L\'evy processes with marked jumps II : Application to a population model with mutations at birth

Consider a population where individuals give birth at constant rate during their lifetimes to i.i.d. copies of themselves. Individuals bear clonally inherited types, but (neutral) mutations may happen at the birth events. The smallest subtree containing the genealogy of all the extant individuals at a fixed time \tau, is called the coalescent point process. We enrich this process with the history of the mutations that appeared over time, and call it the marked coalescent point process. With the help of limit theorems for L\'evy processes with marked jumps established in a previous work (arXiv:1305.6245), we prove the convergence of the marked coalescent point process with large population size and two possible regimes for the mutations - one of them being a classical rare mutation regime, towards a multivariate Poisson point process. This Poisson point process can be described as the coalescent point process of the limiting population at \tau, with mutations arising as inhomogeneous regenerative sets along the lineages. Its intensity measure is further characterized thanks to the excursion theory for spectrally positive L\'evy processes. In the rare mutations asymptotic, mutations arise as the image of a Poisson process by the ladder height process of a L\'evy process with infinite variation, and in the particular case of the critical branching process with exponential lifetimes, the limiting object is the Poisson point process of the depths of excursions of the Brownian motion, with Poissonian mutations on the lineages.Comment: 45 pages, 6 figure

Strong memoryless times and rare events in Markov renewal point processes

Let W be the number of points in (0,t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable \zeta, we construct a ``strong memoryless time'' \hat \zeta such that \zeta-t is exponentially distributed conditional on {\hat \zeta\leq t, \zeta>t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000005

A two-time-scale phenomenon in a fragmentation-coagulation process

Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course that $A$, or $B$, is not empty). The number of balls in $B$ after $n$ steps is of order $\sqrt n$, and this number remains essentially the same after $\sqrt n$ further steps. Observe that each ball in the urn $B$ after $n$ steps has a probability bounded away from $0$ and $1$ to be placed back in the urn $A$ after $\sqrt n$ further steps. So, even though the number of balls in $B$ does not evolve significantly between $n$ and $n+\sqrt n$, the precise contain of urn $B$ does. This elementary observation is the source of an interesting two-time-scale phenomenon which we illustrate using a simple model of fragmentation-coagulation. Inspired by Pitman's construction of coalescing random forests, we consider for every $n\in \N$ a uniform random tree with $n$ vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits subaging in the sense that when it is observed after $k$ steps in the regime $k\sim tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as $n\to\infty$; but different values of $t$ yield distinct pseudo-stationary distributions

Large-deviation principles for connectable receivers in wireless networks

We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connectComment: 29 pages, 2 figure

Complete convergence and records for dynamically generated stochastic processes

We consider empirical multi-dimensional Rare Events Point Processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time and record values point processes. An example where the clustering prevents the convergence of the record times point process is given

Asymptotics for hitting times

In this paper we characterize possible asymptotics for hitting times in aperiodic ergodic dynamical systems: asymptotics are proved to be the distribution functions of subprobability measures on the line belonging to the functional class {6pt} {-3mm}(A){6mm}F={F:R\to [0,1]:\left\lbrack \matrixF is increasing, null on ]-\infty, 0]; \noalignF is continuous and concave; \noalignF(t)\le t for t\ge 0.\right.}. {6pt} Note that all possible asymptotics are absolutely continuous.Comment: Published at http://dx.doi.org/10.1214/009117904000000883 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org