Martingale functional central limit theorems for a generalized Pólya urn.

*(English)*Zbl 0788.60044Consider a generalized Pólya urn scheme with \(W_ n\) white and \(B_ n\) black balls in the urn at the stage \(n\). Balls are drawn at random in succession and replaced in the urn together with other balls. If a white ball is drawn, it is returned to the urn with \(a\) white and \(b\) black balls. If a black ball is drawn, it is replaced with \(c\) white and \(d\) black balls. It is assumed that \(b,c \geq 0\), \(a+b = c + d = s\geq 1\), \(a\neq c\) and that the process does not stop because of impossible removals (in cases \(a < 0\) or \(d < 0\) which are allowed). Denote \(\rho = (a-c)/s\) the ratio of eigenvalues of the replacement matrix. By using martingale techniques there are obtained several weak invariance principles for the urn process \(W_ n\) according to different values of \(\rho\) and the product \(bc\) including also the case \(bc = 0\). The limiting Gaussian processes, the normalizing constants and tools used in the proofs depend on parameters \(\rho\) and \(bc\). The paper is motivated by and extends results of A. Bagchi and A. K. Pal [SIAM J. Algebraic Discrete Methods 6, 394-405 (1985; Zbl 0568.60010)].

Reviewer: T.Inglot (Wrocław)