Journal of Probability http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On the Preservation of Infinite Divisibility under Length-Biasing Mon, 21 Jul 2014 00:00:00 +0000 http://www.hindawi.com/journals/jprob/2014/703697/ The law of has distribution function and first moment . The law of the length-biased version of has by definition the distribution function . It is known that is infinitely divisible if and only if , where is independent of . Here we assume this relation and ask whether or is infinitely divisible. Examples show that both, neither, or exactly one of the components of the pair can be infinitely divisible. Some general algorithms facilitate exploring the general question. It is shown that length-biasing up to the fourth order preserves infinite divisibility when has a certain compound Poisson law or the Lambert law. It is conjectured for these examples that this extends to all orders of length-biasing. Anthony G. Pakes Copyright © 2014 Anthony G. Pakes. All rights reserved. Hitting Times of Walks on Graphs through Voltages Tue, 20 May 2014 12:52:57 +0000 http://www.hindawi.com/journals/jprob/2014/852481/ We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali’s formula for hitting times without making use of the reciprocity principle. In fact this principle follows as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour. José Luis Palacios, Eduardo Gómez, and Miguel Del Río Copyright © 2014 José Luis Palacios et al. All rights reserved. Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables Mon, 07 Apr 2014 16:23:36 +0000 http://www.hindawi.com/journals/jprob/2014/786854/ The notion of -fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions. M. A. Alshanskiy Copyright © 2014 M. A. Alshanskiy. All rights reserved. Marshall-Olkin Discrete Uniform Distribution Mon, 07 Apr 2014 11:17:36 +0000 http://www.hindawi.com/journals/jprob/2014/979312/ We introduce and characterize a new family of distributions, Marshall-Olkin discrete uniform distribution. The natures of hazard rate, entropy, and distribution of minimum of sequence of i.i.d. random variables are derived. First order autoregressive (AR (1)) model with this distribution for marginals is considered. The maximum likelihood estimates for the parameters are found out. Also, the goodness of the distribution is tested with real data. E. Sandhya and C. B. Prasanth Copyright © 2014 E. Sandhya and C. B. Prasanth. All rights reserved.