Stability of the nonlinear filter for slowly switching Markov chains

Exponential stability of the nonlinear filtering equation is revisited, when the signal is a finite state Markov chain. An asymptotic upper bound for the filtering error due to incorrect initial condition is derived in the case of slowly switching signal.Comment: the final versio

Estimation in threshold autoregressive models with correlated innovations

Large sample statistical analysis of threshold autoregressive (TAR) models is usually based on the assumption that the underlying driving noise is uncorrelated. In this paper, we consider a model, driven by Gaussian noise with geometric correlation tail and derive a complete characterization of the asymptotic distribution for the Bayes estimator of the threshold parameter.Comment: to appear in Ann. Inst. Stat. Mat

Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities

Many populations, e.g. of cells, bacteria, viruses, or replicating DNA molecules, start small, from a few individuals, and grow large into a noticeable fraction of the environmental carrying capacity $K$. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from $Z_0=z_0$ in a branching process or Malthusian like, roughly exponential fashion, $Z_t \sim a^tW$, where $Z_t$ is the size at discrete time $t\to\infty$, $a>1$ is the offspring mean per individual (at the low starting density of elements, and large $K$), and $W$ a sum of $z_0$ i.i.d. random variables. It will, thus, become detectable (i.e. of the same order as $K$) only after around $\log K$ generations, when its density $X_t:=Z_t/K$ will tend to be strictly positive. Typically, this entity will be random, even if the very beginning was not at all stochastic, as indicated by lower case $z_0$, due to variations during the early development. However, from that time onwards, law of large numbers effects will render the process deterministic, though initiated by the random density at time log $K$, expressed through the variable $W$. Thus, $W$ acts both as a random veil concealing the start and a stochastic initial value for later, deterministic population density development. We make such arguments precise, studying general density and also system-size dependent, processes, as $K\to\infty$. As an intrinsic size parameter, $K$ may also be chosen to be the time unit. The fundamental ideas are to couple the initial system to a branching process and to show that late densities develop very much like iterates of a conditional expectation operator.Comment: presented at IV Workshop on Branching Processes and their Applications at Badajoz, Spain, 10-13 April, 201

On the emergence of random initial conditions in fluid limits

The paper presents a phenomenon occurring in population processes that start near zero and have large carrying capacity. By the classical result of Kurtz~(1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to infinity, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth and death process