Strong limit of processes constructed from a renewal process (Q6611469)

The authors establish strong convergence to Brownian motion for an extension of the uniform transport process based on a renewal-reward process. Let \(U_1,U_2,\ldots\) be independent and identically distributed random variables, and \(S_k=U_1+\cdots+U_k\). For a positive function \(\beta\) with \(\sum_{n\geq1}\beta(n)<\infty\), define\N\[\NT_{\beta(n)}(t)=\eta_0+\sum_{k=1}^\infty\eta_k\mathbf{1}_{[0,t]}(\beta(n)S_k)\,,\N\]\Nwhere \(\eta_0,\eta_1,\ldots\) are independent and identically distributed Bernoulli random variables with mean \(1/2\), which are also independent of the \(U_i\). The authors study the processes\N\[\Nx_n(t)=\left(\beta(n)\frac{\mathbf{E}[U_1^2]}{\mathbf{E}[U_1]}\right)^{-1/2} \int_0^t(-1)^{T_{\beta(n)}(u)}\,\text{d}u\,,\N\]\Nshowing strong convergence of these processes to Brownian motion as \(n\to\infty\). In the special case where the \(U_i\) have a uniform distribution on \((0,1)\) and \(\beta(n)=n^{-k}\) for \(k>1\), the authors further establish the corresponding rate of convergence, showing that for each \(q>0\) there exists a constant \(\alpha=\alpha(q)\) such that\N\[\N\mathbf{P}\left(\max_{0\leq t\leq1}|x_n(t)-x(t)|>\alpha n^{-k/4}(\log n)^{3/2}\right)=o(n^{-q})\N\]\Nas \(n\to\infty\), where \(\{x(t):t\geq0\}\) is a standard Brownian motion.