## The functional central limit theorem for Baxter sums of fractional Brownian motion.(English. Ukrainian original)Zbl 0986.60026

Theory Probab. Math. Stat. 61, 87-94 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 84-90 (2000).
In Skorokhod’s space $$D([0,1])$$ the sequence of random processes $S_{N}(t)=\frac{1}{\sqrt{\text{Var } S_{N}(1)}}\sum_{i=1}^{[Nt]}H(N^{H/2}X_{i,N}),\quad t\in [0,1],$ is considered. Here $$[x]$$, $$x\in R,$$ is the largest integer, which does not exceed $$x;$$ $$H(x)=x^{2}-1;$$ $$X_{i,N}:=X(i/N)-X((i-1)/N),$$ $$N\geq 1,$$ $$(X(t); t\geq 0)$$ is a fractional Brownian motion, namely, Gausian random process with mean zero and covariance function $$r(s,t)=(1/2)(|t|^{H}+|s|^{H}-|t-s|^{H})$$, $$t,s\geq 0$$, $$0<H<3/2;$$ $$S_{N}(1):= \sum_{i=1}^{N}H(N^{-H/2}X_{i,N}).$$
The author proves the following assertion: The sequence of distributions of random processes $$S_{N}(t)$$ converges weakly in Skorokhod’s space $$D([0,1])$$ to the Wiener measure $$W.$$ The proof of the weak convergence is carried out by two steps: 1) weak convergence of the sequence of finite-dimensional distributions of the processes $$S_{N}(t)$$ to the finite-dimensional distributions of the Wiener measure is proved; 2) tightness (relative compactness) of the sequence of measures generated by the processes $$S_{N}(t)$$ in $$D([0,1])$$ is proved.

### MSC:

 60F17 Functional limit theorems; invariance principles 60G10 Stationary stochastic processes 60G15 Gaussian processes