A functional central limit theorem for a class of urn models
Consider a four-color urn model in which the replacement matrix is a stochastic matrix \({\mathbf R}\). Let \({\mathbf W}_n\) denote the column vector of the number of balls of the four colors up to the \(n\)th trial, where the components of \({\mathbf W}_n\) are nonnegative real numbers. Let \(\pi\) be the stationary distribution of \({\mathbf R}\). Let \(\lambda_i\) be the eigenvalues of \({\mathbf R}\) and \(\xi_i\) the eigenvector corresponding to \(\lambda_i\). Write \[ Z_{i,n}= {{\mathbf W}_n\xi_i\over \prod^{n-1}_{j=0} (1+{\lambda_i\over j+1})}\qquad (i= 1,2,3). \] The authors prove that the sequence of processes \({\mathbf G}_n(t)= (G_{1,n}(t), G_{2,n}(t), G_{3,n}(t))\) defined by \[ G_{i,n}(t)= \sum^{[ne^t]}_{m=n} m^{\lambda_i-{1/2}}(Z_{i,m+1}- Z_{i,m})\quad (i=1,2,4,\,t\geq 0) \] converges to an independent increments Gaussian process \({\mathbf G}(t)\) with covariance \(c_{i,j}(t)=\)\break \(t\lambda_i\lambda_j\langle\xi_i\xi_j, \pi'\rangle\), \(i,j= 1,2,3\), where \(\xi_i\xi_j\) denotes the vector of the coordinate-wise product of the components of \(\xi_i\) and \(\xi_j\) and \(\langle\cdot,\cdot\rangle\) denotes the Euclidian inner product of the vectors. Of course, the result can be extended to urn models with any number of colors.
scientific article; zbMATH DE number 5000441