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In this paper the joint limiting distribution of maximum of the specific sub-sample and maximum of the complete sample from the first-order auto-regressive process with uniform marginal distributions is obtained. There are considered several examples of partial samples, consisted of non-randomly selected terms of the full sample. It is well known that the uniform AR(1) process is strictly stationary random sequence; it doesn't satisfy condition of weak dependency, that prohibits clustering of extremes. As a consequence of this property, some interesting conclusions about joint asymptotic distributions are reached.

New examples of partial samples from the uniform AR(1) process and asymptotic distributions of extremes