ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let X 1, X 2, ... be i.i.d. positive random variables, and let ρ n be the initial rank of X n (that is, the rank of X n among X 1, ..., X n). Those observations whose initial rank is k are collected into a point process N k on ℝ+, called the k-record process. The fact that {itNk; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N 1, ..., N k, the “lifetimes” in rank k of all observations of initial rank at most k are independent geometric random variables. These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a “time-space” Poisson process. Initially, we think of this Poisson process as having values in ℝ+, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00535268
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