Mean passage times for tridiagonal transition matrices and a two-parameter ehrenfest urn model

Abstract

A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].

Generalized Pólya urn Designs with Null Balance

“…Such a class of Markovian processes can be regarded as a randomized extension of the Ehrenfest process and generalizes previous contributions on the topic (e.g. [9], [11], [13], and [21]). By exploiting martingale theory, we can identify sequential urn procedures that are asymptotically balanced and satisfy the central limit property.…”

“…This scheme is quite flexible, since the PDFs can be chosen ad hoc in order to model several kinds of replacements, and generalizes some proposals in the literature (see [9], [10], [11], [13], and [21]). Observe that, if A n = B n = 1 almost surely (a.s.) for any n, the ball that has been drawn is placed in the other urn with probability 1 and {W n,1 } n∈N becomes the classical Ehrenfest urn process (so that the associated design is the ED proposed by Chen [9]).…”

“…From Proposition 1, in order to obtain a sequential procedure which is asymptotically balanced we set s = t. In such a case the stationary distribution of the ergodic chain {W n,1 } n∈N is binomial with π(·) = Bin(2w, 1 2 ). Moreover, from the spectral representation given in [21] the transition matrix P has eigenvalues λ x = 1 − sx/w for x = 0, . .…”

The Continuous-Time Ehrenfest Process in Term Structure Modelling

“…The temperatures are symbolized by the number of fluctuating balls in two urns with a total of N ∈ N balls. For details of the continuous and discrete time versions of the model we refer to [2], [10], [12] and [16]. In this section we shall discuss the continuous time Ehrenfest process.…”

“…, N }, denoting the number of balls in urn I at time t. We call (X t ) t≥0 the (continuous time) Ehrenfest process. A discrete time version of (14) with arbitrary α and β was studied by Kraft and Schaefer [12].…”