An be categorized on the basis of their assumptions on (Wn )n1 . By way of example, [18,19,21,22] assume that Wn and ( X1 , W1 , . . . , Xn-1 , Wn-1 , Xn ) are independent, in which case the process ( Xn )n1 is conditionally identically distributed (c.i.d.) [21], which is, conditionally on current data, all future observations are identically distributed. It follows from [21] that c.i.d. processes preserve a lot of of your Bomedemstat MedChemExpress properties of exchangeable sequences and, in specific, satisfy (two)three). In contrast, [17,20,23] assume that the reinforcement Wn is dependent upon the unique color Xn , and prove a version of (2) exactly where P is concentrated around the set of dominant colors for which the anticipated reinforcement is maximum. In this work, we reconsider the above models inside the framework of RRPPs. For the c.i.d. case, we prove benefits whose analogues have already been established by [23] for the model with dominant colors. In distinct, we extend the convergence in (two) to be in total variation and give a unified central limit theorem. We also examine the amount of distinct values which can be generated by the sequence ( Xn )n1 . In some applications, the definition of an MVPP might be as well restrictive because it assumes that the probability law with the reinforcement R is known. Nevertheless, we can envisage conditions where the law is itself random, so we extend the definition of an MVPP by introducingMathematics 2021, 9,4 ofa random parameter V. The resulting generalized measure-valued P ya urn method (GMVPP) turns out to become a mixture of Markov processes and admits representation (four)five), conditional around the parameter V. When the reinforcement measure R x is concentrated on x, we call ( )n0 a generalized randomly reinforced P ya process (GRRPP). We give a characterization of GRRPPs with exchangeable weights (Wn )n1 and show that the course of action (( Xn , Wn ))n1 is partially conditionally identically distributed (partially c.i.d) [24], which is, conditionally around the previous observations as well as the concurrent observation in the other sequence, the future observations are marginally identically distributed. We also extend a number of the outcomes for RRPPs towards the generalized setting. The paper is structured as follows. In Section two.1, we recall the definition of a measurevalued P ya urn process and prove representation (4)five) for any suitably selected sequence ( Xn )n1 . Section two.2 defines a specific subclass of MVPPs, called randomly reinforced P ya processes (RRPP), which share with exchangeable P ya sequences the home of reinforcing only the observed colour. Section 3 is devoted towards the study on the asymptotic properties of RRPPs. In Section four, we give the definition of GMVPPs and GRRPPs, and get basic final results. two. Definitions and a Representation Outcome Let (X, d) be a complete separable metric space, MCC950 site endowed with its Borel -field X . Denote byMF (X),M (X), FMP (X),the collections of measures on X which can be finite, finite and non-null, and probability measures, respectively. We regard MF (X), M (X) and MP (X) as measurable spaces equipped F together with the -fields generated by B), B X . We additional letKF (X, Y),KP (X, Y),be the collections of transition kernels K from X to Y that happen to be finite and probability kernels, respectively. Any non-null measure M (X) has a normalized version = X). F If f : X Y is measurable, then f : MF (X) MF (Y) denotes the induced mapping of measures, f (( = f -1 , MF (X). All random quantities are defined on a widespread probability space (, H, P), that is a.
