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 $n\ge 1$, 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.

Keywords:

• Central limit theorem
• martingale
• regular variation
• unbalanced urn
• stochastic algorithm
• strong law of large numbers

Acknowledgements

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding their Research group No. (RG-1441-317).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Deanship of Scientific Research at King Saud University for funding their Research group No. (RG-1441-317).