Estimates of some probabilities in multidimensional convex records

ArticleOriginal scientific text

Title

Authors ¹

Institute of Mathematics, Technical University of Łódź, al. Politechniki 11, 90-924 Łódź, Poland

Convex records in Euclidean space are considered. We provide both lower and upper bounds on the probability

p_{n} (k)

that in a sequence of random vectors

X_{1}

,...,

X_{n}

there are exactly k records.

estimates of probabilities, geometric probability, convex records in Euclidean space

I. Bárany and Z. Füredi, On the shape of the convex hull of random points, Probab. Theory Related Fields 80 (1988), 72-87.
W. Feller, An Introduction to Probability Theory and its Applications, Vol. I, Wiley, New York, 1961.
C. M. Goldie and S. Resnick, Records in a partially ordered set, Ann. Probab. 17 (1989), 678-699.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1965.
P. Groeneboom, Limit theorems for convex hulls, Probab. Theory Related Fields 79 (1988), 327-368.
M. G. Kendall, and P. A. P. Moran, Geometrical Probability, Griffin's Statist. Monographs Courses 5, Griffin, London, 1963.
J. F. C. Kingman, Random secants of a convex body, J. Appl. Probab. 6 (1969), 660-672.
V. B. Nevzorov, Records, Teor. Veroyatnost. i Primenen. 32 (2) (1987), 219-252 (in Russian).
S. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, Berlin, 1987.
I. M. Yaglom and V. G. Boltyanskiĭ, Convex Bodies, Nauka, Moscow, 1951 (in Russian).