Central limit theorems for a class of irreducible multicolor urn models (Q2468406)

The authors first consider a four-colour urn model in which the replacement matrix \(R\) is a stochastic one in the manner of \textit{R. Gouet} [J. Appl. Probab. 34, No. 2, 426--435 (1997; Zbl 0884.60028)]. That is, one starts with one ball of arbitrary colour, which is the \(0\)-th trial, and at the \(n\)-trial one arrives at the column vector \(W_{n}\) of the number of balls of the four colours. For any \(n\geq 0\) a colour is observed by random sampling from a multinomial distribution with probabilities \(W_n/(n+1)\). Depending on the colour that is observed, the corresponding row of \(R\) is added to \(W_n^{\prime }\) to give \(W_{n+1}^{\prime }\). Assume that the non-dominant eigenvalues of \(R\) are real and satisfy \(\lambda _{1}<1/2,\;\lambda _{2}=1/2,\) and \(\lambda _{3}>1/2,\) and let \(\xi _{1},\xi _{2},\xi _{3}\) be the corresponding column eigenvectors. Define \[ X_{n}=\frac{W_{n}^{\prime }\xi _{1}}{\sqrt{n}},\quad Y_{n}=\frac{W_{n}^{\prime }\xi _{2}}{\sqrt{n\log n}},\quad Z_{n}=\frac{W_{n}^{\prime }\xi _{3}}{\prod_{j=0}^{n-1}\left( 1+\frac{\lambda _{3}}{j+1}\right) }. \] The authors' main result is that \((X_n,Y_n,Z_n)\) converges in distribution to \((X,Y,Z),\) where \(X,Y,Z\) are independent, \(X\) and \(Y\) are normal with zero means, and the convergence of \( Z_{n}\) to \(Z\) is in the almost sure sense, too. Next, it is shown that this result extends to situations (with more than four colors) where there are (i) more than one eigenvalue(s) of any one or more of the three types above, or (ii) complex eigenvalues. These extensions involve the same technique, but require more calculations related to the Jordan form of the replacement matrix.