A Markov Chain Model for the Evolution of Sex Ratio

A model in the form of a Markov chain is constructed to mimic variations in the human sex ratio. It is illustrated by simulation. The equilibrium distribution is shown to be a simple modification of the binomial distribution. This enables an easy calculation of the variation in sex ratio which could be expected in small populations.

Random Fields with Pólya Correlation Structure

We construct random fields with Pólya-type autocorrelation function and dampened Pólya cross-correlation function. The marginal distribution of the random fields may be taken as any infinitely divisible distribution with finite variance, and the random fields are fully characterized in terms of their joint characteristic function. This makes available a new class of non-Gaussian random fields with flexible correlation structure for use in modeling and estimation.

Estimation theory for growth and immigration rates in a multiplicative process

This paper deals with the simple Galton-Watson process with immigration, {Xn} 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'(1–) < 1), and that 0 <λ≡B'(1–) < ∞, 0 <B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the ratesmandλon the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for bothmandλare obtained, together with associated central limit theorems in relation tomandμ≡λ(1–m)–1Following 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.

Moments for stationary and quasi-stationary distributions of markov chains

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

Towards consensus: some convergence theorems on repeated averaging

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

The simple branching process with infinite mean. I

The simple branching process {Zn} with mean number of offspring per individual infinite, is considered. Conditions under which there exists a sequence {pn} of positive constants such that pn log (1 +Zn) converges in law to a proper limit distribution are given, as is a supplementary condition necessary and sufficient for pn~ constant cn as n→∞, where 0 < c < 1 is a number characteristic of the process. Some properties of the limiting distribution function are discussed; while others (with additional results) are deferred to a sequel.

Some results for quasi-stationary distributions of birth-death processes

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).

A note on simple branching processes with infinite mean

We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn} say, conditions are given ensuring that there exists a sequence of positive constants, {ρn}, such that {ρnU(Zn + 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.

A note on models using the branching process with immigration stopped at zero

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.

Entropy and martingales in Markov chain models

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .

A Gamma Activity Time Process with Noninteger Parameter and Self-Similar Limit

We construct a process with gamma increments, which has a given convex autocorrelation function and asymptotically a self-similar limit. This construction validates the use of long-range dependent t and variance-gamma subordinator models for actual financial data as advocated in Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.

Fitting the variance-gamma model to financial data

This paper has as its main theme the fitting in practice of the variance-gamma distribution, which allows for skewness, by moment methods. This fitting procedure allows for possible dependence of increments in log returns, while retaining their stationarity. It is intended as a step in a partial synthesis of some ideas of Madan, Carr and Chang (1998) and of Heyde (1999). Standard estimation and hypothesis-testing theory depends on a large sample of observations which are independently as well as identically distributed and consequently may give inappropriate conclusions in the presence of dependence.

Notes on “Estimation theory for growth and immigration rates in a multiplicative process”

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.

Stationary-increment Student and variance-gamma processes

A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e−rtPt} a familiar martingale. We then specify the activity time process, {Tt}, for which {Tt−t} is asymptotically self-similar and {τt}, with τt=Tt−Tt−1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return)Xtand a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric)t-distributions or variance-gamma distributions.

On invariant measures for simple branching processes

This paper was initially motivated by a problem raised earlier by the author (Seneta (1969), Section 5.3) viz. that of the existence and uniqueness of an invariant measure for a supercritical Galton-Watson process with immigration; and, indeed, in the sequel we show that such a measure always exists, but is not in general unique.

Peaks and Eulerian numbers in a random sequence

We consider the exact distribution of the number of peaks in a random permutation of the integers 1, 2, ···, n. This arises from a test of whether n successive observations from a continuous distribution are i.i.d. The Eulerian numbers, which figure in the p.g.f., are then shown to provide a link between the simpler problem of ascents (which has been thoroughly analysed) and both our problem of peaks and similar problems on the circle. This link then permits easy deduction of certain general properties, such as linearity in n of the cumulants, in the more complex settings. Since the focus of the paper is on exact distributional results, a uniform bound on the deviation from the limiting normal is included. A secondary purpose of the paper is synthesis, beginning with the more familiar setting of peaks and troughs.

On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

The time to absorption from the setTof transient states of a Markov chain may be sufficiently long for the probability distribution overTto settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

The Galton-Watson process with mean one

Let Zn be the number of individuals in the n-th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 < F(0) < 1. It is well known that the p.g.f. of Zn is Fn(s), the n-th functional iterate of F(s), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n-step transition probabilities are given by

On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.

The critical branching process with immigration stopped at zero

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

Augmented truncations of infinite stochastic matrices

We consider the problem of approximating the stationary distribution of a positive-recurrent Markov chain with infinite transition matrix P, by stationary distributions computed from (n × n) stochastic matrices formed by augmenting the entries of the (n × n) northwest corner truncations of P, as n →∞.

Inequalities for branching processes

If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn(s) = Fn–1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates limn→∞m−n {1−Fn(s)}, 0 ≦ s ≦ 1 where m = F′(1) < 1 and F′′(1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn(s) − Fn(0)} [1 – Fn(0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

Regularly varying functions in the theory of simple branching processes

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Bonferroni-type inequalities and the methods of indicators and polynomials

A difficulty in the application [5] of the method of polynomials as exposited by Galambos is investigated. The method, recast as the method of indicators, in a form due originally to Rényi [6], is applied to the situation of non-constant coefficients.

Functional equations and the Galton-Watson process

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f0 ≡ F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

An inequality from genetics

A class of fitness matrices whose parameters may be varied to give differing stability structure is shown by Chebyshev&s covariance inequality to possess a variance lower bound for the change in mean fitness.

On the theory of Markov set-chains

In the theory of homogeneous Markov chains, states are classified according to their connectivity to other states and this classification leads to a classification of the Markov chains themselves. In this paper we classify Markov set-chains analogously, particularly into ergodic, regular, and absorbing Markov set-chains. A weak law of large numbers is developed for regular Markov set-chains. Examples are used to illustrate analysis of behavior of Markov set-chains.

Coefficients of ergodicity: structure and applications

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

Bonferroni-type inequalities

We derive the Sobel–Uppuluri and Galambos-type extensions of the Bonferroni bounds, and further extensions of the same nature, as consequences of a single non-probabilistic inequality. The methodology follows that of Galambos.

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 πT, may be used to assess sensitivity of πT to perturbation of P.

On Arnold's treatment of Moran's bounds

We prove a conjecture of Arnold (1968) which simplifies the determination of an optimal bound on absorption probability originally due to Moran (1960).

A relaxation view of a genetic problem

Suppose where S is a compact set (of say). Let Φ (mapping S into S) and Ψ be continuous on S and such that Ψ(xk) is monotone (non-decreasing, say) as k increases, and xk+1 = Φ(xk). Put Ψ∗ = lim (k → ∞)Ψ(xk); if {xk(i)}∞i=0 is a convergent subsequence of {xk}, with a limit-point α, then Ψ∗ = Ψ(α), and and