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Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space. (English. Russian original) Zbl 1239.60019

Theory Probab. Appl. 54, No. 4, 609-625 (2010) and Teor. Veroyatn. Primen. 54, No. 4, 750-770 (2009).
The authors consider the joint weak convergence of the empirical process and the rescaled empirical distribution function. They show that the pair of these processes converges to a limit having two independent components, namely, a time-transformed Brownian bridge and a two-sided Poisson process. In the discussion, the authors point out that the asymptotic independence is a rather unexpected result.
To prove the limit statement, they develop a short weak convergence theory for the Skorokhod product space; here, the classical criteria convergence of finite-dimensional distributions and tightness are formulated for the product space.
The paper ends with an example showing how the result can be used in statistics.

MSC:

60F05 Central limit and other weak theorems
60G99 Stochastic processes
62M99 Inference from stochastic processes
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