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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 23, Fasc. 1,
pages 19 - 38
 

LIMIT DISTRIBUTIONS OF DIFFERENCES AND QUOTIENTS OF NON-ADJACENT k -TH RECORD VALUES

Mariusz Bieniek
Dominik Szynal

Abstract: Let   (k)
(Yn ,n > 1) denote the sequence of k -th record values of the sequence (Xn, n > 1) of i.i.d. random variables with an absolutely continuous distribution function F. Fix r  (-  N. We show that, for some very broad class of distributions F, the limit distribution of the sequence

    (k)    (k)
k(Yn+r - Yn ),  k > 1,
is the gamma distribution with pdf

            r
fr,c(x) =---c---xr-1exp(-cx),  x > 0,
        (r -1)!
where c > 0 is a parameter which depends on F. We prove the similar result for k -th lower record values Z(nk). Moreover, we discuss the asymptotic behaviour of quotients of these quantities.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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