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## Über dieses Buch

The Athens Conference on Applied Probability and Time Series in 1995 brought together researchers from across the world. The published papers appear in two volumes. Volume I includes papers on applied probability in Honor of J.M. Gani. The topics include probability and probabilistic methods in recursive algorithms and stochastic models, Markov and other stochastic models such as Markov chains, branching processes and semi-Markov systems, biomathematical and genetic models, epidemilogical models including S-I-R (Susceptible-Infective-Removal), household and AIDS epidemics, financial models for option pricing and optimization problems, random walks, queues and their waiting times, and spatial models for earthquakes and inference on spatial models.

## Inhaltsverzeichnis

### Half-Prophets and Robbins’ Problem of Minimizing the Expected Rank

Let X1, X2,…X n be i.i.d. random variables with a known continuous distribution function. Robbins’ problem is to find a sequential stopping rule without recall which minimizes the expected rank of the selected observation. An upper bound (obtained by memoryless threshold rules) and a procedure to obtain lower bounds of the value are known, but the difficulty is that the optimal strategy depends for all n > 2 in an intractable way on the whole history of preceding observations. The goal of this article is to understand better the structure of both optimal memoryless threshold rules and the (overall) optimal rule. We prove that the optimal rule is a “stepwise” monotone increasing threshold-function rule and then study its property of, what we call, full history-dependence. For each n, we describe a tractable statistic of preceding observations which is sufficient for optimal decisions of decision makers with half-prophetical abilities who can do generally better than we. It is shown that their advice can always be used to improve strictly on memoryless rules, and we determine such an improved rule for all n sufficiently large. We do not know, however, whether one can construct, as n → ∞ asymptotically relevant improvements.

F. Thomas Bruss, Thomas S. Ferguson

### Analysis of recursive algorithms by the contraction method

Several examples of the asymptotic analysis of recursive algorithms are investigated by the contraction method. The examples concern random permutations and binary search trees. In these examples it is demonstrated that the contraction method can be applied successfully to problems with contraction constants converging to one and with nonregular normalizations as logarithmic normalizations, which are typical in search type algorithms. An advantage of this approach is its generality and the possibility to obtain quantitative approximation results.

M. Cramer, L. Rüschendorf

### Comparison of completely positive maps on a C* algebra and a Lebesgue decomposition theorem

From the expositions in [M], [BP], [P2] it is now well understood that Status: Runital completely positive maps on unital C* algebras are the quantum probabilistic analogues of transition probability operators in Markov processes. In [BS], V.P. Belavkin and P.Staszewski introduced three different notions of absolute continuity of one completely positive (c.p.) map with respect to another on a C* algebra and proved the existence of a Radon-Nikodym density under the condition of ‘strong complete absolute continuity’. Here we combine their approach with the Hilbert space - theoretic proof of the classical RadonNikodym theorem (See the exercises in Section 31 of [H] or Section 47 of [P1] and obtain the Lebesgue decomposition of a unit al c.p. map into its absolutely continuous and singular parts with respect to another such map. Analogues of chain rule and martingale properties of Radon-Nikodym derivatives and some examples are also included.

K. R. Parthasarathy

### Abel Expansions and Generalized Abel Polynomials in Stochastic Models

To build expansions, the family of the Abel polynomials $$\left\{ {\left( {x - a} \right){{\left( {x - a - bn} \right)}^{n - 1}}/n!{\text{; n}} \in \mathbb{N}} \right\}$$ can be used as a basis in place of the classical family of monomials $$\left\{ {{x^n}/n!{\text{; }}n \in \mathbb{N}} \right\}$$. In that case we get Abel’s expansions that generalize Taylor’s ones. The purpose of the present paper is to show that these polynomials and expansions are present implicitely in several probability models, and that making explicit their hidden algebraic structure is very useful. More complex stochastic models can then also be considered, after extending the Abelian structure to more general polynomials.

Philippe Picard, Claude Lefevre

### Positive Dependence Orders: A Survey

Notions of positive dependence of two random variables X and Y have been introduced in the literature in an effort to mathematically describe the property that “large (respectively, small) values of X go together with large (respectively, small) values of Y.” Some of these notions are based on some comparison of the joint distribution of X and Y with their distribution under the theoretical assumption that X and Y are independent. Often such a comparison can be extended to general pairs of bivariate distributions with given marginals. This fact led researchers to introduce various notions of positive dependence orders. These orders are designed to compare the strength of positive dependence of the two underlying bivariate distributions. In this survey we describe some such notions.

Marco Scarsini, Moshe Shaked

### A Poisson Limit Theorem on the Number of Appearances of a Pattern in a Markov Chain

A sequence of Markov dependent trials is performed, each one of them producing a letter from a given finite alphabet. Under quite general conditions we prove that the number of non—overlapping occurrences of long patterns approximates a Poisson distribution.

Ourania Chryssaphinou, Stavros Papastavridis

### Palindromes in Random Letter Generation: Poisson Approximations, Rates of Growth,and Erdös-Rényi Laws

Consider a sequence $$\left\{ {{X_j}} \right\}_{j = 1}^n$$ of i.i.d. uniform {0,1,… d −1}-válued random variables, and let M n,k be the number of palindromes of length k counted in an overlapping fashion; a palindrome is any word that is symmetric about its center. We prove that the distribution of M n,k can be well-approximated by that of a Poisson random variable. Similar approximations are obtained for various other random quantities of interest. We also obtain maximal and minimal rates of growth for the length L n of the longest palindrome; an Erdös-Rényi law is derived as a corollary: the length of the longest palindrome is, almost surely, of order logan, where a is the square root of the alphabet size d. Analogous results on partial palindromes i.e. words in which a certain (non-zero) number of “mismatches” prevent symmetry about the center, have been presented by Revelle [R] in a sequel to this paper.

Debashis Ghosh, Anant P. Godbole

### Direct analytical methods for determining quasistationary distributions for continuous-time Markov chains

We shall be concerned with the problem of determining the quasistationary distributions of an absorbing continuous-time Markov chain directly from the transition-rate matrix Q.We shall present conditions which ensure that any finite μ-invariant probability measure for Q is a quasistationary distribution. Our results will be illustrated with reference to birth and death processes.

A. G. Hart, P. K. Pollett

### Explosions in Markov Processes and Submartingale Convergence.

Conditions for nonexplosions and explosions in Markov pure jump processes are given in terms of the rate of change in the process. We show how these conditions follow from a submartingale convergence theorem. As a corollary, new conditions for nonexplosions in Birth-Death processes in terms of the survival rate are obtained.

G. Kersting, F. C. Klebaner

### Probability Bounds for Product Poisson Processes

Consider processes formed as products of independent Poisson and symmetrized Poisson processes. This paper provides exponential bounds for the tail probabilities of statistics representable as integrals of bounded functions with respect to such product processes. In the derivations, tail probability bounds are also obtained for product empirical measures. Such processes arise in tests of independence. A generalization of the Hanson-Wright inequality for quadratic forms is used in the symmetric case. The paper also provides some reasonably tractable approximations to the more general bounds that are derived first.

Joong Sung Kwon, Ronald Pyke

### On the First-Crossing of a Poisson Process in a Lower Boundary

This paper is concerned with the first-crossing of a Poisson process in a general lower boundary. The statistic under investigation is the level of first-crossing N. It is shown that the distribution of N can be expressed in a simple and compact way in terms of a particular family of Abel-Gontcharoff polynomials. Attention is then paid to that specific law, called the Poisson-Gontcharoff distribution. It is pointed out that the law keeps some nice properties of the usual Poisson distribution. Finally, two related first-crossing problems that can be tackled using again AbelGontcharoff polynomials are examined.

Claude Lefevre, Philippe Picard

### Explicit Rates of Convergence of Stochastically Ordered Markov Chains*

Let $$\Phi = \left\{ {{\Phi _n}} \right\}$$be a Markov chain on a half-line [0, ∞) that is stochastically ordered in its initial state. We find conditions under which there are explicit bounds on the rate of convergence of the chain to a stationary limit π: specifically, for suitable rate functions r which may be geometric or subgeometric and “moments” f ≥ 1, we find conditions under which $$r\left( n \right)\mathop {\sup }\limits_{|g| \leqslant f} |{E_x}\left[ {g\left( {{\Phi _n}} \right)} \right] - \pi \left( g \right)| \leqslant M\left( x \right)$$for all n and all x. We find bounds on r ( n ) and M(x) both in terms of geometric and subgeometric “drift functions”, and in terms of behaviour of the hitting times on {0} and on compact sets [0 c ] for c > 0. The results are illustrated for random walks and for a multiplicative time series model.

D. J. Scott, R. L. Tweedie

### Multi-Type Age-Dependent Branching Processes with State-Dependent Immigration

This work continues the study of the age-dependent branching processes allowing two types of immigration, i.e. one in the state zero and another one according to the i.i.d. times of an independent ergodic renewal process. The multidimensional case is considered and asymptotic properties and limit theorems are established. These results generalise both the results of the discrete theory and those for the one-dimensional continuous-time model.

Maroussia N. Slavtchova-Bojkova

### The non homogeneous semi-Markov system in a stochastic environment

In the present we introduce and define for the first time the concept of a non-homogeneous semi Markov system in a stochastic environment (S-NHSMS). We study the problem of finding the expected population structure as a function of the basic parameters of the system. Important properties are established among the basic parameters of a non-homogeneous semi Markov system in a stochastic environment.

P.-C. G. Vassiliou

### Branching Processes with Two Types Emigration and State-Dependent Immigration

We consider branching processes allowing a random migration component. In each generation the following three situations are possible: (i) with probability p - family emigration and individual emigration (possibly dependent); (ii) with probability q - no migration, i.e. the reproduction is as in the classical BGW process; (iii) with probability r - state dependent immigration; p + q + r = 1. In the critical case an additional parameter of recurrence is obtained. The asymptotic behaviour of the hitting zero probability and of the first two moments is investigated. Limiting distributions are also obtained depending on the range of the recurrence parameter.

George P. Yanev, Nickolay M. Yanev

### Remarks on the Law of Succession

Applied probability and time series, the respective areas in which Joe Gani and Ted Hannan made their names, meet in the classical problem of the law of succession. Much of the work on this problem assumes a multinomial distribution for the observations. In this paper assumptions are made which imply a non-multinomial distribution, which in turn implies a rather different approach to that used in the multinomial case to estimating probabilities of novel events.

W. J. Ewens

### Large deviations of the Wright-Fisher Process

In the present paper we will be concerned with the Wright-Fisher process of mathematical genetics in one of its simplest forms: the univariate case with no selection and with either no mutation or only one-way mutation. Consider a large biological population consisting of a single species in which a particular gene appears in two possible forms (alleles) A and a, say. Assume that we are dealing with a one-sex population (as with some plants) and that we are in the haploid case (in which chromosomes occur singly). The Wright-Fisher process models the way in which the proportion of individuals carrying the A form of the gene changes from generation to generation, as the population reproduces itself ([1], [4]). To be specific, let us first consider the case where the proportion of the A allele changes only through the effect of random sampling, with no mutation or selection. If the population consists of N individuals i of whom are of type A and N–i of type a, then the state of the process is the proportion $$y = \frac{i}{N}$$ To produce an offspring generation from this population of “genes” we sample N times with replacement from it, thus keeping the size of the population constant. The probability that the proportion of A-alleles will make a transition from state $$y = \frac{i}{N}$$ in the “current” generation to state $$\tilde y = \frac{j} {N}$$ say, in the following generation is then 1.1$$P\left( {y,\tilde y} \right) = \left( {\begin{array}{*{20}{c}} N \\ j \end{array}} \right){y^j}{\left( {1 - y} \right)^{N - j}}$$

F. Papangeloul

### Threshold behaviour in stochastic epidemics among households

A very general model for the spread of an epidemic among a population consisting of m households, each of size n, is presented. The asymptotic situation in which the number of households in tends to infinity, whilst the household size n remains fixed, is analysed. For large m the process of infected households can be approximated by a branching process. A coupling argument is used to make this approximation precise as in tends to ∞, thus enabling a threshold theorem to be developed. Specialisation to the case where infective individuals during their infectious period make infectious contacts within their own group with rate λw and outside their own group with rate λBis briefly considered. Generalisations to populations with unequal household sizes and different types of individuals are outlined.

Frank Ball

### Reproduction Numbers and Critical Immunity Levels for Epidemics in a Community of Households

Epidemic threshold parameters, also called reproduction numbers, play a central role in computing the vaccination coverage required to prevent epidemics. It is possible to define several different reproduction numbers for infectives in a community of households. To illustrate this we compute four distinct reproduction numbers for infectives for a community consisting of a large number of households of size three, using assumptions similar to the so-called general epidemic model. It is found that when individuals are selected independently for immunization the proportion that needs to be immunized so as to prevent epidemics is vI* = 1 – 1/R30, where R30 is one of these reproduction numbers. When a proportion of households is selected and every member of each selected household is immunized, then the proportion of households that needs to be immunized is vH* = 1 – 1/R40where R40is another of the basic reproduction numbers. The result for vH* applies for an arbitrary household distribution and a disease with an arbitrary infectivity function. However, the result for vI* is more complicated for a community containing larger households.

Niels G. Becker, Klaus Dietz

### Modelling the spread of HIV in prisons

This paper is concerned with models for the spread of HIV in prisons. We first consider a single prison of size N in which there is homogeneous mixing, and where there is an inflow and outflow of n< N prisoners at times t= 0, 1, 2,... Conditions are derived for the stability of the system. The interaction of such a prison with the outside world is then studied, and stability conditions obtained for this case. Finally, a quarantine policy is examined and its cost analyzed, both in the non-stable and stable cases.

J. Gani

### An Algorithmic Study of S-I-R Stochastic Epidemic Models

Algorithms for the S-I-R epidemic with an initial population with m infectives and n susceptibles are examined. We propose efficient algorithms for the distributions of the total and the maximum size of the epidemic, and for the joint distribution of the maximum and the time of its occurrence. We also discuss the joint distribution of the sizes of the epidemic at two epochs. By studying the Markov chain describing an indefinite replication of the epidemic, we obtain new descriptors of the process of infections.

Marcel F. Neuts, Jian Min Li

### Testing the Validity of Value-at-Risk Measures

Value at Risk (<VaR >) is a notion meant to measure the risk linked to the holding of an asset or a portfolio of diverse assets. It is defined as the amount of money that one might lose with a certain confidence level and within a given time horizon. It is used by bankers because it allows them to measure, compare and consolidate “risk” that is linked to their trading activities. As it has recently been recommended to bankers by many financial institutions, it may increasingly become more and more widespread, with the possibility of becoming a market standard. Within a given time horizon, the portfolio return should not drop below a stated VaR number more often than predicted by the confidence interval. This paper aims to test this assertion under different methods of estimating VaR. Much of the current literature is devoted to the prediction of volatility, another measure of risk commonly used by bankers. However, the predictive power of Value-at-Risk has not been tested so far.We show that “classical” ways of estimating VaR are valid only in reasonable ranges of confidence intervals. If bankers are looking at very low quantiles, then they might be advised to employ more sophisticated models.

Bertrand Gamrowski, Svetlozar Rachev

### Option pricing for hyperbolic CRR model

Employing the Rachev and Rüschendorf (1994) idea to generalize Cox—Ross—Rubinstein (1979) binomial model we find the option price formula for the limiting model with hyperbolic distributions. It turns out that the results in the continuous and discrete models lead to markedly different fair prices.

Aleksander Rejman, Aleksander Weron

### A class of shot noise models for financial applications

We describe a class of non-Markov shot noise processes that can be used as models for rates of return on securities, exchange rate processes and other processes in finance. These are continuous time processes that can exhibit heavy tails that become lighter when sampling interval increases, clustering and long memory.

### Why discount? The Rationale of Discounting in Optimisation Problems

The use of a discounted cost function seems to produce inconsistencies if the utility function is nonlinear, in that the optimal policy is then non-stationary, even for a time-invariant model. The reason for this is that, if discounting is motivated by the fact that capital can grow by compound interest, then there is an implication that one has the alternatives of operating the enterprise which is being controlled (and of accepting consequent costs or gains) or of letting one’s capital grow at a constant interest rate (in the bank, say). The complete state variable for this model is then the state variable of the enterpriseplusthe amount held in the bank. If the utility function is linear or the enterprise deterministic then optimisation reduces to the minimisation of expected discounted enterprise costs. In other cases one has a non-degenerate problem of optimal allocation between enterprise and bank account.

P. Whittle

### On Periodic Pollaczek Waiting Time Processes

Present-day modelling of traffic in broadband communication requires the use of rather sophisticated stochastic processes. Although a large class of suitable stochastic processes is known in the literature, their rather complicated structure limits their use because of the laborious numerical evaluation involved. The present study concerns the so-called periodic Pollaczek processes. The characteristics of the arrival process, such as service time τn and interarrival time σ n,are here periodic functions of n i.e. the vector (τnσn)and (τn+N,σn+N)have the same distribution N being the period. The sequence Wn,n= 1, 2,..., defined by $${W_{n + 1}} = {\left[ {{W_n} - {T_n} - {\sigma _n}} \right]^ + }$$ is investigated; as such the queue under consideration is a direct generalisation of the classical Pollaczek GI/G/1queue. It appears that the model is a quite flexible one, and moreover very accessible for numerical evaluation if the distributions of all the service times, or of all the interarrival times, have rational LaplaceStieltjes transforms.

J. W. Cohen

### Random Walk Approach to Relaxation in Disordered Systems

A detailed study of the limiting probability distributions of R(t) — the location at time t — for one-dimensional random walks with waiting-time distributions having long tails, is presented. In the framework of the random walk approach the nonexponential character of relaxation as a function of time, given by the tail of the Mittag-Leffler distribution, has been obtained. As a consequence, the frequency-domain response takes the well-known form of the empirical Cole-Cole function.

Marcin Kotulskil, Karina Weron

### Inference for a Class of Causal Spatial Models

A new class of lattice models, whose joint densities are products of unilateral conditional densities, is introduced. Maximum likelihood estimation, for causal conditional exponential families on a two dimensional lattice, is discussed. The technique proposed enables one to construct a rich class of lattice models with a parsimoneous parameterization.

I. V. Basawa

### Problems in the Modelling and Statistical Analysis of Earthquakes

The purpose of this paper is to set out some of the statistical and probabilistic problems that arise from the work of the first author over the past two decades. Nearly all of this work has been published in geophysical journals, where the major emphasis is on the physical interpretation of the model structure and subsequent statistical analysis. One consequence is that manyof themathematical issues inherent in the model structures and conceptions have never been fully explored (see also Vere-Jones, 1994). In fact they raise many issues for discussion, from the adequacy of existing stochastic models for processes exhibiting self-similar or fractal behaviour, to specific questions concerning the testing of statistical models for random 3-dimensional rotations. Our hope is that by reviewing these papers in the context of the present conference, we may encourage attempts to resolve some of the more mathematical issues which arise, or relate them to relevant recent work.

Y. Y. Kagan, D. Vere-Jones

### On the Existence of UMVU Estimators for Bernoulli Experiments in the Non-Identically Distributed Case with Applications to the Randomized Response Method and the Unrelated Question Model

It is shown that there does not exist any UMVU estimator of the type $${d^*}\left( {\Sigma _{j = 1}^n{X_j}} \right)$$ based on independent random variables X1,...,Xn, where Xj has a Bernoulli distribution with corresponding probability aj p +bj ∈ [0, 1] for p ∈ [0,1],aj≠ 0, j= 1,...n and where $${d^*}:\left\{ {0, \ldots,n} \right\} \to \mathbb{R}$$ is one-to-one, if the condition aj= ak and bj=b k or aj=-akandbj=1-b k is not valid for all $$\left( {j,k} \right) \in {\left\{ {1, \ldots,n} \right\}^2}$$ In particular, it is proved that the model parameterajbjj = 1,...,n, in connection with the unrelated question model must be kept fixed, if there should exist some UMVU estimator forpof this type. Furthermore, it is shown that the existence of some non-constant UMVU estimator forpaccording to the randomized response method implies that the randomization parameter $${p_j} \in {\raise0.7ex\hbox{{\left[ {0,1} \right]}} \!\mathord{\left/ {\vphantom {{\left[ {0,1} \right]} {\left\{ {\frac{1}{2}} \right\}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{{\left\{ {\frac{1}{2}} \right\}}}},j = 1, \ldots,n$$ must satisfy the condition pj=pk orpj+ pk=1 for all $$\left( {j,k} \right) \in {\left\{ {1, \ldots,n} \right\}^2}$$

A. Dannwerth, D. Plachky

### On a Three-Sample Test

In 1958 H.T. David defined a statistic for testing the hypothesis that three samples of equal size n are drawn from the same continuous distribution. In this paper explicit formulas are given for the distribution and the asymptotic distribution of this statistic.

Lajos Takács
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