Rates of clustering in FLIL for weighted partial sum processes (Q1126105)

The author studies the weighted partial sum process \(S(n)=\sum^{n-1}_{k=0}(n-k)^\alpha X_k\) of a sequence of i.i.d. random variables with mean zero and variance 1. He establishes a strong approximation of \((S_n)_n\) by \((\delta_nY(n))_n\) where \(Y(t)=\int^t_0(t-u)^\alpha dW(u)\) and \((\delta_n)_n\subset \mathbb{R}\) a sequence of truncated second moments of \(X\). With \(a_n=\sqrt{2\text{Var}(Y(n))\log\log n}\) a clustering rate of \(\{Y(nt)/a_n, 0\leq t\leq 1\}\) towards the unit ball \(K_\alpha\) of the RKHS of \(\{Y(t),\) \(0\leq t\leq 1\}\) is given. Specifically, it is shown that \[ P(Y(n\cdot)/a_n\in K_\alpha + \varepsilon_n (2\alpha+2)/(2\alpha+3)U\text{ eventually})=1 \] where \(U\) is the unit ball in \(C[0,1]\) and \(\varepsilon_n\) a specific sequence of constants. Together with the strong approximation, this yields a clustering rate for \((S_{n\cdot}/a_n)_n\) around \((\delta_n K_\alpha)_n\).