Research Article
On the Expected Number of Limited Length Binary Strings
Derived by Certain Urn Models
Frosso S. Makri
1
and Zaharias M. Psillakis
2
1
Department of Mathematics, University of Patras, 26500 Patras, Greece
2
Department of Physics, University of Patras, 26500 Patras, Greece
Correspondence should be addressed to Frosso S. Makri; makri@math.upatras.gr
Received 27 July 2014; Accepted 7 October 2014; Published 27 October 2014
Academic Editor: Tae-Sung Kim
Copyright © 2014 F. S. Makri and Z. M. Psillakis. his is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
he expected number of 0-1 strings of a limited length is a potentially useful index of the behavior of stochastic processes describing
the occurrence of critical events (e.g., records, extremes, and exceedances). Such model sequences might be derived by a Hoppe-
Polya or a Polya-Eggenberger urn model interpreting the drawings of white balls as occurrences of critical events. Numerical results,
concerning average numbers of constrained length interruptions of records as well as how on the average subsequent exceedances
are separated, demonstrate further certain urn models.
1. Introduction and Preliminaries
Recently, some researches associated with the number of pat-
terns which consist of runs of zeros (0s) between subsequent
ones (1s) in sequences of binary random variables (RVs)
have appeared in the literature. he 0-1 sequences {
}
≥1
may have several internal structures including among others
sequences of independent but not necessarily identically
distributed (INID) RVs, with (
1
=
1
,
2
=
2
,...,
=
)=∏
=1
,
∈ {0,1}, and
= (
= 1) = 1 −
(
= 0) = 1 −
, = 1,2,...,, and sequences of
exchangeable (EXCH) or symmetrically dependent RVs, the
joint distribution of which is invariant under any permuta-
tion of its arguments; that is, for any >0 and any vector
(
1
,
2
,...,
),
∈ {0,1}, it holds that
() = (
1
=
1
,
2
=
2
,...,
=
)=(
1
=
1
,
2
=
2
,...,
=
) for any permutation (
1
,
2
,...,
) of the set {1,2,...,}
and =∑
=1
. A common ground for both INID and
EXCH sequences is sequences of independent and identically
distributed (IID) RVs with probability of 1s , 0<<1,
since an IID sequence is an INID sequence with
==1−
or an EXCH sequence with
()=
−
.
In population genetics and evolution of species, urn
models are frequently used as probabilistic models/devices
to explain/apply some theories. Among the plethora of such
models (see, e.g., Johnson and Kotz [1], Blom et al. [2], and
Mahmoud [3]) we consider in the sequel two of them: the
Hoppe-Polya urn model (HPUM) and the Polya-Eggenberger
urn model (PEUM). he irst one, introduced by Holst [4, 5]
as a generalization of the Hoppe urn model, is a device to
produce certain INID binary sequences whereas the second
one supplies a mechanism for producing particular EXCH
binary sequences.
Special cases of PEUM are models of a (F/R − TM)
ixed/random threshold (see, e.g., Eryilmaz and Yalcin [6],
Makri and Psillakis [7], and Eryilmaz et al. [8]), whereas a
special case of HPUM is the (RIM) record indicator model
(see, e.g., Holst [5, 9, 10], Demir and Eryılmaz [11], and
Makri and Psillakis [7]). F/R − TM and RIM ind potential
applications in the frequency analysis and risk managing
of the occurrence of critical events (records, extremes, and
exceedances) in several scientiic disciplines like physical
sciences (e.g., seismology, meteorology, and hydrology) and
stochastic inancial analysis (e.g., insurance and inancial
engineering). For details see the cited works.
In this paper we wish to use the expected number of
an overlapping enumerative statistic (RV) as an index of
the average occurrence of some special events over time or
Hindawi Publishing Corporation
Journal of Probability
Volume 2014, Article ID 646140, 6 pages
http://dx.doi.org/10.1155/2014/646140