An approximation in probability of normalized integrals of processes with weak dependence by a set of Wiener processes and its applications (Q2784972)

Let \(\eta(t), t\in [0,+\infty),\) be a strictly stationary centered process which satisfies the Cramér condition \(E|\eta(t)|^{m}\leq\sigma^2H^{m-2}m!/2, m\geq 2,\) and the uniformly strong mixing condition with coefficient of mixing \(\varphi(\tau)\to 0\), \(\tau\to+\infty\). The main result of the article is the following. Let \(\varphi(\tau)<1\), \(\varphi^{1/2}(k)\leq Ak^{-p}\), \(k=2,3,\ldots\), for some \(A>0\), \(p>2\), then for any \(0<\varepsilon<1/2\) there exists a Wiener process \(W_{\varepsilon}(t)\) such that the following inequality holds true NEWLINE\[NEWLINEP\left\{\sup_{0\leq t\leq 1}\left|\zeta_{\varepsilon}(t)-W_{\varepsilon}(t\sigma_0^2)\right|> \varepsilon^{-2\alpha/3+1/9}C_0\right\}\leq \varepsilon^{-7/6-\alpha}C_1(\exp\{-C_2\varepsilon^{-\alpha}\}+ \varphi(C_3\varepsilon^{\alpha-1/3})),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\zeta_{\varepsilon}(t)=\sqrt{\varepsilon}\int_0^{t/\varepsilon}\eta(s)ds, \quad \sigma_0^2=E\left(\int_0^1\eta(s) ds\right)^2+2\sum_{j=2}^{\infty} E\int_0^1\eta(s)ds\int_{j-1}^{j}\eta(s) ds,NEWLINE\]NEWLINE \(0<\alpha<2/15\), constants \(C_{i}>0, i=0,\ldots,3\), depend on \(\varphi,C,\sigma,A,p\). Some applications of this result are presented.