Search

The Dirichlet-Ferguson measure is a random probability measure that has seen widespread use in Bayesian nonparametrics. Our main results can be seen as a first step towards the development of a stochastic analysis of the Dirichlet-Ferguson measure. We define a gradient that acts on functionals of the measure and derive its adjoint. The corresponding integration by parts formula is used to prove a covariance representation formula for square integrable functionals of the Dirichlet-Ferguson measure and to provide a quantitative central limit theorem for the first chaos. Our findings are illustrated by a variety of examples. [ABSTRACT FROM AUTHOR]

Let ℙ and ℙ ~ be the laws of two discrete-time stochastic processes defined on the sequence space S ℕ , where S is a finite set of points. In this paper we derive a bound on the total variation distance d TV (ℙ , ℙ ~) in terms of the cylindrical projections of ℙ and ℙ ~ . We apply the result to Markov chains with finite state space and random walks on ℤ with not necessarily independent increments, and we consider several examples. Our approach relies on the general framework of stochastic analysis for discrete-time obtuse random walks and the proof of our main result makes use of the predictable representation of multidimensional normal martingales. Along the way, we obtain a sufficient condition for the absolute continuity of ℙ ~ with respect to ℙ which is of interest in its own right. [ABSTRACT FROM AUTHOR]