Confidence Bands for Distribution Functions: A New Look at the Law of the Iterated Logarithm

We present a general law of the iterated logarithm for stochastic processes on the open unit interval having subexponential tails in a locally uniform fashion. It applies to standard Brownian bridge but also to suitably standardized empirical distribution functions. This leads to new goodness-of-fit tests and confidence bands which refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter procedures in the tail regions of distributions are essentially preserved while gaining considerably in the central region

Nuts and Bolts of a Realistic Stochastic Geometric Analysis of mmWave HetNets: Hardware Impairments and Channel Aging

© 2019 IEEE.Motivated by heterogeneous network (HetNet) design in improving coverage and by millimeter-wave (mmWave) transmission offering an abundance of extra spectrum, we present a general analytical framework shedding light on the downlink of realistic mmWave HetNets consisting of K tiers of randomly located base stations. Specifically, we model, by virtue of stochastic geometry tools, the multi-Tier multi-user (MU) multiple-input multiple-output (MIMO) mmWave network degraded by the inevitable residual additive transceiver hardware impairments (RATHIs) and channel aging. Given this setting, we derive the coverage probability and the area spectral efficiency (ASE), and we subsequently evaluate the impact of residual transceiver hardware impairments and channel aging on these metrics. Different path-loss laws for line-of-sight and non-line-of-sight are accounted for the analysis, which are among the distinguishing features of mmWave systems. Among the findings, we show that the RATHIs have a meaningful impact at the high-signal-To-noise-ratio regime, while the transmit additive distortion degrades further than the receive distortion the system performance. Moreover, serving fewer users proves to be preferable, and the more directive the mmWaves are, the higher the ASE becomes.Peer reviewedFinal Accepted Versio

BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin

We developed a general non-perturbative framework for the BFKL spectrum of planar N=4 SYM, based on the Quantum Spectral Curve (QSC). It allows one to study the spectrum in the whole generality, extending previously known methods to arbitrary values of conformal spin $n$. We show how to apply our approach to reproduce all known perturbative results for the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron eigenvalue and get new predictions. In particular, we re-derived the Faddeev-Korchemsky Baxter equation for the Lipatov spin chain with non-zero conformal spin reproducing the corresponding BFKL kernel eigenvalue. We also get new non-perturbative analytic results for the Pomeron eigenvalue in the vicinity of $|n|=1,\;\Delta=0$ point and we obtained an explicit formula for the BFKL intercept function for arbitrary conformal spin up to the 3-loop order in the small coupling expansion and partial result at the 4-loop order. In addition, we implemented the numerical algorithm of arXiv:1504.06640 as an auxiliary file to this arXiv submission. From the numerical result we managed to deduce an analytic formula for the strong coupling expansion of the intercept function for arbitrary conformal spin.Comment: 70 pages, 5 figures, 1 txt, 2 nb and 2 mx files; v2: references added, typos fixed and nb file with Mathematica stylesheet attached; v3: more typos fixed; v4: the text edited according to the report of the refere

Singular structure of Toda lattices and cohomology of certain compact Lie groups

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $\tau$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $\tau$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics

Variance Reduced Distributed Non-Convex Optimization Using Matrix Stepsizes

Matrix-stepsized gradient descent algorithms have been shown to have superior performance in non-convex optimization problems compared to their scalar counterparts. The det-CGD algorithm, as introduced by Li et al. (2023), leverages matrix stepsizes to perform compressed gradient descent for non-convex objectives and matrix-smooth problems in a federated manner. The authors establish the algorithm's convergence to a neighborhood of a weighted stationarity point under a convex condition for the symmetric and positive-definite matrix stepsize. In this paper, we propose two variance-reduced versions of the det-CGD algorithm, incorporating MARINA and DASHA methods. Notably, we establish theoretically and empirically, that det-MARINA and det-DASHA outperform MARINA, DASHA and the distributed det-CGD algorithms in terms of iteration and communication complexities.Comment: Major update: The paper now includes an analysis of det-DASHA, which is another variance reduction extension of det-CGD. 63 pages, 12 figure

Chiral Anomaly in SU(2)R{}_R-Axion Inflation and the New Prediction for Particle Cosmology

Upon embedding the axion-inflation in the minimal left-right symmetric gauge extension of the SM with gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$, [arXiv:2012.11516] proposed a new particle physics model for inflation. In this work, we present a more detailed analysis. As a compelling consequence, this setup provides a new mechanism for simultaneous baryogenesis and right-handed neutrino creation by the chiral anomaly of $W_R$ in inflation. The lightest right-handed neutrino is the dark matter candidate. This setup has two unknown fundamental scales, i.e., the scale of inflation and left-right symmetry breaking $SU(2)_R\times U(1)_{B-L}\rightarrow U(1)_{Y}$. Sufficient matter creation demands the left-right symmetry breaking scale happens shortly after the end of inflation. Interestingly, it prefers left-right symmetry breaking scales above $10^{10}~GeV$, which is in the range suggested by the non-supersymmetric SO(10) Grand Unified Theory with an intermediate left-right symmetry scale. Although $W_R$ gauge field generates equal amounts of right-handed baryons and leptons in inflation, i.e. $B-L=0$, in the Standard Model sub-sector $B-L_{SM}\neq 0$. A key aspect of this setup is that $SU(2)_R$ sphalerons are never in equilibrium, and the primordial $B-L_{SM}$ is conserved by the Standard Model interactions. This setup yields a deep connection between CP violation in physics of inflation and matter creation (visible and dark); hence it can naturally explain the observed coincidences among cosmological parameters, i.e., $\eta_{B}\simeq 0.3 P_{\zeta}$ and $\Omega_{DM}\simeq 5\Omega_{B}$. The $SU(2)_R$-axion inflation comes with a cosmological smoking gun; chiral, non-Gaussian, and blue-tilted gravitational wave background, which can be probed by future CMB missions and laser interferometer detectors.Comment: 28+20 Pages, 15 Fig

Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related to the Poisson clumping heuristic and tube formulas of Gaussian random fields, and describe their applications to laws of the iterated logarithm in the form of the Kolmogorov--Erd\H{o}s--Feller integral tests.Comment: Published at http://dx.doi.org/10.1214/009117905000000378 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Half-integer Higher Spin Fields in (A)dS from Spinning Particle Models

We make use of O(2r+1) spinning particle models to construct linearized higher-spin curvatures in (A)dS spaces for fields of arbitrary half-integer spin propagating in a space of arbitrary (even) dimension: the field potentials, whose curvatures are computed with the present models, are spinor-tensors of mixed symmetry corresponding to Young tableaux with D/2 - 1 rows and r columns, thus reducing to totally symmetric spinor-tensors in four dimensions. The paper generalizes similar results obtained in the context of integer spins in (A)dS.Comment: 1+18 pages; minor changes in the notation, references updated. Published versio

Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional.Comment: Accepted version. To appear in Advances in Mat