## Dynamics of an adaptive randomly reinforced urn.(English)Zbl 1417.60011

A randomly reinforced urn model can be defined by $$(Y_{1,n},Y_{2,n})$$ random variables meaning the number of balls of red and white colors. At time $$n=0$$ a ball is drawn randomly from the $$(y_{1,0},y_{2,0})$$ balls. If a red ball is selected, it will be returned together with $$D_{1,1}$$ red balls. If a white ball is selected, it will be returned with $$D_{2,1}$$ while balls. Let $$Y_{1,1}=y_{1,0}+D_{1,1}$$ and $$Y_{2,1}=y_{2,0}$$ be the urn composition when the selected ball is red. Let $$Y_{1,1}=y_{1,0}$$ and $$Y_{2,1}=y_{2,0}+D_{2,1}$$ be the urn composition when the selected ball is white. The process is continued leading to $$\{(Y_{1,n},Y_{2,n});n\ge1\}$$. We have independent collections of i.i.d random variables $$\{D_{1,n};n\ge1\}$$ and $$\{D_{2,n};n\ge1\}$$. Set $$m_1=E(D_{1,n})$$ and $$m_2=E(D_{2,n})$$. Suppose $$\rho_1$$ and $$\rho_2$$ are unknown and rely on the parameters of $$D_{1,1}$$ and $$D_{2,1}$$. Let $$\mathcal{F}_{n-1}$$ be the sigma algebra generated by the information up to time $$n-1$$ and set $$\hat{\rho}_{1,n-1}$$ and $$\hat{\rho}_{2,n-1}$$ be two random variables that are $$\mathcal{F}_{n-1}$$-measurable. If the sequence $$\hat{\rho}_{i,n}$$ converges in probability to $$\rho_i$$ as $$n\rightarrow\infty$$ for $$i=1,2$$, it is shown that the urn proportions have weak consistency results when $$m_1\not=m_2$$. Some second order results are proved for the urn proportions and the proportion of selected ball colors when $$m_1=m_2$$.

### MSC:

 60C05 Combinatorial probability 60F05 Central limit and other weak theorems 60F15 Strong limit theorems
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