Weak convergence for the row sums of a triangular array of empirical processes indexed by a manageable triangular array of functions. (English) Zbl 0936.60036

Arcones, Miguel A.

Electron. J. Probab. 4, Paper No. 7, 17 p. (1999).

From the author’s abstract: “We study the weak convergence for the row sums of a general triangular array of empirical processes indexed by a manageable class of functions converging to an arbitrary limit. As particular cases we consider random series processes and normalized sums of i.i.d. random processes with Gaussian and stable limits. An application to linear regression is presented. In this application, the limit of the row sums of a triangular array of empirical processes is a mixture of a Gaussian process with a random series process.”
Manageable classes refer to classes of functions satisfying uniform entropy bounds. The results are very general and may come handy in applications (besides the interesting application presented in the article, the author applies them to \(M\)-estimators with stable limits in a subsequent article).

Reviewer: E.Gine (Storrs)

MSC:

60F17	Functional limit theorems; invariance principles
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)

Keywords:

empirical processes; triangular arrays