The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807–832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897–906]. First, we study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its the asymptotic distribution. Second, we consider the extension of the ideas in Gassiat [Ann. Inst. Henri Poincaré 38 (2002) 897–906] to autoregressive models with regimes switches according to a Markov chain. In this case, it is shown that the marginal likelihood is no longer a contrast function and cannot be used to select the number of regimes. Some numerical examples illustrate the results and their convergence properties.

The growth exponent α for loop-erased or Laplacian random walkon the integer lattice is defined by saying that the expected time toreach the sphere of radius n is of order nα . We prove thatin two dimensions, the growth exponent is strictly greater than one.The proof uses a known estimate on the third moment of the escapeprobability and an improvement on the discrete Beurling projection theorem.

We establish posterior consistency for non-parametric Bayesian estimation of thedispersion coefficient of a time-inhomogeneous Brownian motion.

We consider the continuous time, one-dimensional random walk in random environmentin Sinai's regime. We show that the probability for theparticle to be, at time t and in a typical environment,at a distance larger than ta (0<a<1)from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus.As a byproduct, the systematic method for constructing entropieswhich we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.

We continue the investigation started in a previous paper, onweak convergence to infinitely divisible distributions with finitevariance. In the present paper, we study this problem for someweakly dependent random variables, including in particularassociated sequences. We obtain minimal conditions expressed interms of individual random variables. As in the i.i.d. case, wedescribe the convergence to the Gaussian and the purelynon-Gaussian parts of the infinitely divisible limit. We alsodiscuss the rate of Poisson convergence and emphasize the specialcase of Bernoulli random variables. The proofs aremainly based on Lindeberg's method.

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle's current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed. The paper realizes the McKean's program for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.

We prove that the large deviation principle holds for a class ofprocesses inspired by semi-Markov additive processes. For theprocesses we consider, the sojourn times in the phase process neednot be independent and identically distributed. Moreover thestate selection process need not be independent of the sojourntimes. We assume that the phase process takes values in a finite set andthat the order in which elements in the set, called states, arevisited is selected stochastically. The sojourn times determinehow long the phase process spends in a state once it has beenselected. The main tool is a representation formula for the samplepaths of the empirical laws of the phase process. Then, based onassumed joint large deviation behavior of the state selection andsojourn processes, we prove that the empirical laws of the phaseprocess satisfy a sample path large deviation principle. From thislarge deviation principle, the large deviations behavior of aclass of modulated additive processes is deduced. As an illustration of the utility of the general results, we providean alternate proof of results for modulated Lévy processes. As apractical application of the results, we calculate the large deviationrate function for a processes that arises as the InternationalTelecommunications Union's standardized stochastic model of two-wayconversational speech.

We have seen in a previous article how the theory of “rough paths”allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructedfor stochastic processes generated by divergence form operators by using time-reversal techniques.

We study the convergence rate of randomly truncated stochastic algorithms, which consistin the truncation of the standard Robbins–Monro procedure on an increasing sequence ofcompact sets. Such a truncation is often required in practice to ensure convergence whenstandard algorithms fail because the expected-value function grows too fast. In this work,we give a self contained proof of a central limit theorem for this algorithm under localassumptions on the expected-value function, which are fairly easy to check in practice.

In many applications, we assume that two random observations x andy are generated according to independent Poisson distributions\hbox{$\PPP(\lambdaS)$}𝒫(λS)and \hbox{$\PPP(\muT)$}𝒫(μT)and we are interested in performing statistical inference on the ratioφ = λ / μ of the twoincidence rates. In vaccine efficacy trials, x and y aretypically the numbers of cases in the vaccine and the control groups respectively,φ is called the relative risk and the statistical model is called‘partial immunity model’. In this paper we start by defining a natural semi-conjugatefamily of prior distributions for this model, allowing straightforward computation of theposterior inference. Following theory on reference priors, we define the reference priorfor the partial immunity model when φ is the parameter of interest. Wealso define a family of reference priors with partial information on μwhile remaining uninformative about φ. We notice that these priors belongto the semi-conjugate family. We then demonstrate using numerical examples that Bayesiancredible intervals for φ enjoy attractive frequentist properties whenusing reference priors, a typical property of reference priors.

We consider a system of fully coupled forward-backward stochasticdifferential equations. First we generalize the results ofPardoux-Tang [7] concerning the regularity of the solutions withrespect to initial conditions. Then, we prove that in some particularcases this system leads to aprobabilistic representation of solutions of a second-order PDE whosesecond order coefficients depend on the gradient of the solution. Wethen give some examples in dimension 1 and dimension 2 for which theassumptions are easy to check.

Let X1,...,Xn1 be arandom sample from a population with mean µ 1 and variance $\sigma_1^2$ , and X1,...,Xn1 be a random sample fromanother population with mean µ 2 and variance $\sigma_2^2$ independent of{Xi,1 ≤ i ≤ n1 }.Consider the twosample t-statistic $ T={{\bar X-\bar Y-(\mu_1-\mu_2)} \over\sqrt{s_1^2/n_1+s_2^2/n_2}}$ . This paper shows thatln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, inaddition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T \geq x)}{1-\Phi(x)} \to 1 $ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

The paper develops an approach to optimal design problems based on application of abstract optimisation principles in the space of measures. Various design criteria and constraints, such as bounded density, fixed barycentre, fixed variance, etc. are treated in a unified manner providing a universal variant of the Kiefer-Wolfowitz theorem and giving a full spectrum of optimality criteria for particular cases. Incorporating the optimal design problems into conventional optimisation framework makes it possible to use the whole arsenal of descent algorithms from the general optimisation literature for finding optimal designs. The corresponding steepest descent involves adding a signed measure at every step and converges faster than the conventional sequential algorithms used to construct optimal designs. We study a new class of design problems when the observation points are distributed according to a Poisson point process arising in the situation when the total control on the placement of measurements is impossible.

In the problem of signal detectionin Gaussian white noisewe show asymptotic minimaxity of kernel-based tests. The test statisticsequal L 2-norms of kernel estimates.The sets of alternatives are essentially nonparametric and are defined asthe sets of all signals such that the L 2-norms of signal smoothed by the kernels exceed some constants pε > 0.The constant pε depends on the power ϵof noise and pε → 0 as ε → 0.Similar statements are proved also if an additional informationon a signal smoothness is given.By theorems on asymptotic equivalence of statistical experimentsthese results are extended to the problems of testing nonparametrichypotheseson density and regression. The exact asymptotically minimaxlower bounds of type II error probabilities are pointed out forall these settings. Similar results are also obtained for the problemsof testing parametric hypotheses versus nonparametric sets of alternatives.

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function $f : \; ]0, +\infty[ \;\to \mathbb{R}$ with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab.4 (1999) 111–118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and L p -norms.

This paper introduces a computationally tractable density estimatorthat has the same asymptotic variance as the classical Nadaraya-Watsondensity estimator but whose asymptotic bias is zero. We achieve this resultusing a two stage estimator that applies a multiplicative bias correctionto an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator.

Let F n be the empirical distribution function (df) pertainingto independent random variables with continuous df F. Weinvestigate the minimizing point $\hat\tau_n$ of the empiricalprocess Fn - F0 , where F 0 is another df which differs fromF. If F and F 0 are locally Hölder-continuous of orderα at a point τ our main result states that $n^{1/\alpha}(\hat\tau_n - \tau)$ converges in distribution. Thelimit variable is the almost sure unique minimizing point of atwo-sided time-transformed homogeneous Poisson-process with adrift. The time-transformation and the drift-function are of thetype |t|α.

In this paper, we give sufficient conditions to establish central limittheorems and moderate deviation principle for a class of support estimates ofempirical and Poisson point processes. The considered estimates are obtained bysmoothing some bias corrected extreme values of the point process. We show howthe smoothing permits to obtain Gaussian asymptotic limits and thereforepointwise confidence intervals. Some unidimensional and multidimensionalexamples are provided.