Bernoulli convolutions associated with certain non-Pisot numbers

The Bernoulli convolution ν_λ measure is shown to be absolutely continuous with L² density for almost all , and singular if λ⁻¹ is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions ν_λ such that their density functions, if they exist, are not L². We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly.     

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space . Specifically, we prove that the Brownian motion belongs to for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces , and indicate the endpoint large deviation estimates.     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

Sharp maximal inequality for nonnegative martingales

Let X be a nonnegative martingale, let H be a predictable process taking values in [−1,1] and let Y be an Itô integral of H with respect to X. We establish the bound and show that the constant 3 is the best possible.     

Lower classes of the Riemann–Liouville process

Let be a Riemann–Liouville process with index H>0. We characterize the lower classes of its sup-norm statistic by a unique integral test and thus measure the influence of the non-stationarity of increments.     

Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

Analytic and numerical analysis of some statistical features of fragmentation processes

In this work, we investigate the following inhomogeneous model of fragmentation with an exponential splitting probability: suppose that an initial particle having a mass x₀>0 (large enough) is broken, with a size-dependent probability , into b>1 pieces. With probability 1−p(x₀), this fragment is left unchanged for ever. The process of fragmentation is repeated recursively for all pieces.Some statistical features are studied, chiefly the fragment size distribution, number of fragments and partition function of the random fragmentation tree. Also, we give details of the numerical analysis of these features.     

Asymptotic cone of semisimple orbits for symmetric pairs

Let G be a reductive algebraic group over C and denote its Lie algebra by g. Let Oh be a closed G-orbit through a semisimple element h∈g. By a result of Borho and Kraft (1979) [4], it is known that the asymptotic cone of the orbit Oh is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer ZG(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair.More precisely, let θ be an involution of G, and K=Gθ a fixed point subgroup of θ. Then we have a Cartan decomposition g=k+s of the Lie algebra g=Lie(G) which is the eigenspace decomposition of θ on g. Let {x,h,y} be a normal sl₂ triple, where x,y∈s are nilpotent, and h∈k semisimple. In addition, we assume , where denotes the complex conjugation which commutes with θ. Then is a semisimple element in s, and we can consider a semisimple orbit Ad(K)a in s, which is closed. Our main result asserts that the asymptotic cone of Ad(K)a in s coincides with , if x is even nilpotent.     

Equilibrium states for factor maps between subshifts

Let π:X→Y be a factor map, where (X,σX) and (Y,σY) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a=(a₁,a₂)∈R² with a₁>0 and a₂?0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes . In particular, taking f≡0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a₁hμ(σX)+a₂hμ°π−1(σY), which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dimHμ=dimHK.     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.     

Bogomolov restriction theorem for Higgs bundles

Let (E,θ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C⊂X be a smooth complete curve with [C]=n⋅H. If where , then we prove that the restriction of (E,θ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.     

Euler–Lehmer constants and a conjecture of Erdös

The Euler–Lehmer constants γ(a,q) are defined as the limits We show that at most one number in the infinite list is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If f:Z/qZ→Q is such that f(a)=±1 and f(q)=0, then Erdös conjectured that If , we show that the Erdös conjecture is true.