Uniform AR(1) Processes and Maxima on Partial Samples

Let (Xn)n ⩾ 1 be the uniform AR(1) process with parameter r ⩾ 2, and (cn)n ⩾ 1 a 0-1 sequence such that the limit limn→∞1n∑k=1nck=p$\lim\nolimits_{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^nc_k=p$ exists. Let M˜n$\widetilde{M}_n$ be the maximum of those Xk’s for which k ⩽ n and ck = 1, and Mn = max {X1, …, Xn}. We prove that the limit distribution of the random vector (M˜n,Mn)$(\widetilde{M}_n,M_n)$ as n → ∞ is not uniquely determined by the limit value p. A simulation study and analysis of a simulated data set are presented. The cases when the partial maximum M˜n$\widetilde{M}_n$ is determined by certain point processes are included in the simulation study.