No abstract

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

In this work we consider moments of exponential functionals of Lévy processes on deterministic horizon. We derive two convolutional identities regarding those moments. The first one relates the complex moments of general exponential functional of Lévy process up to a deterministic time to those of the dual one. The second convolutional identity links the complex moments of the exponential functional of a general Lévy process to those of the exponential functionals of its ascending/descending ladder heights taken on a random horizon determined by the respective local times. As a consequence of both we deduce a universal expression for the one half negative moment of the exponential functional of symmetric Lévy process which is reminiscent of the universality for the passage time of symmetric random walk. In this work, under extremely mild conditions, we also obtain series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. This extends substantially previous results and offers neat expressions for the negative real moments. In a special case, it turns out that the Riemann zeta function is the minus first moment of the exponential functional of Gamma subordinator indexed in time.