Strong Laws for Urn Models with Balanced Replacement Matrices

We consider an urn model, whose replacement matrix has all entries nonnegative and is balanced, that is, has constant row sums. We obtain the rates of the counts of balls corresponding to each color for the strong laws to hold. The analysis requires a rearrangement of the colors in two steps. We first reduce the replacement matrix to a block upper triangular one, where the diagonal blocks are either irreducible or the scalar zero. The scalings for the color counts are then given inductively depending on the Perron-Frobenius eigenvalues of the irreducible diagonal blocks. In the second step of the rearrangement, the colors are further rearranged to reduce the block upper triangular replacement matrix to a canonical form. Under a further mild technical condition, we obtain the scalings and also identify the limits. We show that the limiting random variables corresponding to the counts of colors within a block are constant multiples of each other. We provide an easy-to-understand explicit formula for them as well. The model considered here contains the urn models with irreducible replacement matrix, as well as, the upper triangular one and several specific block upper triangular ones considered earlier in the literature and gives an exhaustive picture of the color counts in the general case with only possible restrictions that the replacement matrix is balanced and has nonnegative entries.Comment: The final version. To appear in Electronic Journal of Probabilit

A functional central limit theorem for a class of urn models

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.Comment: 6 page

Multicolor urn models with reducible replacement matrices

Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak limits are known.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ150 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Random Walks in I.I.D. Random Environment on Cayley Trees

We consider the random walk in an \emph{i.i.d.} random environment on the infinite $d$-regular tree for $d \geq 3$. We consider the tree as a Cayley graph of free product of finitely many copies of \Zbold and \Zbold_2 and define the i.i.d. environment as invariant under the action of this group. Under a mild non-degeneracy assumption we show that the walk is always transient.Comment: This version has been revised significantly to make the exposition better, some typing errors corrected and more details have been added to the proofs. Comparison with earlier literature has also been included and the reference list has been expanded. The title and the abstract have been suitably changed as wel

Strong laws for balanced triangular urns

Consider an urn model whose replacement matrix is triangular, has all entries nonnegative and the row sums are all equal to one. We obtain the strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use the strong laws obtained to study further behavior of certain three color urn models

Stationarity and mixing properties of replicating character strings

Abstract: In this article, some models for random replication of character strings are considered that involve random mutations, deletions and insertions of characters. We derive some sufficient conditions on the replication process and the ancestor chain that ensure stationarity and mixing properties of the replicated chain. We also give examples of replication processes which lead to descendant chains not having any mixing properties even if the ancestor chain is i.i.d. in nature. Stationarity and mixing properties are two properties of dependent processes that are of fundamental importance and well studied in the literature. These properties are quite useful in generalizing many asymptotic results for i.i.d. processes to dependent processes and, in many situations, they are useful in justifying statistical estimation and inference based on dependent data. The presence of random deletions and insertions makes our stochastic replication model considerably different from simpler models that involve only mutations, and it leads to some interesting theoretical problems