Polynomial Tails Research Articles

Testing for sparse idiosyncratic components in factor-augmented regression models

We propose a novel bootstrap test of a dense model, namely factor regression, against a sparse plus dense alternative model augmented with sparse idiosyncratic components. The asymptotic properties of the test are established under time series dependence and polynomial tails. We outline a data-driven rule to select the tuning parameter and prove its theoretical validity. In simulation experiments, our procedure exhibits high power against sparse alternatives and low power against dense deviations from the null. Moreover, we apply our test to various datasets in macroeconomics and finance and often reject the null. This suggests the presence of sparsity — on top of a dense component — in commonly studied economic applications. The R package ‘FAS’ implements our approach.

Fast pseudorandom quantum state generators via inflationary quantum gates

We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishable from Haar random, with shallow log-n depth quantum circuits, where n is the number of qudits. We argue that logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log n$$\\end{document} depth 2-qubit-gate-based generic random quantum circuits that are claimed to provide a lower bound on the speed of information scrambling, cannot produce computationally pseudorandom quantum states. This conclusion is connected with the presence of polynomial (in n) tails in the stay probability of short Pauli strings that survive evolution through such shallow circuits. We show, however, that stay-probability-tails can be eliminated and pseudorandom quantum states can be accomplished with shallow logn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log n$$\\end{document} depth circuits built from a special universal family of “inflationary” quantum (IQ) gates. We prove that IQ-gates cannot be implemented with 2-qubit gates, but can be realized either as a subset of 2-qudit-gates in U(d2) with d ≥ 3 and d prime, or as special 3-qubit gates.

Mixing Rates of the Geometrical Neutral Lorenz Model

The aim of this paper is to obtain polynomial decay of correlations of a Lorenz-like flow where the hyperbolic saddle at the origin is replaced by a neutral saddle. To do that, we take the construction of the geometrical Lorenz flow and proceed by changing the nature of the saddle fixed point at the origin by a neutral fixed point. This modification is accomplished by changing the linearised vector field in a neighbourhood of the origin for a neutral vector field. This change in the nature of the fixed point will produce polynomial tails for the Dulac times, and combined with methods of Araújo and Melbourne (used to prove exponential mixing for the classical Lorenz flow) this will ultimately lead to polynomial upper bounds of the decay of correlations for the modified flow.

Erdős-Rényi laws for exponentially and polynomially mixing dynamical systems

Erdős-Rényi limit laws give the length scale of a time-window over which time-averages in Birkhoff sums have a non-trivial almost-sure limit. We establish Erdős-Rényi type limit laws for Hölder observables on dynamical systems modeled by Young Towers with exponential and polynomial tails. This extends earlier results on Erdős-Rényi limit laws to a broad class of dynamical systems with some degree of hyperbolicity.

Polynomial Convergence Rates of Piecewise Deterministic Markov Processes

We consider piecewise-deterministic Markov processes such as the Bouncy Particle sampler, on target densities with polynomial tails. Using direct drift condition methods, we provide bounds on the polynomial order of the processes’ convergence rate to stationary, on both one-dimensional and high-dimensional state spaces, in both total variation distance and f-norm.

A multi-agent approach to the impact of epidemic spreading on commercial activities

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of the amount of consumer spending transactions of a multi-agent society, that, in the presence of social distancing laws consequent to an epidemic spreading, can reduce the volume of a consistent part of economic commercial activities requiring the presence. At the Boltzmann level, the microscopic variation of the number of transactions around a desired commercial target, is built up by joining a random risk with a deterministic change. Both the risk and target value parameters are here dependent of the number of social contacts of the multi-agent system, which can vary in time in reason of the governments decisions to control the epidemic spreading by introducing restrictions on the social activities. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck-type equation with variable coefficients of diffusion and drift, characterized by the details of the elementary interaction. The economic description is then coupled with the evolution equations of a new SIR-type compartmental epidemic system suitable to describe both the classical epidemic spreading and the social contacts evolution in dependence of the multi-agent social heterogeneity. The (local in time) steady states of the statistical distribution of the amount of economic transactions predicted by the Fokker–Planck equation are shown to be inverse Gamma densities, with time-dependent polynomial tails at infinity which characterize the consequences of the control policies on the time evolution of the Pareto index.

One-Dimensional Fokker–Planck Equations and Functional Inequalities for Heavy Tailed Densities

We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on mathbb R_+, and Cauchy-type densities, taking values on mathbb R.

Genealogy and spatial distribution of the N-particle branching random walk with polynomial tails

The N-particle branching random walk is a discrete time branching particle system with selection. We have N particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent’s location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Revisiting Jeffreys’ Example: Bayes Test of the Normal Mean

We revisit the classical problem of testing whether a normal mean is zero against all possible alternatives within a Bayesian framework. Jeffreys showed that the Bayes factor for this problem has a drawback with normal priors for the alternatives. He showed also that this deficiency is rectified when one uses a Cauchy prior instead. Noting that a Cauchy prior is an example of a scale-mixed normal prior, we want to examine whether or not scale-mixed normal priors can always overcome the deficiency of the Bayes factor. It turns out though that while mixing priors with polynomial tails can overcome this deficiency, those with exponential tails fail to do so. Examples are provided to illustrate this point.

Dynamic optimization of an emulsion copolymerization process for product quality using a deterministic kinetic model with embedded Monte Carlo simulations

We present the dynamic optimization of an emulsion copolymerization process described by a deterministic kinetic ordinary differential equation model including a stochastic Monte Carlo submodel, describing the growth of polymer chains within particles. For the considered semi-batch operation, the time dependent input trajectories for monomer and initiator flow rates are optimized along with the isothermal reactor temperature. We use the surrogate-model-based optimizer MATSuMoTo for the optimization to avoid the need to compute derivatives of the stochastic model. Radial basis functions with linear polynomial tails are selected as surrogate functions which are updated during optimization by the newly evaluated points. A relevant application problem formulation together with results for two case studies are presented. Qualitatively similar input trajectories are obtained for different optimization runs due to the stochastic process and the limited number of iterations. All solutions reduce the batch time significantly.

Limit theorems for some skew products with mixing base maps

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps$T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$together with a$T$-invariant measure whose base map$\unicode[STIX]{x1D703}$satisfies certain topological and mixing conditions and the maps$T_{\unicode[STIX]{x1D714}}$on the fibers are certain non-singular distance-expanding maps. Our results hold true when$\unicode[STIX]{x1D703}$is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by$T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems.Probab. Theory Related Fields162(2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of$T$(with respect to the invariant measure) does not seem to have a spectral gap.

The pressure function for infinite equilibrium measures

Assume that (X, f) is a dynamical system and φ: X → [−∞, ∞) is a potential such that the f-invariant measure μφ equivalent to the φ-conformal measure is infinite, but that there is an inducing scheme F = fτ with a finite measure $$\mu_\phi^-$$ and polynomial tails $$\mu_\phi^-$$ (τ ≥ n) = O(n−β), β ∈ (0, 1). We give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system as $$P(\phi + s\psi ) = (CP{(\overline {\phi + s\psi )} )^{1/\beta }}(1 + o(1)),$$ together with estimates for the o(1)-error term. This extends results from Sarig [S06] to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes [L93] for the Pomeau-Manneville map with potential φt = −tlogf′, as well as on the results by Bruin and Todd [BTo09, BTo12] on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures μφ+sψ as s → 0 are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.

Detection of sparse mixtures: higher criticism and scan statistic

We consider the problem of detecting a sparse mixture as studied by Ingster (1997) and Donoho and Jin (2004). We consider a wide array of base distributions. In particular, we study the situation when the base distribution has polynomial tails, a situation that has not received much attention in the literature. Perhaps surprisingly, we find that in the context of such a power-law distribution, the higher criticism does not achieve the detection boundary. However, the scan statistic does.

Asymptotics for scalar perturbations from a neighborhood of the bifurcation sphere

In our previous work (Angelopoulos 2018 Adv. Math. 323 529–621) we showed that the coefficient in the precise leading-order late-time asymptotics for solutions to the wave equation with smooth, compactly supported initial data on Schwarzschild backgrounds is proportional to the time-inverted Newman–Penrose constant (TINP), that is the Newman–Penrose constant of the associated time integral. The time integral (and hence the TINP constant) is canonically defined in the domain of dependence of any Cauchy hypersurface along which the stationary Killing field is non-vanishing. As a result, an explicit expression of the late-time polynomial tails was obtained in terms of initial data on Cauchy hypersurfaces intersecting the future event horizon to the future of the bifurcation sphere.In this paper, we extend the above result to Cauchy hypersurfaces intersecting the bifurcation sphere via a novel geometric interpretation of the TINP constant in terms of a modified gradient flux on Cauchy hypersurfaces. We show, without appealing to the time integral construction, that a general conservation law holds for these gradient fluxes. This allows us to express the TINP constant in terms of initial data on Cauchy hypersurfaces for which the time integral construction breaks down.

Analytical study of kinklike structures with polynomial tails

This work deals with models described by a single real scalar field in two-dimensional spacetime. The aim is to propose potentials that support massless minima and investigate the presence of kinklike structures that engender polynomial tails. The results unveil the presence of families of asymmetric solutions with energy density and linear stability that behave adequately, enhancing the importance of the analytical study. We stress that the novel topological structures which we find in this work engender long range interactions that are of current interest to statistical mechanics, dipolar quantum gases and the study of quantum information with Rydberg atoms.

Matching polynomial tails to the cut-off Woods–Saxon potential

Cutting off the tail of the Woods–Saxon (WS) and generalized WS (GWS) potentials changes the distribution of the poles of the [Formula: see text]-matrix considerably. Here, we modify the tail of the cut-off WS (CWS) and cut-off generalized WS (CGWS) potentials by attaching Hermite polynomial tails to them beyond the cut. The tails reach the zero value more or less smoothly at the finite ranges of the potential. Reflections of the resonant wave functions can take place at different distances. The starting points of the pole trajectories have been reproduced not only for the real values and the moduli of the starting points, but also for the imaginary parts.

Planelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces

This paper contains three types of results: 1. the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, 2. the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, 3. the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

Joint Estimation of Precision Matrices in Heterogeneous Populations.

We introduce a general framework for estimation of inverse covariance, or precision, matrices from heterogeneous populations. The proposed framework uses a Laplacian shrinkage penalty to encourage similarity among estimates from disparate, but related, subpopulations, while allowing for differences among matrices. We propose an efficient alternating direction method of multipliers (ADMM) algorithm for parameter estimation, as well as its extension for faster computation in high dimensions by thresholding the empirical covariance matrix to identify the joint block diagonal structure in the estimated precision matrices. We establish both variable selection and norm consistency of the proposed estimator for distributions with exponential or polynomial tails. Further, to extend the applicability of the method to the settings with unknown populations structure, we propose a Laplacian penalty based on hierarchical clustering, and discuss conditions under which this data-driven choice results in consistent estimation of precision matrices in heterogenous populations. Extensive numerical studies and applications to gene expression data from subtypes of cancer with distinct clinical outcomes indicate the potential advantages of the proposed method over existing approaches.

Maxwellian velocity distributions in slow time

Abstract We extend Maxwellian velocity distributions to long observational timescales in much the same way that short timescale statistical mechanics distributions are averaged to yield normal laboratory timescale thermodynamic distributions. This long timescale view has several novel effects: Fluctuating overall velocities (i.e. “wind”) thermalizes into an additional component of temperature, while returning a Maxwellian velocity distribution. However, fluctuating temperature results in a new distribution with a Gaussian core but heavy polynomial tails. The power of the polynomial tail is either -3 or -2 depending on whether the precision of the temperature is allowed to extend to ± infinity or is required to remain strictly positive. The distribution is also interesting in the way it remains almost exactly Gaussian up to a certain velocity after which it quickly breaks off to become polynomial. The distributions are carefully analyzed mathematically, and physical consequences are drawn.

Fractional decay bounds for nonlocal zero order heat equations

In this paper, we obtain bounds for the decay rate for solutions to the nonlocal problem ∂ t u ( t , x ) = ∫ R n J ( x , y ) [ u ( t , y ) − u ( t , x ) ] d y . Here we deal with bounded kernels J but with polynomial tails, that is, we assume a lower bound of the form J ( x , y ) ⩾ c 1 | x − y | − ( n + 2 σ ) , for | x − y | > c 2 . Our estimates takes the form ∥ u ( t ) ∥ L q ( R n ) ⩽ C t − ( n / 2 σ ) ( 1 − 1 / q ) for t large.