On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Estimating the ground state energy of the Schrödinger equation for convex potentials

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε$\epsilon$. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d$d$. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$ and ε⁻¹${\epsilon }^{-1}$, while the number of qubits is polynomial in d$d$ and logε⁻¹$log{\epsilon }^{-1}$. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1)$1-\delta ,\delta \in (0,1)$, with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε$\epsilon$. The cost of the algorithm is polynomial in d$d$, ε⁻¹${\epsilon }^{-1}$ and δ⁻¹${\delta }^{-1}$, while the number of qubits is polynomial in d$d$, logε⁻¹$log{\epsilon }^{-1}$ and logδ⁻¹$log{\delta }^{-1}$.     

Transformation formulas for fractional Brownian motion

We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index K$K$ into fractional Brownian motion of index H$H$. Integration is carried out over [0,t]$[0,t]$, t>0$t>0$. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L²(P)$L^2\left(\mathbb{P}\right)$-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t]$(-\infty ,t]$, t>0$t>0$.     

Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

The on-line nearest-neighbour graph on a sequence of n$n$ uniform random points in (0,1)^d${(0,1)}^d$ (d∈N$d\in \mathbb{N}$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α∈(0,d/2]$\alpha \in (0,d/2]$, we prove O(max{n^1−(2α/d),logn})$O(max\{n^{1-(2\alpha /d)},logn\})$ upper bounds on the variance. On the other hand, we give an n→∞$n\to \infty$ large-sample convergence result for the total power-weighted edge-length when α>d/2$\alpha >d/2$. We prove corresponding results when the underlying point set is a Poisson process of intensity n$n$.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.     

On probabilistic results for the discrepancy of a hybrid-Monte Carlo sequence

In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s$s$-dimensional sequence m$m$, whose elements are vectors obtained by concatenating d$d$-dimensional vectors from a low-discrepancy sequence q$q$ with (s−d)$(s-d)$-dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε>0$\epsilon >0$ the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$ is bounded by ε$\epsilon$ with probability at least 1−2exp(−ε²N/2)$1-2exp(-{\epsilon }^{2}N/2)$ for N$N$ sufficiently large. The authors did not study how large N$N$ actually has to be and if and how this actually depends on the parameters s$s$ and ε$\epsilon$. In this note we derive a lower bound for N$N$, which significantly depends on s$s$ and ε$\epsilon$. Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first N$N$ points of m$m$ and q$q$, which holds without any restrictions on N$N$. In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes N$N$. We compare this bound to other known bounds.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.     

Comparison inequalities on Wiener space

We define a covariance-type operator on Wiener space: for F$F$ and G$G$ two random variables in the Gross–Sobolev space D^1,2${\mathbb{D}}^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let _ΓF,G?〈DF,−DL⁻¹G〉${\Gamma }_{F,G}?〈{DF,-DL}^{-1}G〉$, where D$D$ is the Malliavin derivative operator and L⁻¹$L^{-1}$ is the pseudo-inverse of the generator of the Ornstein–Uhlenbeck semigroup. We use Γ$\Gamma$ to extend the notion of covariance and canonical metric for vectors and random fields on Wiener space, and prove corresponding non-Gaussian comparison inequalities on Wiener space, which extend the Sudakov–Fernique result on comparison of expected suprema of Gaussian fields, and the Slepian inequality for functionals of Gaussian vectors. These results are proved using a so-called smart-path method on Wiener space, and are illustrated via various examples. We also illustrate the use of the same method by proving a Sherrington–Kirkpatrick universality result for spin systems in correlated and non-stationary non-Gaussian random media.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N$N$ independent copies of AR(1) process with random-coefficient a∈[0,1)$a\in [0,1)$ when N$N$ and time scale n$n$ increase at different rate. Assuming that a$a$ has a density, regularly varying at a=1$a=1$ with exponent −1<β<1$-1<\beta <1$, different joint limits of normalized aggregated partial sums are shown to exist when N^1/(1+β)/n$N^{1/(1+\beta )}/n$ tends to (i) ∞$\infty$, (ii) 0$0$, (iii) 0<μ<∞$0<\mu <\infty$. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R)$(0,\infty )\times C\left(\mathbb{R}\right)$ and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).     

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn^−β$\alpha n^{-\beta }$, with α>0$\alpha >0$, β∈[0,+∞]$\beta \in [0,+\infty ]$ and n$n$ is the scaling parameter. Depending on the regime of β$\beta$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1$\beta =1$, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α$\alpha$, interpolating a fractional Brownian motion of Hurst exponent 1/4$1/4$ and the degenerate process equal to zero.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.