Unbalanced multi-drawing urn with random addition matrix II

Abstract

In this paper, we give some results about a multi-drawing urn with random addition matrix. The process that we study is described as: at stage n ≥ 1 , we pick out at random m balls, say k white balls and m−k black balls. We inspect the colours and then we return the balls, according to a predefined replacement matrix, together with ( m − k ) X n white balls and k Y n black balls. Here, we extend the classical assumption that the random variables ( X n , Y n ) are bounded and i.i.d. We prove a strong law of large numbers and a central limit theorem on the proportion of white balls for the total number of balls after n draws under the following more general assumptions: (i) a finite second-order moment condition in the i.i.d. case; (ii) regular variation type for the first and second moments in the independent case.