On the Expected Number of Limited Length Binary Strings Derived by Certain Urn Models

The 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 patterns which consist of runs of zeros (s) between subsequent ones (s) in sequences of binary random variables (RVs) have appeared in the literature. The 0-1 sequences may have several internal structures including among others sequences of independent but not necessarily identically distributed (INID) RVs, with , , and , , and sequences of exchangeable (EXCH) or symmetrically dependent RVs, the joint distribution of which is invariant under any permutation of its arguments; that is, for any and any vector , , it holds that for any permutation of the set and . A common ground for both INID and EXCH sequences is sequences of independent and identically distributed (IID) RVs with probability of s , , since an IID sequence is an INID sequence with 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 () and the Polya-Eggenberger urn model (). The first 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 are models of a () fixed/random threshold (see, e.g., Eryilmaz and Yalcin [6], Makri and Psillakis [7], and Eryilmaz et al. [8]), whereas a special case of is the (RIM) record indicator model (see, e.g., Holst [5, 9, 10], Demir and Ery？lmaz [11], and Makri and Psillakis [7]). and find potential applications in the frequency analysis and risk managing of the occurrence of critical events (records, extremes, and exceedances) in several scientific disciplines like physical sciences (e.g., seismology, meteorology, and hydrology) and stochastic