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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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