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

Kurchenko, O. O.

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.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

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

Keywords:

fractional Brownian motion; Baxter’s sum; central limit theorem