Regular variation in a fixed-point problem for single- and multi-class branching processes and queues

Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

Stationary measures for multitype branching processes

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.

Branching processes with immigration

Consider a branching process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, ···) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where, with probability bj (j = 0, 1, ···) j objects enter the population at each generation. Then letting Xn (n = 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that {Xn} defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual, , satisfies α < 1 or α > 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.

The Galton-Watson process with infinite mean

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z 0 = 1. If E(Z 1) &lt; ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z 1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.

The multi-type Galton-Watson process with immigration

We consider the limiting behaviour of a k-type (k &lt; ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.

The Galton-Watson process with infinite mean

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z0 = 1. If E(Z1) < ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.

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.

On dams with Markovian inputs

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, p ij , are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [p ij xj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.

Branching processes with immigration

Consider a branching process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, ···) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where, with probability bj (j = 0, 1, ···) j objects enter the population at each generation. Then letting Xn (n = 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that {Xn } defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual, , satisfies α &lt; 1 or α &gt; 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.

On the age distribution of a Markov chain

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).

A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.

Conditional limit theorems for a left-continuous random walk

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.

Some central limit analogues for supercritical Galton-Watson processes

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, ifXi, i= 1, 2, 3, ··· are independent and identically distributed random variables withEXi=μ, varXi= σ2&lt; ∞ andthen the central limit theorem can be written in the formThis provides information on the rate of convergence in the strong lawas. (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.

On the limit of a supercritical branching process

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

Almost sure convergence in Markov branching processes with infinite mean

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

The simple branching process: a note on convergence when the mean is infinite

Let denote the simple branching process with Z 0 = 1 and let G denote the distribution function of Z 1 . Suppose G satisfies x −α−γ(x)≦1 − G(x) ≦ x −α+γ(x) for large x, where (i) 0 &lt; α &lt; 1, (ii) γ (x) is non-negative and non-increasing, (iii) xγ (x) is non-decreasing and (iv) Then lim n→∞ α n log (Zn + 1) converges almost surely to a non-degenerate finite random variable W satisfying P(W = 0) = q = probability of extinction of the process.

On dams with Markovian inputs

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, pij, are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [pijxj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.

Asymptotic behaviour of continuous time, continuous state-space branching processes

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

The Galton-Watson process revisited: some martingale relationships and applications

A martingale is used to study extinction probabilities of the Galton-Watson process using a stopping time argument. This same martingale defines a martingale function in its arguments; consequently, its derivative is also a martingale. The argumentscan be classified as regular or irregular and this classification dictates very different behavior of the Galton-Watson process. For example, it is shown that irregularity of a pointsis equivalent to the derivative martingale sequence atsbeing closable, (i.e., it has limit which, when attached to the original sequence, the martingale structure is retained). It is also shown that for irregular points the limit of the derivative is the derivative of the limit, and two different types of norming constants for the asymptotics of the Galton-Watson process are asymptotically equivalent only for irregular points.

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 &lt; m ≡ F'(1–) &lt; 1), and that 0 &lt; λ ≡ B'(1–) &lt; ∞, 0 &lt; B(0) &lt; 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 {Xn } 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.

The simple branching process: a note on convergence when the mean is infinite

Letdenote the simple branching process withZ0= 1 and letGdenote the distribution function ofZ1.SupposeGsatisfiesx−α−γ(x)≦1 −G(x) ≦x−α+γ(x)for largex, where (i) 0 < α < 1, (ii)γ(x) is non-negative and non-increasing, (iii)xγ(x)is non-decreasing and (iv)Then limn→∞αnlog (Zn+ 1) converges almost surely to a non-degenerate finite random variableWsatisfyingP(W= 0) =q= probability of extinction of theprocess.

Relative extinction times for independent branching processes

The conditional probability that a Galton-Watson process with initial population j dies out before an independent Galton-Watson process initially i, is computed asymptotically as i or j gets large, given that both processes eventually die out. The results are interpreted qualitatively to discuss parameters of fitness for biological populations, especially with respect to the spread of mutant genes under evolutionary models.

Almost sure convergence in Markov branching processes with infinite mean

If {Zn } is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn } and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt } with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α &gt; 0.

On branching processes allowing immigration

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

Asymptotic behaviour of continuous time, continuous state-space branching processes

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

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.

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 &lt; c &lt; 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.

On the limit of a supercritical branching process

LetWbe the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

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.

Asymptotic properties of supercritical branching processes I: The Galton-Watson process

We obtain results connecting the distributions of the random variables Z 1 and W in the supercritical Galton-Watson process. For example, if a &gt; 1, and converge or diverge together, and regular variation of the tail of one of Z 1, W with non-integer exponent α &gt; 1 is equivalent to regular variation of the tail of the other.

A time series approach to the study of the simple subcritical Galton–Watson process with immigration

The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

A branching process with a state dependent immigration component

Consider the well known Galton-Watson branching process (Harris (1963)) in which individuals reproduce independently of each other and have probability a j (j = 0, 1, · · ·) of giving rise to j individuals in the next generation. In recent years some attention has been given to the branching process in which there is an independent immigration component at each generation, b j (j = 0, 1, · · ·) being the probability of j immigrants entering each generation. For a review of this work see Seneta (1969), and Pakes (1971 a, b and c) for further results.

Heavy traffic approximations for the Galton-Watson process

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).

On the asymptotic behaviour of branching processes with infinite mean

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn }). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn }, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn } such that In Part 1 we study the properties of {Zn /Cn } and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn (Zn )}, where {yn } is a suitable chosen sequence of functions related to {Zn }. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn )/Cn }, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn } it is possible to construct explicitly functions U, such that U(Zn )/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.

Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part

We consider immigration-branching processes constructable from an inhomogeneous Poisson process, a sequence of population probability distributions, and a homogeneous branching transition function. The set of types is arbitrary, and the process parameter is allowed to be discrete or continuous. For the branching part a weak form of positive regularity, criticality, and the existence of second moments are assumed. Varying the conditions on the immigration law, we obtain several results concerning asymptotic extinction, the rate of extinction, and limiting distribution functions of properly normalized, vector-valued counting processes associated with the immigration branching process. The proofs are based on the generating functional method.

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.

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

Fixed points of a generalized smoothing transformation and applications to the branching random walk

Let {Ai:i≥ 1} be a sequence of non-negative random variables and letMbe the class of all probability measures on [0,∞]. Define a transformationTonMby lettingTμ be the distribution of ∑i=1∞AiZi, where theZiare independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly whenThas a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε &gt; 0, we findallthe fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (byT) with an appropriate initial distribution; convergence to the trivial fixed points δ0and δ∞is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

A note on Markov branching processes

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

Heavy traffic approximations for the Galton-Watson process

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).

Some limit theorems for the total progeny of a branching process

We consider a branching process in which each individual reproduces independently of all others and has probability a j (j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 &lt; a 0, a 0 + a 1 &lt; 1.

Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.

The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

Weibull-type limiting distribution for replicative systems

The Weibull function is widely used to describe skew distributions observed in nature. However, the origin of this ubiquity is not always obvious to explain. In the present paper, we consider the well-known Galton-Watson branching process describing simple replicative systems. The shape of the resulting distribution, about which little has been known, is found essentially indistinguishable from the Weibull form in a wide range of the branching parameter; this can be seen from the exact series expansion for the cumulative distribution, which takes a universal form. We also find that the branching process can be mapped into a process of aggregation of clusters. In the branching and aggregation process, the number of events considered for branching and aggregation grows cumulatively in time, whereas, for the binomial distribution, an independent event occurs at each time with a given success probability.