The functional central limit theorem for Baxter sums of fractional Brownian motion (Q2740448)

In Skorokhod's space \(D([0,1])\) the sequence of random processes NEWLINE\[NEWLINES_{N}(t)=\frac{1}{\sqrt{\text{Var } S_{N}(1)}}\sum_{i=1}^{[Nt]}H(N^{H/2}X_{i,N}),\quad t\in [0,1],NEWLINE\]NEWLINE 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 \(r(s,t)=(1/2)(|t|^{H}+|s|^{H}-|t-s|^{H})\), \(t,s\geq 0\), \(0<H<3/2;\) \(S_{N}(1):= \sum_{i=1}^{N}H(N^{-H/2}X_{i,N}).\)NEWLINENEWLINENEWLINEThe 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.