×

Characteristics of normal samples. (English) Zbl 0712.60006

B. V. Gnedenko’s law of large numbers for the maximum of i.i.d. normal random variables [Ann. Math., II. Ser. 44, 423-453 (1943)] is generalized to a multidimensional space case. Namely, the following theorem is proved.
Theorem 2.1: Let \(Z_ 1,Z_ 2,..\). be i.i.d. random variables with a centered Gaussian distribution \(\nu\) on a real separable Banach space and let K denote the unit ball of the reproducing kernel Hilbert space for \(\nu\). Then \[ \max_{i\leq n}d(Z_ i,\quad \sqrt{2 \log n}K)\to 0,\quad \max_{y\in K}d(\sqrt{2 \log n}y,\quad \{Z_ 1,Z_ 2,...,Z_ n\})\to 0, \] as \(n\to \infty\) with probability 1. Here d(\(\cdot,\cdot)\) denotes the Banach norm distance from a point to a set.
As a consequence of this theorem, the main result of R. LePage and B. M. Schreiber [Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, No.3, 341-344 (1985; Zbl 0573.60031)] is improved.

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G15 Gaussian processes
60D05 Geometric probability and stochastic geometry
60F10 Large deviations

Citations:

Zbl 0573.60031
PDF BibTeX XML Cite
Full Text: DOI