Amites Dasgupta

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 replace ment matrices which are not irreducible. For three- and four-color ums, 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.

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.

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.

A single channel on/off model with TCP-like control

We model behavior of a TCP-like source transmitting over a single channel to a server that processes work at a constant rate τ. Transmission by the source follows an on/off mechanism. When the overall load in the system is below a critical constant γ, transmission rates increase linearly but when the load exceeds γ, then transmission rates decrease geometrically fast. We study the system by means of an embedded Markov chain, which gives the buffer content at the start of transmissions. Attention is paid to the time necessary to transmit a file of size L and both the tail behavior and expectation of the distribution of file transmission time are considered.

MAXIMA OF DIRICHLET AND TRIANGULAR ARRAYS OF GAMMA VARIABLES

Consider a row-wise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation.

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.

Nash Twist and Gaussian Noise Measure for Isometric C1 Maps

Starting with a short map

Invariant Measure and a Limit Theorem for Some Generalized Gauss Maps

f_0:I\to \mathbb R^3

Central limit theorems for a class of irreducible multicolor urn models

on the unit interval

Stationarity and mixing properties of replicating character strings

I