An extension of Billingsley's uniform boundedness theorem to higher-dimensional M-processes

For random functions belonging to the C[0,1] or D[0,1] space, Theorem 12.2 of Billingsley's monograph [2] relates to a probability inequality for the supremum, and it plays a vital role in the proof of the tightness of these processes. In the context of tightness of multi-parameter processes (i.e., for random functions belonging to the D[0,1]* space, for some q g 1), various extensions of the Billingsley inequality have been considered by a host of workers (viz., Bickel and Wichura [1] and references cited therein). For robust estimation in general linear models (viz., Jureckova and Sen [5] and the references cited therein), it may be convenient to consider some general multi-parameter M-processes and to exploit their asymptotic linearity results in the study of the properties of the derived estimators. In this context, a basic requirement is the uniform boundedness in probability of such M-processes. Such a result can, of course, be derived through the weak convergence of such processes (viz., Jureckova and Sen [3], [4] and others). However, this may demand comparatively more stringent regularity conditions. For this reason, for a general class of multi-dimensional M-processes, an extension of Billingsley's uniform boundedness theorem is considered under less stringent regularity conditions, and applications of this result in statistical inference are stressed. Along with the preliminary notions, the main theorem is presented in Section 2. Applications are considered in the last section.