probab. Math. Stat. 32, 125-130 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 103-113 (1985).

Consider a Gaussian process (X(t),0\(\leq t\leq 1)\) with bounded mean function \(m(t)=EX(t)\) and covariance function r(s,t). (X(t),0\(\leq t\leq 1)\) is a Gladyshev process if r(s,t) is continuous on \([0,1]^ 2\); r(s,t) has second derivatives on \(\{\) (s,t):s\(\neq t\}\); \(| \partial^ 2r(s,t)/\partial s\partial t| =O(| t-s|^{- \gamma}\}\) for some \(0<\gamma <2\); and \[ (r(t,t)-2r(t,t-h)+r(t-h,t- h))/h^{2-\gamma}\to f(t)\quad as\quad h\to \infty \] uniformly on [0,1] for some function f. For a partition \(\pi =\{0=t_ 0<t_ 1<...<t_{c(\pi)}=1\}\), let \[ m(\pi)=\max \{t_ i-t_{i-1},\quad 1\leq i\leq c(\pi)\}\quad and \]
\[ B(\pi)=\sum^{c(\pi)}_{k=1}[(X(t_ k)- X(t_{k-1}))/(t_ k-t_{k-1})^{(1-\gamma)/2}]^ 2. \] R. Klein and E. Giné [Ann. Probab. 3, 716-721 (1975; Zbl 0318.60031)] showed that \(B(\pi_ n)\to \int^{1}_{0}f(t)dt\) almost surely for any sequence \(\{\pi_ n\}\) such that \(m(\pi_ n)=o(1/\log n)\). It is now shown that this result is not true in general in the case where \(m(\pi_ n)=o(1/\log n)\).

Reviewer: R.J.Tomkins

MSC:

60F15	Strong limit theorems
60G15	Gaussian processes

Keywords:

Lévy-Baxter theorem; almost sure convergence; Gladyshev process

Citations:

Zbl 0318.60031