Eugene Seneta

Non-negative Matrices and Markov Chains

Finite Non-Negative Matrices.- Fundamental Concepts and Results in the Theory of Non-negative Matrices.- Some Secondary Theory with Emphasis on Irreducible Matrices, and Applications.- Inhomogeneous Products of Non-negative Matrices.- Markov Chains and Finite Stochastic Matrices.- Countable Non-Negative Matrices.- Countable Stochastic Matrices.- Countable Non-negative Matrices.- Truncations of Infinite Stochastic Matrices.

The Variance Gamma (V.G.) Model for Share Market Returns

A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance that is distributed as a gamma variate. Its advantages include long tailedness, continuous-time specification, finite moments of all orders, elliptical multivariate unit period distributions, and good empirical fit. The process is pure jump, approximable by a compound Poisson process with high jump frequency and low jump magnitudes. Applications to option pricing show differential effects for options on the money, compared to in or out of the money. Copyright 1990 by the University of Chicago.

Regularly varying sequences

Abstract : A simple necessary and sufficient condition is developed for a sequence (theta(n)) , n = 0,1,2,.... of positive terms, to satisfy theta(n) = R(n), n > or = 0 , where R(.) is a regularly varying function on (0, infinity). The condition given in the report leads to a Karamata-type exponential representation for theta(n). Various associated difficulties are also discussed. The results are of relevance in connection with limit theorems in various branches of probability theory. (Author)

ESTIMATION THEORY FOR GROWTH AND IMMIGRATION RATES IN A MULTIPLICATIVE PROCESS

This paper deals with the simple Galton-Watson process with immigration, {X n } with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F’(l−) < 1), and that 0 < » ≡ B’(l−) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {X n } is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1 − m)−1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales ; and discusses relation of the above theory to that of a first order autoregressive process.

Slowly varying functions and asymptotic relations

Abstract : A survey of basic properties of slowly varying functions is given. The notion of quasi monotone functions is introduced and it is shown that a quasi monotone slowly varying function can be represented as a quotient of two non decreasing functions. The same problem of representation is also considered for some subclasses of quasi monotone slowly varying functions. (Author)

Perturbation of the stationary distribution measured by ergodicity coefficients

It is shown that an easily calculated ergodicity coefficient of a stochastic matrix P with a unique stationary distribution π T , may be used to assess sensitivity of π T to perturbation of P.

Sensitivity of finite Markov chains under perturbation

Meyer (1992) has developed inequalities in terms of the non-unit eigenvalues [lambda]j, j = 2,...,n, of a stochastic matrix P containing a single irreducible set of states, for the condition number maxa#ij, where A# = {a#ij} is the group generalized inverse of A = I - P. In this note we derive, succinctly, analogous inequalities for the alternative condition number, the ergodicity coefficient [tau]1(A#), using the properties of ergodicity coefficients: (min1 - [lambda]j)-1

Relative entropy under mappings by stochastic matrices

The relative g-entropy of two finite, discrete probability distributions x = (x1,…,xn) and y = (y1,…,yn) is defined as Hg(x,y) = Σkxkg (yk/kk - 1), where g:(-1,∞)→R is convex and g(0) = 0. When g(t) = -log(1 + t), then Hg(x,y) = Σkxklog(xk/yk), the usual relative entropy. Let Pn = {x ∈ Rn : σixi = 1, xi > 0 ∀i}. Our major results is that, for any m × n column-stochastic matrix A, the contraction coefficient defined as ηg(A) = sup{Hg(Ax,Ay)/Hg(x,y) : x,y ∈ Pn, x ≠ y} satisfies ηg(A) ⩽1 - α(A), where α(A) = minj,kΣi min(aij, aik) is Dobrushins coefficient of ergodicity. Consequently, ηg(A) < 1 if and only if A is scrambling. Upper and lower bounds on αg(A) are established. Analogous results hold for Markov chains in continuous time.

Monotone infinite stochastic matrices and their augmented truncations

Let P be a positive-recurrent, stochastically monotone, stochastic matrix on the positive integers, with stationary vector [pi]. Let (n)P be an (n x n) stochastic matrix where 1, and (n)P is the (n x n) northwest corner truncation of P, and suppose (n)[pi] is any stationary vector of (n)P. We show that (n)[pi] --> [pi] elementwise as n --> [infinity]. One corollary is the convergence to [pi] of quasistationary distributions of the (n)P. Another is that the conditions on P itself can be relaxed to domination of P by a positive-recurrent, stochastically monotone matrix R.

Computing the stationary distribution for infinite Markov chains

Abstract In a situation where the unique stationary distribution vector of an infinite irreducible positive-recurrent stochastic matrix P is not analytically determinable, numerical approximations are needed. This paper partially synthesizes and extends work on finite-vector approximative solutions obtained from nxn northwest corner truncations (n)P of P, from the standpoints of (pointwise convergence) algorithms as n→∞, and the manner of their computer implementation with a view to numerical stability and conditioning. The problem for finite n is connected with that of finding the unique stationary distribution of the finite stochastic matrix (n)P obtained from (n)P by augmenting a column.