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

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

Zbl 0318.60031