We investigate the initial behavior of a deterministic model of parasitic infection, appropriate to transmission between homogeneously mixing hosts, where the amount of infection which is transferred from one host to another at a single contact depends on the number of parasites in the infecting host. In this model, $R_0$ can be defined to be the lifetime expected number of offspring of an adult parasite under ideal conditions, but it does not necessarily contain the information needed to separate growth from extinction of infection; nor need the growth rates of parasite numbers and numbers of infected hosts be the same. Similar phenomena are observed if real time is replaced by generation number, and the overlap of generations as time passes need not correspond to that found, for instance, in the supercritical linear birth and death process. The proofs involve martingale methods, applied to a Markov chain associated with the deterministic differential equation system.

We derive the implications of default risk for valuation of securities in an abstract setting in which the fractional default recovery rate and the hazard rate for default may depend on the market value of the instrument itself, or on the market values of other instruments issued by the same entity (which are determined simultaneously). A key technique is the use of backward recursive stochastic integral equations. We characterize the dependence of the market value on the manner of resolution of uncertainty, and in particular give conditions for monotonicity of value with respect to the information filtration.

We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with $A^+ = \int_0^1 B^+ (t) dt$ and $A_0^+ = \int_0^1 U^+ (t) dt$, we use excursion theory to show that the Laplace transforms $\Psi^+ (s) = E \exp (-sA^+)$ and $\Psi_0^+ (s) = E \exp (-sA_0^+)$ of $A^+$ of $A_0^+$ satisfy $$\int_0^{\infty} e^{-\lambda s \Psi +} (\sqrt{2} s^{3/2}) ds = \frac{\lambda^{-1/2} Ai(\lambda) + (1/3 - \int_0^{\lambda} Ai(t) dt)}{\sqrt{\lambda} Ai(\lambda) - Ai (\lambda)}.$$ and $$\int_0^{\infty} \frac{e^{-\lambda s}{\sqrt{s}} \Psi_0^+ (\sqrt{2} s^{3/2}) ds = 2 \sqrt{\pi} \frac{Ai(\lambda)}{\sqrt{\lambda} Ai'(\lambda) - Ai(\lambda)}.$$ where Ai is Airy's function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takács for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the "positive part" areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, $\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)$.

Let $(X^{(t)})_{t \geq 0}$ be a family of inhomogeneous Markov processes on a finite set M, whose jump intensities at the time $s \geq 0$ are given by $\exp(-\beta_s^{(t)} V(x, y))q(x, y)$ for all $x \not= y \epsilon M$, where the evolutions of the inverse of the temperature $\mathbb{R}_+ \ni s \mapsto \beta_s^{(t)} \epsilon \mathbb{R}_+$ take in some ways greater and greater values with t. We study by using semigroup techniques the asymptotic behavior of the couple consisting of the renormalized exit time and exit position from sets which are a little more general than the cycles associated with the cost function V. We obtain a general criterion for weak convergence, for which we describe explicitly the limit law. Then we are interested in the particular case of evolution families satisfying $\forall t, s \geq 0, \beta_s^{(t)} = \beta_{t+s}^{(0)}$, for which we show there are only three kinds of limit laws for the renormalized exit time (this is relevant for the limit theorems satisfied by renormalized occupation times of generalized simulated annealing algorithms, but this point will not be developed here).

Soit $(X^{(t)})_{t \geq 0}$ une famille de processus de Markov inhomogènes sur un ensemble fini M dont les intensités de sauts à l'instant $s \geq 0$ sont de la forme $\exp(-\beta_s^{(t)} V(x, y))q(x, y)$ pour tous $x \not= y \epsilon M$, où les évolutions de l'inverse de la température $\mathbb{R}_+ \ni s \mapsto \beta_s^{(t)} \epsilon \mathbb{R}_+$ ont tendance à prendre des valeurs de plus en plus élevées avec t grand. On étudie par des techniques de semi-groupes, le comportement asymptotique pour t grand du couple de sortie (constitué du temps de sortie convenablement renormalisé et de la position de sortie) d'ensembles un peu plus généraux que les cycles associés à la fonction de coût V. On obtient notamment des critères généraux de convergence étroite pour lesquels on décrit explicitement la loi limite. On s'intéresse ensuite plus particulièrement aux familles d'évolutions se déduisant par translations d'une évolution donnée (i.e. satisfaisant $\forall t, s \geq 0, \beta_s^{(t)} = \beta_{t+s}^{(0)}$, et qui apparaissent naturellement lors de l'étude des théorèmes limites satisfaits par les temps d'occupations renormalisés des algorithmes de recuit simulé généralisés, mais ceci ne sera que bièvement mentionné ici), en montrant que pour celles-ci il n'a que trois formes possibles de lois limites des temps de sortie renormalisés.

This paper continues the study of time series models generated by nonnegative innovations which was begun by Feigin and Resnick. We concentrate on moving average processes. Estimators for moving average coefficients are proposed and consistency and asymptotic distributions established for the case of an order-one moving average assuming either the right or the left tail of the innovation distribution is regularly varying. The rate of convergence can be superior to that of the Yule-Walker or maximum likelihood estimators.

We consider a simple stationary bilinear model $X_t = cX_{t-1} Z_{t-1} + Z_t, t = 0, \pm 1, \pm 2, \dots,$ generated by heavy-tailed noise variables ${Z_t}$. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy-tailed models by means of the sample correlation function.

Point processes on the positive real axis which are positively self-exciting in a sense expressed by their martingale dynamics are studied in this paper. It is shown that such processes can be realized as increasing mappings of Poisson processes and are therefore associated in appropriate manners. Some examples are presented, including Hawkes, renewal, Pólya-Lundberg, Markov dependent, semi-Markov, in addition to other point processes. As corollaries an extension of the Burton-Waymire association result and a solution of the Glasserman conjecture are obtained. Some results on dependence in stochastic processes of interest in queueing are given as a by product.

The state of the self-organizing cluster process is a finite subset of points in a bounded region. This subset represents an evolving discrete approximation to a continuous probability distribution in the region. The dynamics of the process is determined by an independent sequence of random points in the region chosen according to the distribution. At each time step the random point attracts the nearest point in the finite set. In this way the subset learns to approximate its environment. It is shown that initial states approach each other exponentially fast for all time with probability one. Thus all memory of the initial state is lost; the environment alone determines future history.

"Oscar's system" is a gambling system in which the aim is to win one betting unit, at least with high probability, and then start over again. The system can be modeled by an irreducible Markov chain in a subset of the two-dimensional integer lattice. We show that the Markov chain, which depends on a parameter p representing the single-trial win probability, is transient if $p < 1/2$ and positive recurrent if $p \geq 1/2$.

We investigate the duration of an elimination process for identifying a winner by coin tossing or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations.

In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization and de-Poissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis methods are indispensable for deriving asymptotic expressions for the mean and variance, which also contain periodic terms of small magnitude.

Using the concept of a "tree-growing" search strategy, we prove that for most practical insertion sorting algorithms, the number of comparisons needed to sort n keys has asymptotically normal behavior. We prove and apply a sufficient condition for asymptotically normal behavior. The condition specifies a relationship between the variance of the number of comparisons and the rate of growth in height of the sequence of trees that the search strategy "grows."