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Martingale functional central limit theorems for a generalized Pólya urn. (English) Zbl 0788.60044
Consider 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)].

##### MSC:
 60F17 Functional limit theorems; invariance principles 60K99 Special processes
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