In the setting of the Heath-Jarrow-Morton model, this paper presents sufficient conditions to assure that the stochastic forward rates are strictly positive while maintaining the martingale property of the discounted bond price processes in the case where the stochastic forward rates are described as stochastic differential equations with explicitly state dependent stochastic volatility. Moreover, the stochastic development of the term structure of interest rates is generalized to be described by a class of continuous local martingales instead of Wiener processes. An example showing that this is a true extension of the Heath-Jarrow-Morton model is provided.

Let $\mathbf{Z}$ be a normal random vector in $R^k$ and let $\mathbf{1}$ be the element of $R^k$ with equal components 1. Let $X$ be a random variable that is independent of $\mathbf{Z}$ and consider the sum $\mathbf{Z} + X\mathbf{1}$. The latter has a normal distribution in $R^k$ if and only if $X$ has a normal distribution in $R^1$. The first result of this paper is a formula for a uniform bound on the difference between the density function of $\mathbf{Z} + X\mathbf{1}$ and the density function in the case where $X$ has a suitable normal distribution. This is applied to a problem in the theory of stationary Gaussian processes which arose from the author's work on a stochastic model for the CD4 marker in the progression of HIV.

Recent results for geometrically ergodic Markov chains show that there exist constants $R < \infty, \rho < 1$ such that $\sup_{|f|\leq V}\big|\int P^n(x, dy)f(y) - \int \pi(dy)f(y)\big| \leq RV(x)\rho^n,$ where $\pi$ is the invariant probability measure and $V$ is any solution of the drift inequalities $\int P(x, dy)V(y) \leq \lambda V(x) + b \mathbb{l}_C(x),$ which are known to guarantee geometric convergence for $\lambda < 1, b < \infty$ and a suitable small set $C$. In this paper we identify for the first time computable bounds on $R$ and $\rho$ in terms of $\lambda, b$ and the minorizing constants which guarantee the smallness of $C$. In the simplest case where $C$ is an atom $\alpha$ with $P(\alpha, \alpha) \geq \delta$ we can choose any $\rho > \vartheta$, where $\lbrack 1 - \vartheta\rbrack^{-1} = \frac{1}{(1 - \lambda)^2} \lbrack 1 - \lambda + b + b^2 + \zeta_\alpha(b(1 - \lambda) + b^2)\rbrack$ and $\zeta_\alpha \leq \big(\frac{32 - 8 \delta^2}{\delta^3}\big) \big(\frac{b}{1 - \lambda}\big)^2,$ and we can then choose $R \leq \rho/(\rho - \vartheta)$. The bounds for general small sets $C$ are similar but more complex. We apply these to simple queuing models and Markov chain Monte Carlo algorithms, although in the latter the bounds are clearly too large for practical application in the case considered.

We study the convergence of nonsymmetric annealing processes, extending the classical Dirichlet form approach to a broad class of Markov chains with exponentially vanishing transition functions. We show that both the true and symmetrized spectral gaps are logarithmically equivalent, and give robust estimates for the gap using geometric methods.

A Beardwood-Halton-Hammersley type of limit theorem is established for a broad class of Euclidean functionals which arise in stochastic optimization problems on the $d$-dimensional unit cube. The result, which applies to all functionals having a certain "quasiadditivity" property, involves minimal structural assumptions and holds in the sense of complete convergence. It extends Steele's classic theorem and includes such functionals as the length of the shortest path through a random sample, the minimal length of a tree spanned by a sample, the length of a rectilinear Steiner tree spanned by a sample and the length of a Euclidean matching. A rate of convergence is proved for these functionals.

Given two i.i.d. sequences of $n$ letters from a finite alphabet, one can consider the length $L_n$ of the longest sequence which is a subsequence of both the given sequences. It is known that $EL_n$ grows like $\gamma n$ for some $\gamma \in \lbrack 0, 1\rbrack$. Here it is shown that $\gamma n \geq EL_n \geq \gamma n - C(n \log n)^{1/2}$ for an explicit numerical constant $C$ which does not depend on the distribution of the letters. In simulations with $n = 100,000, EL_n/n$ can be determined from $k$ such trials with 95% confidence to within $0.0055/\sqrt k$, and the results here show that $\gamma$ can then be determined with 95% confidence to within $0.0225 + 0.0055/\sqrt k$, for an arbitrary letter distribution.

This paper investigates the deformation of an acoustic pulse travelling in a slab of random medium when its width is large compared to the size of the random inhomogeneities of the medium. A limit theorem is shown that explains how the shape of the transmitted pulse can be obtained as a result of a deterministic Gaussian convolution of the initial pulse. Since the random fluctuations are not supposed to be small, this gives a new rigorous formulation of the O'Doherty-Anstey result, which is well known in geophysical literature theory.

Let $(r_1, s_1), \ldots, (r_n, s_n)$ be a sequence of requests to place arcs on the unit circle, where $0 \leq r_i, s_i \leq 1$ are endpoints relative to some origin on the circle. The first request is always satisfied by reserving, or parking, the shorter of the two arcs between $r_1$ and $s_1$ (either arc can be parked in case of ties). Thereafter, one of the two arcs between $r_i$ and $s_i$ is parked if and only if it does not overlap any arc already parked by the first $i - 1$ requests. Assuming that the $r_i, s_i$ are $2n$ independent uniform random draws from $\lbrack 0, 1\rbrack$, what is the expected number $E(N_n)$ of parked arcs as a function of $n$? By an asymptotic analysis of a relatively complicated exact formula, we prove the estimate for large $n$: $E\lbrack N_n\rbrack = cn^\alpha + o(1), \quad n \rightarrow \infty,$ where $\alpha = (\sqrt{17} - 3)/4 = 0.28078\ldots$ and where the evaluation of an exact formula gives $c = 0.98487\ldots$. We also derive a limit law for the distribution of gap lengths between parked arcs as $n\rightarrow \infty$. The problem arises in a model of one-dimensional loss network: The circle is a continuous approximation of a ring network and arcs are paths between communicating stations. The application suggests open problems, which are also discussed.

In this paper we study a stochastic network model introduced recently in the analysis of neural networks. In this model the interaction between the nodes of the network is local: with each node is associated some real number (the inhibition in the language of neural networks) which is decreasing linearly with time. When this number reaches 0, it sends out some random input to its neighbors (a spike) and restarts with some random value. The state of our network is described as a Markov process. We are interested in the stability of this network, that is, under which conditions the associated Markov process is ergodic. As we will see, when the network is not stable, some of the nodes die, that is, almost surely after a given time, their inhibition never returns to 0 and grows arbitrarily. When these stability conditions are not satisfied, we analyze the set of nodes which are likely to die. We consider networks with a finite number of nodes and two kinds of topologies, the fully connected network and related graphs, and the linear network where the nodes are located on a line. A quantity $\rho$ is associated with this network and the stability properties of the network depend only on it. For the fully connected network, we prove that if $\rho < 1$, then the network is stable, and in this case we give the explicit expression for the invariant measure of the Markov process associated with this model. When $\rho > 1$ this network is shown to be not stable. For the stability of the linear network of size $N$, there is a critical value for $\rho$ which is $1/2$ if $N$ is odd and $1/(2 \cos \pi/(N + 1))$ if $N$ is even. We prove that if $\rho$ is strictly less than this critical value, then the network is stable, and if $\rho$ is strictly greater, it is not. In this last case, the set of possible asymptotic states is analyzed.

We study a system of infinitely many queues with Poisson arrivals and exponential service times. Let the net output process be the difference between the departure process and the arrival process. We impose certain ergodicity conditions on the underlying Markov chain governing the customer path. These conditions imply the existence of an invariant measure under which the average net output process is positive and proportional to the time. Starting the system with that measure, we prove that the net output process is a Poisson process plus a perturbation of order 1. This generalizes the classical theorem of Burke which asserts that the departure process is a Poisson process. An analogous result is proven for the net input process.

A fundamental problem that led to the development of queueing theory is the probabilistic modelling of the number of busy lines in telephone trunk groups. Based on the behavior of real telephone systems, a natural model to use would be the $M_t/G/s/0$ queue, which has $s$ servers, no extra waiting space and a nonhomogeneous Poisson arrival process $(M_t)$. Unfortunately, so far queueing theory has provided an exact analysis for only the $M/G/s/0$ queue in steady state, which yields the Erlang blocking formula, and the $M_t/G/\infty$ queue, which treats nonstationary arrivals at the expense of having infinitely many servers. However, these results can be synthesized to create a modified offered-load (MOL) approximation for the $M_t/G/s/0$ queue: the distribution of the number of busy servers in the $M_t/G/s/0$ queue at time $t$ is approximated by the steady-state distribution of the stationary $M/G/s/0$ queue with an offered load (arrival rate times mean service time) equal to the mean number of busy servers in the $M_t/G/\infty$ queue at time $t$. In addition to being a simple effective approximation scheme, the MOL approximation makes all of the exact results for infinite server queues relevant to the analysis of nonstationary loss systems. In this paper, we provide a rigorous mathematical basis for the MOL approximation. We find an expression for the difference between the $M_t/G/s/0$ queue length distribution and its MOL approximation. From this expression we extract bounds on the error and deduce when one distribution stochastically dominates the other.

In this paper we analyze the following loss network: When a customer arrives at a node of the network, it is served by this node if the node is not occupied; otherwise it is transmitted to some empty node where it will be served at a different rate. For the simplest systems of this type with a very large number of nodes and with global sharing, we show the existence of second order phase transitions and present explicit formulas for probability characteristics. For local sharing, we study the case of an infinite network and present some convergence results. Formulas for small and large loads are obtained.

In the present paper, compound Poisson approximation by Stein's method is considered. A general theorem analogous to the local approach for Poisson approximation is proved. It is then applied to a reliability problem involving the number of isolated vertices in the rectangular lattice on the torus.

The influence of noise on a class of discrete time systems arising from models of density-dependent branching processes is investigated. By considering iterates of the basic map, the time to escape from a stable orbit is investigated as a (nonstandard) problem of exit from a domain.

We define a stochastic process $\mathscr{X} = \{X_n, n = 0,1,2, \ldots\}$ in terms of cumulative sums of the sequence $K_1, K_2, \ldots$ of integer-valued random variables in such a way that if the $K_i$ are independent, identically distributed and nonnegative, then $\mathscr{X}$ is a Bienayme-Galton-Watson branching process. By exploiting the fact that $\mathscr{X}$ is in a sense embedded in a random walk, we show that some standard branching process results hold in more general settings. We also prove a new type of limit result.

Let $X_1, X_2, \ldots$ be independent and identically distributed positive random variables with $S_n = X_1 + \cdots + X_n$, and for nonnegative $b$ define $R_b = \inf\{S_n - b: S_n > b\}$. Then $R_b$ is called the overshoot at $b$. In terms of the moments of $X_1$, Lorden gave bounds for the moments of $R_b$ that hold uniformly over all $b$. Using a coupling argument, we establish stochastic ordering inequalities that imply the moment inequalities of Lorden. In addition to simple new proofs of Lorden's inequalities, we provide new inequalities for the tail probabilities $P\{R_b > x\}$ and moments of $R_b$ that improve upon those of Lorden. We also present conjectures for sharp moment inequalities and describe an application to the first ladder height of random walks.

A new approach to the least squares procedure for correlation dimension estimation is suggested. Consistency of the new estimator is established for a class of dynamical systems that includes the Cantor map and the logistic map with parameter value 4. Unlike the proofs of consistency for other estimation procedures, no assumptions are made about the Grassberger-Procaccia spatial correlation integral beyond the existence of the correlation dimension.