Consider supercritical general branching processes, where, however, individual reproduction may be influenced by the history of the population, in particular by the total population size. Assume that reproductions approach those of a classical, possibly multitype, supercritical process either from above or from below as the population grows. Conditions are established for the population-history dependent populations to display balanced exponential growth with the same Malthusian parameter as the limiting population. Applications are made to a tumor model with quiescence.

The Poisson-skip model introduced in this paper generalizes the chi-square model of crossover interference. Both models are constructed from the random points of a Poisson process occurring along a meiotic bundle of four chromatids. The points of the Poisson process are divided into $\chi$ points and o points, with $\chi$ points corresponding to crossovers. In the chi-square model, a fixed number of o points intervene between every adjacent pair of $\chi$ points; in the Poisson-skip model, a random number of o points intervene. Both of these renewal models permit reasonably straightforward calculation of gamete and tetrad probabilities for multiple linked markers. We illustrate the data analysis possibilities of the Poisson-skip model by fitting it to classical recombination data on Drosophila, the mouse, and Neurospora. We also describe conditions on the discrete skip distribution that guarantee positive interference.

In this paper we present some interesting results which follow from the celebrated determinant formulas for noncoincidence probabilities of Markov processes discovered by Karlin and McGregor. The first theorem is a determinant formula for the probability that a Markov jump process will avoid a certain finite set of points. From this theorem a simple solution of the moving boundary problem for certain types of Markov processes can be obtained. The other theorems deal with noncoincidence probabilities of sets of random walks which need not be identically distributed. These formulas have interesting applications, especially in the theory of queues.

This paper deals with first passage heights of sums of renewal sequences, random walks, and Lévy processes. We prove that the joint age and excess (and therefore, the current life) stationary distributions of these heights are stochastically increasing (in the usual first-order sense) in the passage levels. As a preliminary tool, which is also of independent interest, a new decomposition of the stationary excess distribution, as a convolution of two other distributions, is developed. As a consequence of these results, certain monotonicity results are concluded for ratios involving convex functions. This paper is motivated by problems related to control of queues with removable servers which model single-machine produce-to-order manufacturing systems. Applications to these problems are provided.

For n points placed uniformly at random on the unit square, suppose $M_n$ (respectively, $M'_n$) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of $n \pi M_n^2 - \log n$ converges weakly to the double exponential; we give a new proof of this. We show that $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .

We apply an extended contraction principle and superexponential convergence in probability to show that a functional large deviation principle for a sequence of stochastic processes implies a corresponding functional large deviation principle for an associated sequence of first-passage-time or inverse processes. Large deviation principles are established for both inverse processes and centered inverse processes, based on corresponding results for the original process. We apply these results to obtain functional large deviation principles for renewal processes and superpositions of independent renewal processes.

The $M/M/1$ queue is considered in the case in which customers are served in random order. A formula is obtained for the distribution of the waiting time w in the stationary state. The formula is used to show that $P9w > t) \sim \alpha t^{-5/6} \exp (-\beta t - \gamma t^{1/3})$ as $t \to \infty$, with the constants $\alpha, \beta$, and $\gamma$ expressed as functions of the traffic intensity $\rho$. The distribution of w for the random order discipline is compared to that of the first in, first out discipline.

We study the continuous-time problem of hedging a generalized call option of the European and of the American type, in the presence of transaction costs. We show that if the price process of the relevant stock fluctuates with positive probability, then the only hedge that is possible for the American option is the trivial one. If the price of the stock, in addition to fluctuating with positive probability, is also stable with positive probability, then the same is true for the European option. We also show that in some sense, stable price with positive probability is a necessary condition for having only a trivial hedge for the European option. Our basic idea is to work with an appropriate discrete-time version of the problem which is transaction costs free. The mathematical tools that we use are elementary. A related result appears in Soner, Shreve and Cvitanic.

In this paper we consider a class of stochastic approximation procedures that arises in linear filtering and regression theory. Our main result asserts that the stochastic approximation process satisfies an almost sure invariance principle (with a certain rate of convergence) if the partial sums of the errors do.

We describe the mean rate at which a general absolutely continuous stationary $S \alpha S$ process crosses a high level. Only nondegeneracy assumptions are imposed in the case $1 < \alpha < 2$. The same results hold for $0 < \alpha \leq 1$ under certain conditions, ensuring existence of the required conditional moments and the applicability of the classical integral formula for the expected number of level crossings.

A matrix-valued counting process that allows for modeling of multivariate failure-time data is presented. the inclusion of covariates in a Cox-type regression model is considered, and asymptotic properties of the estimates of regression parameters appearing in the model are studied.

Consider a class of nonstationary time series with the form $Y_t = \mu_t + \xi_t$ where ${\xi_t}$ is a sequence of infinite moving averages of independent random variables with regularly varying tail probabilities and different scale parameters. In this article, the extreme value theory of ${Y_t}$ is studied. Under mild conditions, convergence results for a point process based on the moving averages are proved, and extremal properties of the nonstationary time series, including the convergence of maxima to extremal processes and the limit point process of exceedances, are derived. The results are applied to the analysis of tropospheric ozone data in the Chicago area. Probabilities of monthly maximum ozone concentrations exceeding some specific levels are estimated, and the mean rate of exceedances of daily maximum ozone over the national standard 120 ppb is also assessed.

We are given a set of n elements, some of them red, the others blue, but their colors are hidden. We are to determine the composition of this set, or to determine an element of the majority color, by making pairwise comparisons of elements from which we obtain the information "the colors of these two elements are the same," or "they are different." Let $\tau_n$, respectively, $\mu_n$, be the optimal average number of comparisons needed to solve these two problems. We give an explicit expression of the limit of $\tau_n /n$, respectively, of $\mu_n /n$, in terms of the probabilities of being red or blue. We also discuss quasi-optimal algorithms in both cases: when these probabilities are known and when they are unknown.

We propose a new empirical procedure for detecting phase transition from a single sample of a Gibbs-Markov random field. The method is based on frequencies for large deviations when the whole sample is divided in smaller blocks and estimates for the rate function. We relate our approach to an almost sure large deviation principle.