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An extension of Billingsley’s uniform boundedness theorem to higher dimensional M-processes. (English) Zbl 0633.60008
Let T be the unit hypercube in $$R^ p$$ (i.e., $$T=[0,1]^ p)$$, and $$\{X_ n(t)=(X^ 1_ n(t),X^ 2_ n(t),...,X_ n^ p(t)):$$ $$t\in T\}^ a$$sequence of multiparameter random processes with values in $$D[0,1]^ p$$. Define $X^*_ n=\max_{1\leq j\leq p}\sup_{t\in T}| X^ j_ n(t)|.$ When there exist positive constants $$r>1/2$$ and $$L_ p$$ such that for every $$n\geq 1$$ $P\{X^*_ n\geq \lambda \}\leq L_ p\lambda^{-2r}\text{ for every } \lambda >0,$ we say that the sequence $$\{X_ n\}$$ is uniformly bounded in probability. For robust estimation in general linear models, it may be convenient to consider some general multiparameter M-processes and to exploit their asymptotic linearity results in the study of properties of the derived estimators. In this context, the uniform boundedness in probability is a basic requirement, but the tightness is not.
In this paper the authors give a sufficient condition which assures the uniform boundedness in probability of M-processes with $$r\geq 1$$. The condition is less stringent than that one of P. J. Bickel and M. J. Wichura [Ann. Math. Stat. 42, 1656-1670 (1971; Zbl 0265.60011)], which assures the tightness for general multiparameter stochastic processes.
Reviewer: H.Takahata

##### MSC:
 60B10 Convergence of probability measures 60E15 Inequalities; stochastic orderings 62F35 Robustness and adaptive procedures (parametric inference) 62E20 Asymptotic distribution theory in statistics
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##### References:
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