Alexander von Humboldt Foundation Conference
«Technologies of the 21st century: biological, physical, informational and socialaspects»
Saint-Petersburg, Russia, September 27-29, 2005
On the approximation of the sample by a Poisson point process
Andrei Yu. Zaitsev
St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences
Saint-Petersburg, Russia
It is shown [1] that the results obtained earlier on the approximation of distributions of sums of independent summands by accompanying infinitely divisible laws may be interpreted as substantial quantitative estimates for the closeness of a sample containing independent observations and a Poisson point process, obtained after a Poissonization of the initial sample. The most interesting results are obtained for a scheme of rare events.
Let Xi , i=1,...,n, be independent random elements of a measurable space Ω. We consider the problem of approximating the sample (X1 ,..., Xn ), by a Poisson point
process (Y1 ,..., Yη ) with intensity measure which is the sum of distributions of Xi .
Let f : Ω→Rd, be a measurable mapping. We shall approximate the probabilities of events of the form {∑i f(Xi) < x} which may be treated as the probabilities of that the cumulative influences of the risk factors fj do not exceed fixed critical values xj . We show that, for arbitrary f, the probabilities of these events may be well approximated by the probabilities of events {∑i (f(Yi) – ani) < x – ∑i ani}, where ani are non-random vectors, which depend on the distributions of Xi.
In conclusion, we shall consider a scheme of independent non-identically distributed rare events, which, being actually a particular case of the situation considered above, admits substantial estimates without additional centering for actually arbitrary mappings f. The rare events are similar to extreme incidents, each of them is unique and has his own individual distribution. Thus, the results may be useful, for example,
in the insurance theory to estimate the probabilities that the cumulative influence of the risk factors fj do not exceed fixed critical values xj (see, for example, [2]).
[1] A.Yu. Zaitsev. On the approximation of the sample by a Poisson point process, Zap. Nauchn. Semin. POMI, 298, 111-125 (2003).
[2] H.U. Gerber. An introduction to mathematical risk theory, Huebner Foundation Monograph (1981).
Dr. Andrei Yu. ZAITSEV
St.-Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences
Fontanka 27, Saint-Petersburg, 191023, Russia
Tel.: +07 812 3124058, FAX: +07 812 3105377
E-mail: