# 2002

#### 2002 Volume 8 Issue 1

D. Svensson
A Class of Renewal Processes Driven by a Birth and Death Process pp.1-42 This paper deals with a generalization of the class of renewal processes with absolutely continuous life length distribution, obtained by allowing a random environment to modulate the stochastic intensity of the renewal process. The random environment is a birth and death process with a finite state space. The modulation is based on a set of deterministic failure rate functions, which are associated with the different environment states. Renewal processes in this environment (RPRE's) are constructed by using a certain Poisson embedding technique. The coupling method is the main tool in this paper, and it turns out to be particularly useful when the underlying deterministic failure rates are increasing or decreasing. For such processes, domination results and stochastic monotonicity properties are established. The existence of a stationary RPRE process is investigated by considering an embedded regenerative process, and asymptotics, rate results and versions of Blackwell's theorem are investigated by establishing exact couplings. Particular attention is paid to properties not present in the standard renewal theory but which are due to the introduction of a random environment. Asymptotic normality and some expansions of the generalized renewal function are also considered. Keywords: renewal and point processes, random environments, coupling, Poisson embedding, stochastic intensity, asymptotics, stochastic domination, stochastic monotonicity
T.V. Dudnikova, A.I. Komech and H. Spohn
On a Two-Temperature Problem for Wave Equation pp.43-80 Consider the wave equation with constant or variable coefficients in $R^3$. The initial datum is a random function with a finite mean density of energy that also satisfies a Rosenblatt- or Ibragimov - Linnik-type mixing condition. The random function converges to different space-homogeneous processes as $x_3\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution $\mu_t$ of the random solution at a time $t\in R$. The main result is the convergence of $\mu_t$ to a Gaussian translation-invariant measure as $t\to\infty$ that means central limit theorem for the wave equation. The proof is based on the Bernstein room-corridor' argument. The application to the case of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$ is given. Limiting mean energy current density formally is $-\infty\cdot (0,0,T_+ - T_-)$ for the Gibbs measures, and it is finite and equals $-C(0,0,T_+ - T_-)$ with $C>0$ for the convolution with a nontrivial test function. Keywords: wave equation, random function, Gaussian measure, correlation function, characteristic functional, Radon transformation
C. Boldrighini, R.A. Minlos and A. Pellegrinotti
Directed Polymers in Markov Random Media pp.81-105 We consider a model of directed polymers in discrete space and time assuming a Markov dependence of the environment in time. We extend results on the almost-sure validity of the Central Limit Theorem for small randomness in space dimension $\nu\geq 3$ which were previously obtained for independent environment by relying on two main technical tools: the analysis of the spectrum of a kind of transfer matrix which allows to treat the averaged model, and the explicit construction of a multiplicative orthonormal basis in the appropriate $L_2$ space, together with cluster estimates of cumulants of the basis functions. Keywords: random walk, random media, Markov processes, Central Limit Theorem
C. Takacs
Strong Law of Large Numbers for Branching Markov Chains pp.107-116 We consider the Markov chains whose index-sets are infinite trees. Given a certain state, we calculate its relative frequency up to generation $n$ of the tree and consider the limit $n \rightarrow \infty$. In this setting we prove a strong law of large numbers for Markov chains with finite state space and irreducible aperiodic transition matrix, and trees with uniformly bounded degree. Keywords: strong law of large numbers, infinite tree, branching Markov chain, population composition
A. Lejay
On the Decomposition of Excursions Measures of Processes Whose Generators Have Diffusion Coefficients Discontinuous at One Point pp.117-126 The coefficients in the decomposition of the excursions measure as convex combination of excursions measures of reflected processes are computed in order to characterize the discontinuity at one point of the diffusion coefficient. In some sense, this result extends to general diffusions a similar one for the skew Brownian motion, and we advocate it may be used in Monte Carlo methods for discontinuous media. Keywords: scale function and speed measure, excursions theory, skew Brownian motion, Monte Carlo methods
A.B. Varakin and A.Yu. Veretennikov
On Parameter Estimation for "Polynomial Ergodic" Markov Chains with Polynomial Growth Loss Functions pp.127-144 We establish the Hajek - Le Cam asymptotic efficiency of maximum likelihood estimators for "polynomial ergodic" Markov regular experiments in the class of loss functions with a polynomial growth. Keywords: asymptotic normality, polynomial loss function, maximum likelihood estimation, Hajek - Le Cam efficiency

#### 2002 Volume 8 Issue 2

Finite-Dimensional Functional Analysis Applied to Transfer Operators for Infinite-Dimensional Maps pp.149-154 We describe a simple approach to perturbative analysis of Perron - Frobenius operators and study the Floquet spectrum of the transfer operators of weakly coupled analytic maps on an infinite lattice: we are able to go beyond the first spectral gap and to exhibit smooth curves of eigenvalues and eigenvectors as functions of the crystal momenta. This talk given on January 23, 2001, at the session on Rapidity of convergence to equilibrium or stationary states, Journees Systemes Aleatoires Inhomogenes (Universite de Cergy-Pontoise, France) describes joint work with H.H. Rugh, Cergy-Pontoise. The detailed proofs of the results announced here, as well as further statements and references, may be found in [V. Baladi and H.-H. Rugh, Floquet spectrum of weakly coupled map lattices, Commun. Math. Phys., 2001, v. 220, 561-582]. Keywords: coupled map lattices, transfer operator, Floquet spectrum
C. Liverani
Computing the Rate of Decay of Correlations in Expanding and Hyperbolic Systems pp.155-162 I discuss a general approach allowing to accurately investigate the statistical properties of expanding and hyperbolic dynamical systems. Keywords: Dynamical systems, transfer operators, statistical properties
D. Talay
Stochastic Hamiltonian Systems: Exponential Convergence to the Invariant Measure, and Discretization by the Implicit Euler Scheme pp.163-198 In this paper we carefully study the large time behaviour of $$u(t,x,y) := E_{x,y}f(X_t,Y_t)-\int f d\mu,$$ where $(X_t,Y_t)$ is the solution of a stochastic Hamiltonian dissipative system with non globally Lipschitz coefficients, $\mu$ its unique invariant law, and $f$ a smooth function with polynomial growth at infinity. Our aim is to prove the exponential decay to 0 of $u(t,x,y)$ and all its derivatives when $t$ goes to infinity, for all $(x,y)$ in $R^{2d}$. We apply our precise estimates on $u(t,x,y)$ to analyze the convergence rate of a probabilistic numerical method based upon the implicit Euler discretization scheme which approximates $\int f d\mu$. Keywords: stochastic differential equations, stochastic Hamiltonian systems, parabolic partial differential equation, invariant measure, Euler method, simulation
J.C. Mattingly and A.M. Stuart
Geometric Ergodicity of Some Hypo-Elliptic Diffusions for Particle Motions pp. 199-214 Two degenerate SDEs arising in statistical physics are studied. The first is a Langevin equation with state-dependent noise and damping. The second is the equation of motion for a particle obeying Stokes' law in a Gaussian random field; this field is chosen to mimic certain features of turbulence. Both equations are hypo-elliptic and smoothness of probability densities may be established. By developing appropriate Lyapunov functions and by studying the necessary control problems, geometric ergodicity is proved. Keywords: geometric ergodicity, stochastic differential equations, Langevin equation, synthetic turbulence, hypoelliptic and degenerate diffusions
W. Krauth
Disks on a Sphere and Two-Dimensional Glasses pp.215-219 Talk given at the conference on Inhomogeneous Random Systems' at the University of Cergy-Pontoise, France, (23 January 2001). I describe the classic circle-packing problem on a sphere, and the analytic and numerical approaches that have been used to study it. I then present a very simple Markov-chain Monte Carlo algorithm, which succeeds in finding the best solutions known today. The behavior of the algorithm is put into the context of the statistical physics of glasses. Keywords: Monte Carlo methods, packing of circles, glass transition
E. Janvresse
Approach to Equilibrium for Kac Master Equation pp. 221-232 We consider the random walk on $S^{n-1}(1)$, the $(n-1)$-dimensional sphere of radius 1, generated by random rotations on randomly selected coordinate planes i,j with $1 \le i < j \le n$. This dynamics was used by M. Kac as a model for the spatially homogeneous Boltzmann equation. If we assume that the initial distribution is of product form, Kac proved that this property remains valid for all time in the limit $n \rightarrow \infty$. In modern terminology, Kac proved the "propagation of chaos". Once propagation of chaos is proved, it is straightforward to show that the marginal density of a particle satisfies the analog of a Boltzmann equation. Clearly, the spectral properties of the collision operator of the Boltzmann equation is of critical importance to understand it. Since this collision operator is generated by a Kac process, a very basic property is the size of the spectral gap, which Kac conjectured to be of order $1/n$. After recalling the idea of the proof of the Kac conjecture and the generalization to the same walk on $SO(n)$, we discuss other ways of measuring the rate of convergence to equilibrium for the Kac master equation (L.S.I., entropy dissipation bound). Keywords: convergence to equilibrium, spectral gap, Kac model, Boltzmann equation
S.G. Bobkov, I. Gentil and M. Ledoux
Hypercontractivity of Hamilton - Jacobi Equations pp.233-235 Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton - Jacobi equations. Given a bounded Lipschitz function $f$, solutions of the Hamilton - Jacobi initial value problem $$\left \{ \begin{array}{rcll} \frac{\partial v}{\partial t} + \frac{1}{2} \, |\nabla v|^2 & = & 0& {\rm in}\,\, R^n \times(0,\infty ), \\ v & = & f& {\rm on} \,\, R^n \times \{t=0\}, \end{array} \right.$$ are described by the Hopf - Lax representation formula as the infimum-convolution of $f$ with the quadratic cost $$Q_t f(x) = \inf _{y \in R^n} \Big [ f(y) + \frac{1}{2t} \, |x-y|^2 \Big ], \quad t>0, \, x \in R^n.$$ Our main result is the following theorem. Let $\mu$ be a probability measure on the Borel sets of $R^n$ absolutely continuous with respect to Lebesgue measure such that for some $\rho >0$ and all smooth enough functions $\varphi$ on $R^n$ with $\int \varphi ^2 d\mu =1$, $$\label{(1)} \rho \int \varphi ^2 \log \varphi ^2 d\mu \leq 2 \int |\nabla \varphi |^2 d\mu .$$ Then for every bounded measurable function $f$ on $R^n$, every $t > 0$ and every $a \in R$, $$\label{(2)} {\big\| \,{\rm e}^{Q_t f} \big\|}_{a + \rho t} \leq {\big\|\, {\rm e}^f\big\|}_a$$ (where the norms are understood with respect to $\mu$). Conversely, if (\ref{(2)}) holds for all $t> 0$ and some $a \not= 0$, then the logarithmic Sobolev inequality (\ref{(1)}) holds. When $a=0$, (\ref{(2)}) actually amounts to the infimum-convolution inequality $$\int {\rm e}^{\,\rho \,Q_1 f} d\mu \leq {\rm e}^{\,\rho \int f d\mu }$$ holding for every bounded (or integrable) function $f$. This inequality is known to be the Monge - Kantorovitch - Rubinstein dual version of the transportation cost inequality $$\rho \, W_2(\mu ,\nu)^2 \leq H(\nu \, | \, \mu )$$ holding for all probability measures $\nu$ absolutely continuous with respect to $\mu$ with Radon - Nikodym derivative $d\nu / d\mu$. Here $W_2$ is the Wasserstein distance with quadratic cost $$W_2 (\mu , \nu)^2 = \inf \int \int {\frac{1}{2}} \, |x-y|^2 d\pi (x,y)$$ where the infimum is running over all probability measures $\pi$ on $R^n \times R^n$ with respective marginals $\mu$ and $\nu$ and $H(\nu \, | \, \mu )$ is the relative entropy, or informational divergence, of $\nu$ with respect to $\mu$. This approach thus provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. Extensions to the Riemannian setting and applications to transportation cost and concentration inequalities, HWI inequalities and isoperimetry complete this work. Keywords: hypercontractivity, logarithmic Sobolev inequality, transportation cost inequality, Hamilton - Jacobi equations
C. Villani
On the Trend to Equilibrium for Kinetic Equations pp.237-250 We summarize recent works about the convergence to thermodynamical equilibrium for the Boltzmann equation modelling dilute gases. Keywords: kinetic equations, Boltzmann equation, long-time behavior, logarithmic Sobolev inequalities
N. Cancrini
Relaxation to Equilibrium of Spin Exchange Dynamics for Lattice Gases pp.251-270 We review recent results on the relaxation to equilibrium of spin exchange dynamics reversible with respect to the canonical Gibbs measure of a lattice gas model. Keywords: Kawasaki dynamics, spectral gap, logarithmic Sobolev constant
F. Baffioni, I. Merola and E. Presutti
The Liquid and Vapor Phases in Particle Models with Kac Potentials pp. 271-318 We consider here the model introduced in [J.L. Lebowitz, A. Mazel and E. Presutti, Liquid-vapor phase transitions for systems with finite-range interactions, J. Stat. Phys., 1999, v. 94, N5-6, 955-1025] to study phase transitions for point particles in the continuum. The mean field phase diagram of the model in the $(\beta,\lambda)$-plane, $\lambda$ the chemical potential, $\beta$ the inverse temperature, consists of a smooth curve $\lambda=\lambda(\beta)$, $\beta>\beta_c>0$, where two phases (liquid and vapor) coexist, elsewhere the phase is unique. In the mentioned paper it has been proved that phase transitions (in the sense of non uniqueness of DLR measures) persist when the mean field interaction is replaced by a Kac potential with small but fixed scaling parameter $\gamma$ (mean field being derived in the limit $\gamma\to 0$). In particular it is shown that for any $\beta>\beta_c$, there is a phase transition at $\lambda=\lambda(\beta,\ga)$ for any $\gamma$ small enough with $\lambda(\beta,\gamma)\to\lambda(\beta)$ as $\gamma\to 0$. Here we complete the analysis of the phase diagram by showing that if $\lambda\ne\lambda(\beta)$, then, for all $\gamma$ small enough, there is a unique DLR measure. Keywords: Pirogov - Sinai theory, Peierls estimates, Vaserstein distance, disagreement percolation
T. Bodineau
On the van der Waals Theory of Surface Tension pp.319-338 In this paper, the works on the justification of the van der Waals theory of surface tension in the context of the Kac Ising models are reviewed. The second part of the paper is devoted to a coarse grained definition of the surface tension for Kac Ising models which is appropriate for the $L^1$ approach of phase coexistence. Keywords: Ising model
L. Triolo
Free Energy Functionals in the Mesoscopic Description of Phase Transitions pp.339-350 A ferromagnetic spin system and a system of point particles in the continuum, both interacting with a long-range Kac-type potentials, are considered in a suitable scaled space coordinates (mesoscopic limit). The main issue is the description of the connection between the two phases which are present at low temperature, and the role of associated free-energy functionals is pointed out. Keywords: nonlocal equations, fronts between phases
O. Penrose
Statistical Mechanics of Nonlinear Elasticity pp.351-364 A method is suggested for defining a deformation-dependent free energy in microscopic terms for a deformed elastic solid and applied to a simple microscopic model of such a solid. Some of the convexity and continuity properties of this free energy function are derived. Keywords: nonlinear elasticity, restricted ensembles in statistical mechanics
L. Bertini
Dynamic Fluctuations for Kac and Related Models pp.365-381 This note is an extended version of a talk given at the conference Homogeneisation en mecanique statistique pour des potentiels a longue portee, Universite de Cergy-Pontoise (2001). We recall the definition of dynamical models with Kac potential and review their scaling limits; particular emphasis is dedicated to the fluctuations phenomena which are also analyzed for the (simpler) Ginzburg - Landau models. Some open problems are also discussed. Keywords: Kac potential, dynamical fluctuation phenomena, Ginzburg - Landau models, interface dynamics
Cluster Expansions of Small Contours in Abstract Pirogov - Sinai Models p.381 We develop the Pirogov - Sinai theory for "abstract Pirogov - Sinai models" where configuration is already represented as a system of matching compatible contours. The Hamiltonian is then expressed in terms of contour energies and an additional external field, the latter acting not on single spins but more generally on their "clusters", in the space ("colored" by various "local ground states") outside of contours. This general model is formulated in such a way that i) its general structure does not change when cluster expansion of its "smallest possible" contours resp. contour systems is applied; ii) it includes, after suitable preparation, many interesting lattice spin models, notably the reformulation of the Ising Kac ferromagnet studied in the paper [A. Bovier and M. Zahradnik, Cluster Expansions and Pirogov - Sinai Theory for Long Range Spin Systems, Markov Processes Relat. Fields, 2002, v. 8, N 3, 443-478]. In fact the result of the latter paper - a model with expanded restricted ensembles living outside of suitably defined contours - is the main testing example for the usefulness of this general method, further elaborating and simplifying the method of the paper [M. Zahradnik, A short course in the Pirogov - Sinai theory, Rendiconti Mat. Appl., 1997, v. 18, N 7, 411-486]. Keywords: low temperature Gibbs states, contours, cluster expansion, contour functional, abstract Pirogov - Sinai model with cluster field, Pirogov - Sinai theory

#### 2002 Volume 8 Issue 3

Cluster Expansions of Small Contours in Abstract Pirogov - Sinai Models pp.383-441 We develop the Pirogov - Sinai theory for "abstract Pirogov - Sinai models" where configuration is already represented as a system of matching compatible contours. The Hamiltonian is then expressed in terms of contour energies and an additional external field, the latter acting not on single spins but more generally on their "clusters", in the space ("colored" by various "local ground states") outside of contours. This general model is formulated in such a way that i) its general structure does not change when cluster expansion of its "smallest possible" contours resp. contour systems is applied; ii) it includes, after suitable preparation, many interesting lattice spin models, notably the reformulation of the Ising Kac ferromagnet studied in the paper [A. Bovier and M. Zahradnik, Cluster Expansions and Pirogov - Sinai Theory for Long Range Spin Systems, Markov Processes Relat. Fields, 2002, v. 8, N 3, 443-478]. In fact the result of the latter paper - a model with expanded restricted ensembles living outside of suitably defined contours - is the main testing example for the usefulness of this general method, further elaborating and simplifying the method of the paper [M. Zahradnik, A short course in the Pirogov - Sinai theory, Rendiconti Mat. Appl., 1997, v. 18, N 7, 411-486]. Keywords: low temperature Gibbs states, contours, cluster expansion, contour functional, abstract Pirogov - Sinai model with cluster field, Pirogov - Sinai theory
Cluster Expansions and Pirogov - Sinai Theory for Long Range Spin Systems pp.443-478 We investigate the low temperature phases of lattice spin systems with interactions of Kac type, that is interactions that are weak but long range in such a way that the total interaction of one spin with all the others is of order unity. In particular we develop a systematic approach to convergent low temperature expansions in situations where interactions are weak but long range. This leads to a reformulation of the model in terms of a generalized abstract Pirogov - Sinai model, that is a representation in terms of contours interacting through cluster fields. The main point of our approach is that all quantities in the contour representation satisfy estimates that are uniform in the range of the interaction and depend only on the overall interaction strength. The extension of the Pirogov - Sinai theory to such models developed in [M. Zahradnik, Cluster expansions of small contours in abstract Pirogov - Sinai models, Markov Processes Relat. Fields, 2002, v.8, N3, 383-441] allows then the investigation of the low-temperature phase diagram of these models. Keywords: low temperature Gibbs states, discrete spin lattice models of Kac - Ising type restricted ensembles with low density constraints, cluster expansion, contours, Pirogov - Sinai theory
A.C.D. van Enter, I. Medved' and K. Netocny
Chaotic Size Dependence in the Ising Model with Random Boundary Conditions pp.479-508 We study the nearest-neighbour Ising model with a class of random boundary conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions 4 and higher that almost surely the only limit points for a sequence of increasing cubes are the plus and the minus state. For d=2 and d=3 we prove a similar result for sparse sequences of increasing cubes. This question was raised by Newman and Stein. Our results imply that the Newman - Stein metastate is concentrated on the plus and the minus state. Keywords: random boundary conditions, metastates, contour models, local limit theorem, central limit theorem
A.I. Petrov
Critical S0L systems pp.509-526 L systems were first introduced by Aristid Lindenmayer as models in developmental biology in the late 1960s. One can find surveys on L systems in journals in biology, molecular genetics, semiotics, artificial intelligence, theory of formal languages. A stochastic version of the 0L systems was introduced in [P. Eichhorst and W.J. Savitch, Growth functions of stochastic Lindenmayer systems. Information and Control, 1980, v.45, 217-228] and [T. Yokomori, Stochastic characterizations of EOL languages, Information and Control, 1980, v.45, 26-33] but statistical physics point of view (thermodynamic limit, cluster expansions technique, etc.) started only recently, see [V.A. Malyshev, Random grammars, Russian Math. Surveys, 1998, v.53, N2, 107-134]. In this paper we undertake more detailed study of the long time behaviour of the critical S0L systems. The supercritical case was considered in [F.I. Karpelevich, V.A. Malyshev, A.I. Petrov, S.A. Pirogov, A.N. Rybko, Context free evolution of words, Rapport de Recherche INRIA, 2002, No. 4413 and [A.I. Petrov, Context free random grammars: supercritical case with nonzero extinction probability. To appear in Theory Probab. and Appl., 2002]. This paper can be read independently of these two papers. Keywords: random grammars, 0L systems, context free, branching process, critical area, thermodynamic limit
B.L. Granovsky
The Simplest Nearest-Neighbour Spin Systems on Regular Graphs: Time Dynamics of the Mean Coverage Function pp.527-546 We establish a characterization of the class of the simplest nearest neighbour spin systems possessing the mean coverage function (mcf) that obeys a second order differential equation, and derive explicit expressions for the mcf's of the above models. Based on these expressions, the problem of ergodicity of the models is studied and bounds for their spectral gaps are obtained. Keywords: interacting spin systems, mean coverage function, voter models, ergodicity, spectral gap

#### 2002 Volume 8 Issue 4

T.M. Liggett
Negative Correlations and Particle Systems
pp.547-564
We consider the symmetric exclusion process and systems of independent Markov chains. For each of these, we prove that certain classes of distributions with negative dependence are preserved by the evolution. We also show by example that the class of negatively associated distributions is not preserved by the symmetric exclusion process.
Keywords: correlation inequalities, symmetric exclusion process, negative association, independent particle systems

G. Ben Arous, O. Hryniv and S. Molchanov
Phase Transition for the Spherical Hierarchical Model
pp.565-594
We present the whole spectrum of the limit theorems for the total magnetization in the hierarchical version of the spherical model in dimensions dim > 2.
Keywords: spherical model, hierarchical interactions, phase transition

P. Mathieu
Log-Sobolev and Spectral Gap Inequalities for the Knapsack Markov Chain
pp.595-610
We study a random walk moving on a discrete cube of large dimension and subject to a linear constraint. Under the assumption that more than the half of the points in the cube satisfy the constraint, we prove log-Sobolev and Poincare inequalities with constants of the correct order.
Keywords: knapsack problem, log-Sobolev inequalities, spectral gap, generalized Poincare inequalities

R. Liptser, V. Spokoiny and A.Yu. Veretennikov
Freidlin - Wentzell Type Large Deviations for Smooth Processes
pp.611-636
We establish large deviation principle for the family of vector-valued random processes $X^\varepsilon,\varepsilon\to 0$ defined by ordinary differential equations (under $0<\kappa<1/2$) $$\dot{X}_t^\varepsilon=F(X_t^\varepsilon)+\varepsilon^{1/2-\kappa} G(X_t^\varepsilon)\dot{W}^\varepsilon_t,$$ where $\dot{W}^\varepsilon_t={\varepsilon}^{-1/2}g(\xi_{t/\varepsilon})$, $\xi_t$ is a vector-valued ergodic diffusion satisfying, so called, recurrence condition'' and $g$ is a vector-function with zero barycenter with respect to the invariant measure of $(\xi_t)$. A choice of $\kappa<1/2$ provides the rate function of Freidlin - Wentzell type.
Keywords: moderate deviations, Poisson decomposition, Puhalskii theorem

D. Svensson
A Random Environment Generalization of Lorden's Renewal Inequality
pp.637-649
The aim of this paper is to obtain a generalization of Lorden's renewal inequality for a class of renewal processes in random environments. These processes (called RPRE's) are generalizations of the classical renewal processes with absolutely continuous life length distribution, and are obtained by allowing a random environment to modulate the stochastic intensity. The first part of this paper presents a probabilistic proof of Lorden's classical renewal inequality. The main ideas from this proof are generalized for regenerative point processes and a certain Lorden type inequality is obtained. Finally, this inequality is applied to RPRE's and the aimed generalization of Lorden's inequality is obtained. It takes a particularly transparent form when the RPRE's considered are of DFR or IFR type.
Keywords: renewal processes, regenerative point processes, random environments, coupling, Poisson embedding, failure rates, stochastic intensity, DFR, IFR

M. Gonzalez, M. Molina and M. Mota
Bisexual Galton - Watson Branching Processes with Immigration of Females and Males. Asymptotic Behaviour
pp.651-663
A bisexual Galton - Watson branching process allowing the immigration of females and males is introduced and its asymptotic behaviour is investigated. Under certain assumptions on the immigration probability distribution, conditions for the almost sure convergence of the suitably normed underlying Markov chains are provided.
Keywords: bisexual Galton - Watson process, branching process with immigration, almost sure convergence

A. Chen and K. Liu
Piecewise Birth-Death Processes
pp.665-698
A natural generalization of the ordinary birth-death processes, the Piecewise Birth-Death Process is considered. The close relationship between this kind of processes and the well-known ordinary birth-death processes is revealed. Regularity criteria for such processes are established. Properties of such structure are further presented. In particular, conditions for recurrence and positive recurrence are obtained. Equilibrium distributions are given for the ergodic case. Some useful properties of the transient Piecewise Birth-Death process are also provided. Probabilistic meanings of the analytic results are explained. These results are then illustrated through several examples.
Keywords: birth-death processes, piecewise birth-death processes, uniqueness, recurrence, positive recurrence, equilibrium distributions, regularity, Feller minimal processes