Zaharias M. Psillakis

Success run statistics defined on an urn model

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs

The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N n, k, l, s denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of l-overlapping success runs of length k is denoted by . Finally, W r, k, l, s denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N n, k,l, s , W r, k, l, s , and . These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.

On ℓ-overlapping Runs of Ones of Length k in Sequences of Independent Binary Random Variables

Consider a finite sequence of independent binary (zero-one) random variables ordered on a line or on a circle. The number of the ℓ-overlapping runs of ones of a fixed length k is studied for both types of the concerned ordering. Recurrences for the exact probability mass functions for these numbers are obtained via simple probabilistic arguments. Exact closed formulae, for the mean and variance of the studied numbers are obtained via their representations through properly defined indicators. Two application case studies, concerning record sequences and reliability of consecutive systems, clarify further the theoretical results.

Distributions of statistics describing concentration of runs in non homogeneous Markov-dependent trials

ABSTRACT In the first n, n ⩾ 3, trials of a non homogeneous zero-one Markov chain of first order, we consider runs of ones of length exceeding a threshold. The article deals with statistics denoting, the length and the position of the shortest segment of the chain in which all such runs of ones are concentrated. The study provides recursive schemes for conditional distributions of these statistics. Numerical examples illustrate the theoretical results.

On the concentration of runs of ones of length exceeding a threshold in a Markov chain

ABSTRACT Consider a homogeneous two state (failure-success or zero-one) Markov chain of first order. The paper deals with the position and the length of the shortest segment of the first n, , trials of the chain in which all runs of ones of length greater than or equal to a fixed number are concentrated. Accordingly, we define random variables denoting the starting/ending position of the first/last such runs in the chain as well as the implied distance between them. The paper provides exact closed form expressions for the probability mass function of these random variables given that the number of the considered runs in the chain is at least two. An application concerning DNA sequences is discussed. It is accompanied by numerics which exemplify further the theoretical results.

On Limited Length Binary Strings with an Application in Statistical Control

In a 0 - 1 sequence of Markov dependent trials we consider a statistic which counts strings of a limited length run of 0s between subsequent 1s. Its probability mass function is used to determine the chance that a stochastic process remains or not in statistical control. Illustrative numerics are presented.

Corrigendum to On success runs of a fixed length in Bernoulli sequences

The authors regret that there are a few corrections on pages 762–764, 770–771 and apologize for any inconvenience caused. On page 762, in Eq. (2) it must be written ‘‘implies’’ instead of ‘‘iff’’. On page 763, on line 10 it must be ‘‘F(x− 1; n, k, p) ≥ P(Wx,k > n)’’ instead of ‘‘F(x− 1; n, k, p) = P(Wx,k > n)’’ and on line 22 ‘‘P(En,k = 0) = 1 − E(En,k) ≥ P(W1,k > n)’’ instead of ‘‘P(En,k = 0) = P(W1,k > n) = 1 − E(En,k)’’. On page 764, Eq. (10) remains as it is if t = r(k + 1) − 1 and it holds P(Wr,k = t) < r−1 x=0 P(Et−1,k = x), if t ≥ r(k + 1). In Eq. (11) it must be ‘‘<’’ instead of ‘‘=’’. On page 770, in the title of Table 9, the second ‘‘=’’ must be ‘‘≥’’. On page 771, the first entry of the last column of Table 10 it should be ‘‘524288’’ instead of ‘‘544288’’.

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.