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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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