This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.

We consider two processes that have been used to study gene duplication, Watterson’s [Genetics 105 (1983) 745–766] double recessive null model and Lynch and Force’s [Genetics 154 (2000) 459–473] subfunctionalization model. Though the state spaces of these diffusions are two and six-dimensional, respectively, we show in each case that the diffusion stays close to a curve. Using ideas of Katzenberger [Ann. Probab. 19 (1991) 1587–1628] we show that one-dimensional projections converge to diffusion processes, and we obtain asymptotics for the time to loss of one gene copy. As a corollary we find that the probability of subfunctionalization decreases exponentially fast as the population size increases. This rigorously confirms a result Ward and Durrett [Theor. Pop. Biol. 66 (2004) 93–100] found by simulation that the likelihood of subfunctionalization for gene duplicates decays exponentially fast as the population size increases.

Consider Z_t^f(u)=∫₀^tuf(N_s) ds, t>0, u∈[0, 1], where N=(N_t)_t∈ℝ is a normal process and f is a measurable real-valued function satisfying Ef(N₀)²<∞ and Ef(N₀)=0. If the dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002) 191–216] showed that Z_t^f/t^1/2 converges in distribution to a multiple of standard Brownian motion as t→∞. If the dependence is sufficiently strong, then Z_t/(EZ_t(1)²)^1/2 converges in distribution to a higher order Hermite process as t→∞ by a result by Taqqu [Wahrsch. Verw. Gebiete 50 (1979) 53–83]. When passing from weak to strong dependence, a unique situation encompassed by neither results is encountered. In this paper, we investigate this situation in detail and show that the limiting process is still a Brownian motion, but a nonstandard norming is required. We apply our result to some functionals of fractional Brownian motion which arise in time series. For all Hurst indices H∈(0, 1), we give their limiting distributions. In this context, we show that the known results are only applicable to H<3/4 and H>3/4, respectively, whereas our result covers H=3/4.

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is 0 (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.

This paper proves a Berry–Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be O(n^−1/2) as n→∞, where n denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate O(n^−1/2) is not known even under an exponential strong mixing rate. The main result of the paper has applications in finance and econometrics as financial time series data often are heavy-tailed and quantile based methods play an important role in various problems in finance, including hedging and risk management.

This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form E_x(e^{−λX_t−∫₀tg(X_s) ds}) can be reduced to evaluating a single integral of known functions. Given a drift f we determine the functions g for which the corresponding functional can be calculated by symmetry. Conversely, given g, we can determine precisely those drifts f for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.

Nearest neighbor cells in R^d, d∈ℕ, are used to define coefficients of divergence (φ-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In d=1, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic k-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other.

Let f(⋅, t) be the probability density function which represents the solution of Kac’s equation at time t, with initial data f₀, and let g_σ be the Gaussian density with zero mean and variance σ², σ² being the value of the second moment of f₀. This is the first study which proves that the total variation distance between f(⋅, t) and g_σ goes to zero, as t→+∞, with an exponential rate equal to −1/4. In the present paper, this fact is proved on the sole assumption that f₀ has finite fourth moment and its Fourier transform ϕ₀ satisfies |ϕ₀(ξ)|=o(|ξ|^−p) as |ξ|→+∞, for some p>0. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.

In this paper, we consider certain σ-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in σ-finite measures, which in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of i.i.d. random variables, to products of independent random variables, and to stochastic integrals with respect to Lévy motions.

We consider a stochastic model of Internet congestion control, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows in a network where bandwidth is shared among document transfers. In contrast to an earlier work by Kelly and Williams [Ann. Appl. Probab. 14 (2004) 1055–1083], the present paper allows interarrival times and document sizes to be generally distributed, rather than exponentially distributed. Furthermore, we allow a fairly general class of bandwidth sharing policies that includes the weighted α-fair policies of Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556–567], as well as certain other utility based scheduling policies. To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes of all flows through the network. We propose a fluid model (or formal functional law of large numbers approximation) associated with the stochastic flow level model. Under mild conditions, we show that the appropriately rescaled measure valued processes corresponding to a sequence of such models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution. For the special case of weighted α-fair policies, we also characterize the invariant states of the fluid model.

We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle θ of the cone: for θ≥180°, the direction is deterministic, while for θ<180°, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

In this paper the reflection and transmission of waves by a three-dimensional random medium are studied in a white-noise and paraxial regime. The limit system derives from the acoustic wave equations and is described by a coupled system of random Schrödinger equations driven by a Brownian field whose covariance is determined by the two-point statistics of the fluctuations of the random medium. For the reflected and transmitted fields the associated Wigner distributions and the autocorrelation functions are determined by a closed system of transport equations. The Wigner distribution is then used to describe the enhanced backscattering phenomenon for the reflected field.

Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute the limiting distributions of the system and determine the time evolution of the credit quality indicators of the firms, deriving moreover the dynamics of a global financial health indicator. We finally describe a suitable version of the “Central Limit Theorem” useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis.

We propose an adaptive independent Metropolis–Hastings algorithm with the ability to learn from all previous proposals in the chain except the current location. It is an extension of the independent Metropolis–Hastings algorithm. Convergence is proved provided a strong Doeblin condition is satisfied, which essentially requires that all the proposal functions have uniformly heavier tails than the stationary distribution. The proof also holds if proposals depending on the current state are used intermittently, provided the information from these iterations is not used for adaption. The algorithm gives samples from the exact distribution within a finite number of iterations with probability arbitrarily close to 1. The algorithm is particularly useful when a large number of samples from the same distribution is necessary, like in Bayesian estimation, and in CPU intensive applications like, for example, in inverse problems and optimization.

In this work we define two kinds of crested product for reversible Markov chains, which naturally appear as a generalization of the case of crossed and nested product, as in association schemes theory, even if we do a construction that seems to be more general and simple. Although the crossed and nested product are inspired by the study of Gelfand pairs associated with the direct and the wreath product of two groups, the crested products are a more general construction, independent from the Gelfand pairs theory, for which a complete spectral theory is developed. Moreover, the k-step transition probability is given. It is remarkable that these Markov chains describe some classical models (Ehrenfest diffusion model, Bernoulli–Laplace diffusion model, exclusion model) and give some generalization of them.

As a particular case of nested product, one gets the classical Insect Markov chain on the ultrametric space. Finally, in the context of the second crested product, we present a generalization of this Markov chain to the case of many insects and give the corresponding spectral decomposition.

We consider the model of random interlacements on ℤ^d introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability that the origin belongs to the infinite component of the vacant set at level u in the supercritical phase u<u_*.

For independent X and Y in the inequality P(X≤Y+μ), we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).