We present two approaches for linear prediction of long-memory time series. The first approach consists in truncating the Wiener-Kolmogorov predictor by restricting the observations to the last k terms, which are the only available data in practice. We derive the asymptotic behaviour of the mean-squared error as k tends to +∞. The second predictor is the finite linear least-squares predictor i.e.  the projection of the forecast value on the last k observations. It is shown that these two predictors converge to the Wiener Kolmogorov predictor at the same rate k -1.

Consider testing whether F = F0 for a continuous cdf on R = (-∞,∞)and for a random sample X 1,..., X n from F.We derive expansions of the associated asymptotic power basedon the Cramer-von Mises, Kolmogorov-Smirnov and Kuiper statistics. We provide numerical illustrations using a double-exponential example with a shifted alternative.


Functionals of spatial point process often satisfy a weak spatial dependencecondition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which isa level-3 result for empirical point fields as well as a level-2 resultfor empirical point measures. The level-3 rate function coincides withthe so-called specific information. We show that the general resultcan be applied to prove MDPs for various particular functionals,including random sequential packing, birth-growth models, germ-grainmodels and nearest neighbor graphs. 


Let $(\xi_n)_{n\ge 1}$ be the polygonal partial sums processes built on the linear processes $X_n=\sum_{i\ge 0}a_i(\epsilon_{n-i})$ , n ≥ 1, where $(\epsilon_i)_{i\in\mathbb{Z}}$ are i.i.d., centered random elements in some separable Hilbert space $\mathbb{H}$ and the a i 's are bounded linear operators $\mathbb{H}\to \mathbb{H}$ , with $\sum_{i\ge 0}\lVert a_i\rVert<\infty$ . We investigate functional central limit theorem for $\xi_n$ in the Hölder spaces $\mathrm{H}^o_\rho(\mathbb{H})$ of functions $x:[0,1]\to\mathbb{H}$ such that ||x(t + h) - x(t)|| = o(p(h)) uniformly in t, where p(h) = hαL(1/h), 0 ≤ h ≤ 1 with 0 ≤ α ≤ 1/2 and L slowly varying at infinity. We obtain the $\mathrm{H}^o_\rho(\mathbb{H})$ weak convergence of $\xi_n$ to some $\mathbb{H}$ valued Brownian motion under the optimal assumption that for any c>0, $tP(\lVert \epsilon_0\rVert>ct^{1/2}\rho(1/t))=o(1)$ when t tends to infinity, subject to some mild restriction on L in the boundary case α = 1/2. Our result holds in particular with the weight functions p(h) = h1/2 lnβ(1/h), β > 1/2>.

The aim of this short note is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously.

In this paper, we apply the technique of decoupling to obtain some exponential inequalities for semi-bounded martingale, which extend the results of de la Peña, Ann. probab.27 (1999) 537–564.

On développe une approche générale du théorème limite centrale presque-sûre pour les martingales vectorielles quasi-continues à gauche convenablement normalisées dont on dégage une extension quadratique et un nouveau théorème de la limite centrale. L'application de ce résultat à l'estimation de la variance d'un processus à accroissements indépendants et stationnaires illustre l'usage qu'on peut en faire en statistique.

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

This paper deals with uniform consistency and uniform confidence bands for the quantile function and its derivatives. We describe a kernel local polynomial estimator of quantile function and give uniform consistency. Furthermore, we derive its maximal deviation limit distribution using an approximation in the spirit of Bickel and Rosenblatt [P.J. Bickel and M. Rosenblatt, Ann. Statist. 1 (1973) 1071–1095].

We use multivariate total positivity theory to exhibit new families of peacocks. As theauthors of [F. Hirsch, C. Profeta, B. Roynette and M. Yor, Peacocks and associatedmartingales vol. 3. Bocconi-Springer (2011)], our guiding example is the resultof Carr−Ewald−Xiao [P. Carr, C.-O. Ewald and Y. Xiao,Finance Res. Lett. 5 (2008) 162–171]. We shall introducethe notion of strong conditional monotonicity. This concept is strictly more restrictivethan the conditional monotonicity as defined in [F. Hirsch, C. Profeta, B. Roynette and M.Yor, Peacocks and associated martingales, vol. 3. Bocconi-Springer (2011)] (see also [R.H.Berk, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42(1978) 303–307], [A.M. Bogso, C. Profeta and B. Roynette, Lect. Notes Math.Springer, Berlin (2012) 281–315.] and [M. Shaked and J.G. Shanthikumar,Probab. Math. Statistics. Academic Press, Boston (1994)].). There aremany random vectors which are strongly conditionally monotone (SCM). Indeed, we shallprove that multivariate totally positive of order 2 (MTP2) random vectors are SCM. As aconsequence, stochastic processes with MTP2 finite-dimensional marginals are SCM. This familyincludes processes with independent and log-concave increments, and one-dimensionaldiffusions which have absolutely continuous transition kernels.

We consider the random vector $u(t,\underlinex)=(u(t,x_1),\dots,u(t,x_d))$ , where t > 0, x1,...,xd aredistinct points of $\mathbb{R}^2$ and u denotes the stochastic process solution to a stochastic waveequation driven bya noise white in time and correlated in space. In a recent paper byMillet and Sanz–Solé[10], sufficient conditions are given ensuring existence andsmoothness ofdensity for $u(t,\underline x)$ . We study here the positivity of suchdensity. Usingtechniques developped in [1] (see also [9]) basedon Analysis on anabstract Wiener space, we characterize the set of points $y\in\mathbb{R}^d$ where the density ispositive and we prove that, under suitable assumptions, this set is $\mathbb{R}^d$ .

We study in this paper the convergence rate of the Swendsen-Wang dynamics towards its equilibrium law, when the energy belongs to a large family of energies used in image segmentation problems. We compute the exponential equivalents of the transitions which control the process at low temperature, as well as the critical constant which gives its convergence rate. We give some theoretical tools to compare this dynamics with Metropolis, and develop an experimental study in order to calibrate both dynamics performances in image segmentation problems.

The problem of predicting integrals of stochastic processes is considered. Linear estimators have been constructed by means of samples at N discrete times for processes having a fixedHölderian regularity s > 0 in quadratic mean. It is known that the rate of convergence of the mean squared error is of order N-(2s+1). In the class of analytic processes H p , p ≥ 1, we show that among all estimators, the linear ones are optimal. Moreover, using optimal coefficient estimators derived through the inversion of the covariance matrix, the corresponding maximal error has lower and upper bounds with exponential rates. Optimal simple nonparametric estimators with optimal sampling designs are constructed in H² and H ∞ and have also bounds with exponential rates.

This paper is a corrigendum to paper Toldo, ESAIM, P&S10 (2006) 141–163 where we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time.

In practice, it is well known that hedging a derivative instrumentcan never be perfect. In the case of credit derivatives (e.g.synthetic CDO tranche products), a trader will have to face somespecific difficulties. The first one is the inconsistence betweenmost of the existing pricing models, where the risk is theoccurrence of defaults, and the real hedging strategy, where thetrader will protect his portfolio against small CDS spreadmovements. The second one, which is the main subject of thispaper, is the consequence of a wrong estimation of some parametersspecific to credit derivatives such as recovery rates orcorrelation coefficients. We find here an approximation of thedistribution under the historical probability of the final Profit& Loss of a portfolio hedged with wrong estimations of theseparameters. In particular, it will depend on a ratio between thesquare root of the historical default probability and therisk-neutral default probability. This result is quite general andnot specific to a given pricing model.

We generalize the results of Komlós, Major and Tusnády concerning the strong approximation of partial sums of independent and identically distributed random variables with a finite r-th momentto the case when the parameter set is two-dimensional. The most striking result is that the rates of convergence are exactly the same as in the one-dimensional case.

This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k,m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m. We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models).

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. These filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. However, in several cases, non-monotonic sequences of filters may appear more appropriate. In this paper, we study a hard-thresholding regularization method that extends the spectral cut-off procedure to non-monotonic sequences. We provide several oracle inequalities, showing the method to be nearly optimal under mild assumptions. Contrary to similar methods discussed in the literature, we use here a non-linear threshold that appears to be adaptive to all degrees of irregularity, whether the problem is mildly or severely ill-posed. Finally, we extend the method to inverse problems with noisy operator and provide efficiency results in a conditional framework.

The transition kernel of the well-known Metropolis-Hastings (MH) algorithm has a point mass at the chain’s current position, which prevent direct smoothness properties to be derived for the successive densities of marginals issued from this algorithm. We show here that under mild smoothness assumption on the MH algorithm “input” densities (the initial, proposal and target distributions), propagation of a Lipschitz condition for the iterative densities can be proved. This allows us to build a consistent nonparametric estimate of the entropy for these iterative densities. This theoretical study can be viewed as a building block for a more general MCMC evaluation tool grounded on such estimates.


We introduce the fractional mixed fractional Brownian motion and fractionalBrownian sheet, and investigate the small ball behavior of its sup-norm statistic.Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.