In this paper, we propose a new class of discrete time stochastic processes generated by a two-color generalized Pólya urn, that is reinforced every time. A single urn contains a white balls, b black balls and evolves as follows: at discrete times we sample 1, 2, , n   

We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries X k,j and independently distributed rows. We study the distribution of S n = r j =1 n k=1 X k,j X k+1,j , which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of S n for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

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Let X(t) denote the remaining useful lifetime of a machine, and Y(t) be a standard Brownian motion. Assume that the derivative [X(t), Y(t)] of X(t) is a deterministic function of (at least) Y(t). We consider the two-dimensional degenerate diffusion process (X(t), Y(t)). We obtain explicit expressions for the expected value of the random variable T(x, y) denoting the first time the machine must be replaced, or repaired, for various functions [X(t), Y(t)].

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