## Recursos de colección

1. #### Change-point estimation from indirect observations. 2. Adaptation

Goldenshluger, A.; Juditsky, A.; Tsybakov, A.; Zeevi, A.
We focus on the problem of adaptive estimation of signal singularities from indirect and noisy observations. A typical example of such a singularity is a discontinuity (change-point) of the signal or of its derivative. We develop a change-point estimator which adapts to the unknown smoothness of a nuisance deterministic component and to an unknown jump amplitude. We show that the proposed estimator attains optimal adaptive rates of convergence. A simulation study demonstrates reasonable practical behavior of the proposed adaptive estimates.

2. #### Change-point estimation from indirect observations. 1. Minimax complexity

Goldenshluger, A.; Juditsky, A.; Tsybakov, A. B.; Zeevi, A.
We consider the problem of nonparametric estimation of signal singularities from indirect and noisy observations. Here by singularity, we mean a discontinuity (change-point) of the signal or of its derivative. The model of indirect observations we consider is that of a linear transform of the signal, observed in white noise. The estimation problem is analyzed in a minimax framework. We provide lower bounds for minimax risks and propose rate-optimal estimation procedures.

3. #### On suprema of Lévy processes and application in risk theory

Song, Renming; Vondraček, Zoran
Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ is reached by a jump of the subordinator C. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.

4. #### Near-minimal spanning trees: A scaling exponent in probability models

Aldous, David J.; Bordenave, Charles; Lelarge, Marc
We study the relation between the minimal spanning tree (MST) on many random points and the “near-minimal” tree which is optimal subject to the constraint that a proportion δ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as 1+Θ(δ2). We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.

5. #### Hitting time of a corner for a reflected diffusion in the square

Delarue, F.
We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. ¶ We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish...

6. #### Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case

Erlihson, Michael M.; Granovsky, Boris L.
We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak∼Ckp−1, k→∞, p>0, where C is a positive constant. The measures considered are associated with the generalized Maxwell–Boltzmann models in statistical mechanics, reversible coagulation–fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional...

7. #### A lattice gas model for the incompressible Navier–Stokes equation

Beltrán, J.; Landim, C.
We recover the Navier–Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier–Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.

8. #### Random permutations and unique fully supported ergodicity for the Euler adic transformation

Frick, Sarah Bailey; Petersen, Karl
There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

9. #### Asymptotic Feynman–Kac formulae for large symmetrised systems of random walks

We study large deviations principles for N random processes on the lattice ℤd with finite time horizon [0, β] under a symmetrised measure where all initial and terminal points are uniformly averaged over random permutations. That is, given a permutation σ of N elements and a vector (x1, …, xN) of N initial points we let the random processes terminate in the points (xσ(1), …, xσ(N)) and then sum over all possible permutations and initial points, weighted with an initial distribution. We prove level-two large deviations principles for the mean of empirical path measures, for the mean of paths and...

10. #### Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

Loukianova, D.; Loukianov, O.
Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. ¶ The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of...

11. #### A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs

For large N, we consider the ordinary continued fraction of x=p/q with 1≤p≤q≤N, or, equivalently, Euclid’s gcd algorithm for two integers 1≤p≤q≤N, putting the uniform distribution on the set of p and qs. We study the distribution of the total cost of execution of the algorithm for an additive cost function c on the set ℤ+* of possible digits, asymptotically for N→∞. If c is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing diophantine conditions on the cost, we are able to control the speed of convergence in the...

12. #### Joint continuity of the local times of fractional Brownian sheets

Ayache, Antoine; Wu, Dongsheng; Xiao, Yimin
Let BH={BH(t), t∈ℝ+N} be an (N, d)-fractional Brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional Brownian sheet B0H. We prove that if d<∑ℓ=1NH−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). ¶ We also establish sharp local and global Hölder conditions for the local times of BH. These results are applied to study analytic and geometric properties of the sample paths of BH.

13. #### On mean central limit theorems for stationary sequences

Dedecker, Jérôme; Rio, Emmanuel
In this paper, we give estimates of the minimal ${\mathbb{L}}^{1}$ distance between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary sequences satisfying projective criteria in the style of Gordin or weak dependence conditions.

14. #### On homogenization of space-time dependent and degenerate random flows II

Rhodes, Rémi
We study the long time behavior (homogenization) of a diffusion in random medium with time and space dependent coefficients. The diffusion coefficient may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance principle is proved for the critical rescaling of the diffusion. Here, we generalize this approach to diffusions whose space-time scaling differs from the critical one.

15. #### Limit laws for the energy of a charged polymer

Chen, Xia
In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy ¶ Hn=∑1≤jωjωk1{Sj=Sk} ¶ of the polymer {S1, …, Sn} equipped with random electrical charges {ω1, …, ωn}. Our approach is based on comparison of the moments between Hn and the self-intersection local time ¶ Qn=∑1≤j1{Sj=Sk} ¶ run by the d-dimensional random walk {Sk}. As partially needed for our main objective and partially motivated by their independent interest, the central limit theorems and exponential integrability for Qn are also investigated in the case d≥3.

16. #### Identification of periodic and cyclic fractional stable motions

We consider an important subclass of self-similar, non-Gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar stable mixed moving averages related to dissipative flows have already been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is...

17. #### Skew-product representations of multidimensional Dunkl Markov processes

Chybiryakov, Oleksandr
In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. Séminaire de Probabilités XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.

18. #### Invariance principles for random walks conditioned to stay positive

Caravenna, Francesco; Chaumont, Loïc
Let {Sn} be a random walk in the domain of attraction of a stable law $\mathcal{Y}$ , i.e. there exists a sequence of positive real numbers (an) such that Sn/an converges in law to $\mathcal{Y}$ . Our main result is that the rescaled process (S⌊nt⌋/an, t≥0), when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable Lévy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

19. #### Stabilité du comportement des marches aléatoires sur un groupe localement compact

Gretete, Driss
Dans cet article nous démontrons un théorème de stabilité des probabilités de retour sur un groupe localement compact unimodulaire, séparable et compactement engendré. Nous démontrons que le comportement asymptotique de F*(2n)(e) ne dépend pas de la densité F sous des hypothèses naturelles. A titre d’exemple nous établissons que la probabilité de retour sur une large classe de groupes résolubles se comporte comme exp(−n1/3).

20. #### LAMN property for hidden processes: The case of integrated diffusions

Gloter, Arnaud; Gobet, Emmanuel
In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither Markovian nor Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for...

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