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Probability

New submissions

[ total of 37 entries: 1-37 ]
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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06361 [pdf, ps, other]
Title: Halving the bounds for the Markov, Chebyshev, and Chernoff Inequalities using smoothing
Authors: Mark Huber
Subjects: Probability (math.PR)

The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equality. Here we will show that through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.

[2]  arXiv:1803.06543 [pdf, ps, other]
Title: The parametrix method for parabolic SPDEs
Comments: Submitted to Stochastics and Partial Differential Equations: Analysis and Computations
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in H\"older classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the It\^o-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

[3]  arXiv:1803.06677 [pdf, ps, other]
Title: A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
Authors: Dmitry Ostrovsky
Comments: 64 pages
Subjects: Probability (math.PR)

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of $1/f$ noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model.

[4]  arXiv:1803.06702 [pdf, ps, other]
Title: On Infinite Divisibility of the Distribution of Some Inverse Subordinators
Comments: 11 pages; submitted for publication
Subjects: Probability (math.PR)

We consider the infinite divisibility of the distributions of some well known inverse subordinators. Using a tail probability bound, we establish that the distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of fractional Poisson process is not infinitely divisible.

[5]  arXiv:1803.06728 [pdf, ps, other]
Title: A non-intersecting random walk on the Manhattan lattice and SLE_6
Authors: Tom Kennedy
Comments: 14 figures
Subjects: Probability (math.PR); Mathematical Physics (math-ph)

We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces in a certain percolation model. So it is natural to conjecture that the scaling limit is SLE$_6$. We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.

[6]  arXiv:1803.06739 [pdf, ps, other]
Title: A Construction of the Stable Web
Subjects: Probability (math.PR)

We provide a process on the space of coalescing cadlag stable paths and show convergence in the appropriate topology for coalescing stable random walks on the integer lattice.

[7]  arXiv:1803.06820 [pdf, ps, other]
Title: Rescaled weighted determinantal random balls
Authors: Adrien Clarenne (IRMAR)
Subjects: Probability (math.PR)

We consider a collection of weighted Euclidian random balls in R^d distributed according a determinantal point process. We perform a zoom-out procedure by shrinking the radii while increasing the number of balls. We observe that the repulsion between the balls is erased and three different regimes are obtained, the same as in the weighted Poissonian case.

[8]  arXiv:1803.06899 [pdf, ps, other]
Title: Limit Theorems for Cylindrical Martingale Problems associated with Lévy Generators
Authors: David Criens
Subjects: Probability (math.PR)

We derive limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients and limit theorems for solution measures to stochastic (partial) differential equations.

[9]  arXiv:1803.06930 [pdf, ps, other]
Title: Explicit formula for the density of local times of Markov Jump Processes
Subjects: Probability (math.PR)

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov Chains on finite graphs, which involves the modified Bessel function of the first type.

[10]  arXiv:1803.06947 [pdf, ps, other]
Title: Differentiability of SDEs with drifts of super-linear growth
Comments: 41 Pages
Subjects: Probability (math.PR)

We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.3 for this setting. Several examples illustrating the range and scope of our results are presented.
We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.

[11]  arXiv:1803.06953 [pdf, ps, other]
Title: Entropy solutions for stochastic porous media equations
Comments: 25 pages
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\Delta (|u|^{m-1}u)$ for all $m\in(1,\infty)$, and H\"older continuous diffusion nonlinearity with exponent $1/2$.

[12]  arXiv:1803.07024 [pdf, ps, other]
Title: A note on vague convergence of measures
Comments: 15 pages
Subjects: Probability (math.PR)

We propose a notion of convergence of measures with intention of generalizing and unifying several frequently used types of vague convergence. We explain that by general theory of boundedness due to Hu (1966), in Polish spaces, this notion of convergence can be always formulated as follows: $\mu_n \stackrel{v}{\longrightarrow} \mu$ if $\int f d\mu_n \to \int f d\mu$ for all continuous bounded functions $f$ with support bounded in some suitably chosen metric. This brings all the related types of vague convergence into the framework of Daley and Vere-Jones (2003) and Kallenberg (2017). In the rest of the note we discuss the vague topology and the corresponding notion of convergence in distribution, complementing the theory developed in those two references.

[13]  arXiv:1803.07027 [pdf, other]
Title: Brownian Motions on Star Graphs with Non-Local Boundary Conditions
Authors: Florian Werner
Subjects: Probability (math.PR)

Brownian motions on star graphs in the sense of It\^o-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for infinite jump measures, the pathwise solution of It\^o-McKean for the half line is analyzed and extended to the star graph. These processes can then be used as main building blocks for Brownian motions on general metric graphs with non-local boundary conditions.

Cross-lists for Tue, 20 Mar 18

[14]  arXiv:1803.06457 (cross-list from math.FA) [pdf, ps, other]
Title: Generalization of a Real-Analysis Result to a Class of Topological Vector Spaces
Authors: Leonard T. Huang
Comments: 6 pages, no figures
Subjects: Functional Analysis (math.FA); Probability (math.PR)

In this paper, we generalize an elementary real-analysis result to a class of topological vector spaces. We also give an example of a topological vector space to which the result cannot be generalized.

[15]  arXiv:1803.06620 (cross-list from math.ST) [pdf, ps, other]
Title: Characterizations of the Logistic and Related Distributions
Comments: 17 pages, Journal of Mathematical Analysis and Applications (2018)
Subjects: Statistics Theory (math.ST); Probability (math.PR)

It is known that few characterization results of the logistic distribution were available before, although it is similar in shape to the normal one whose characteristic properties have been well investigated. Fortunately, in the last decade, several authors have made great progress in this topic. Some interesting characterization results of the logistic distribution have been developed recently. In this paper, we further provide some new results by the distributional equalities in terms of order statistics of the underlying distribution and the random exponential shifts. The characterization of the closely related Pareto type II distribution is also investigated.

[16]  arXiv:1803.06636 (cross-list from math.CO) [pdf, ps, other]
Title: Complexity problems in enumerative combinatorics
Authors: Igor Pak
Comments: 30 pages; an expanded version of the ICM 2018 paper (Section 4 added, refs expanded)
Subjects: Combinatorics (math.CO); Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); History and Overview (math.HO); Probability (math.PR)

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.

[17]  arXiv:1803.06716 (cross-list from math.ST) [pdf, ps, other]
Title: High Dimensional Linear Regression using Lattice Basis Reduction
Subjects: Statistics Theory (math.ST); Probability (math.PR); Machine Learning (stat.ML)

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $\beta^*$ from $n$ noisy linear observations $Y=X\beta^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and unknown $W \in \mathbb{R}^n$. Unlike most of the literature on this model we make no sparsity assumption on $\beta^*$. Instead we adopt a regularization based on assuming that the underlying vectors $\beta^*$ have rational entries with the same denominator $Q \in \mathbb{Z}_{>0}$. We call this $Q$-rationality assumption.
We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the $Q$-rationality assumption, our algorithm recovers exactly the vector $\beta^*$ for a large class of distributions for the iid entries of $X$ and non-zero noise $W$. We prove that it is successful under small noise, even when the learner has access to only one observation ($n=1$). Furthermore, we prove that in the case of the Gaussian white noise for $W$, $n=o\left(p/\log p\right)$ and $Q$ sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

[18]  arXiv:1803.06727 (cross-list from cs.LG) [pdf, other]
Title: Aggregating Strategies for Long-term Forecasting
Comments: 20 pages, 4 figures
Subjects: Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST); Machine Learning (stat.ML)

The article is devoted to investigating the application of aggregating algorithms to the problem of the long-term forecasting. We examine the classic aggregating algorithms based on the exponential reweighing. For the general Vovk's aggregating algorithm we provide its generalization for the long-term forecasting. For the special basic case of Vovk's algorithm we provide its two modifications for the long-term forecasting. The first one is theoretically close to an optimal algorithm and is based on replication of independent copies. It provides the time-independent regret bound with respect to the best expert in the pool. The second one is not optimal but is more practical and has $O(\sqrt{T})$ regret bound, where $T$ is the length of the game.

[19]  arXiv:1803.06808 (cross-list from math-ph) [pdf, other]
Title: Local martingales associated with SLE with internal symmetry
Authors: Shinji Koshida
Comments: 45 pages
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR); Quantum Algebra (math.QA); Representation Theory (math.RT)

We consider Schramm-Loewner evolutions with internal degrees of freedom that are associated with representations of affine Lie algebras, following the group theoretical formulation of SLE. We observe that SLEs considered by Bettelheim et al. [PRL 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243-261 (2011)] in correlation function formulation are reconstrunced. We also explicitly write down stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. Our formulation enables to write down several local martingales associated with the solution of SLE from computation on a representation of an affine Lie algebra. Indeed, we write down local martingales associated with solution of SLE for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also find affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales for the affine $\mathfrak{sl}_{2}$, which can be extended to other affine Lie algebras.

[20]  arXiv:1803.06829 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
Title: Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process
Comments: 6 pages, 3 figures
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)

We consider current statistics for a two species exclusion process of particles hopping in opposite directions on a one-dimensional lattice. We derive an exact formula for the Green's function as well as for a joint current distribution of the model, and study its long time behavior. For a step type initial condition, we show that the limiting distribution is a product of the Gaussian and the GUE Tracy-Widom distribution. This is the first analytic confirmation for a multi-component system of a prediction from the recently proposed non-linear fluctuating hydrodynamics for one dimensional systems.

[21]  arXiv:1803.06847 (cross-list from math.MG) [pdf, ps, other]
Title: The square negative correlation on l_p^n balls
Subjects: Metric Geometry (math.MG); Probability (math.PR)

In this paper we prove that for any $p\in[2,\infty)$ the $\ell_p^n$ unit ball, $B_p^n$, satisfies the square negative correlation property with respect to every orthonormal basis, while we show it is not always the case for $1\le p\le 2$. In order to do that we regard $B_p^n$ as the orthogonal projection of $B_p^{n+1}$ onto the hyperplane $e_{n+1}^\perp$. We will also study the orthogonal projection of $B_p^n$ onto the hyperplane orthogonal to the diagonal vector $(1,\dots,1)$. In this case, the property holds for all $p\ge 1$ and $n$ large enough.

[22]  arXiv:1803.06914 (cross-list from math.CO) [pdf, ps, other]
Title: Mixing Time of Markov chain of the Knapsack Problem
Comments: 9 pages, 2 figures
Subjects: Combinatorics (math.CO); Data Structures and Algorithms (cs.DS); Probability (math.PR)

To find the number of assignments of zeros and ones satisfying a specific Knapsack Problem is $\#P$ hard, so only approximations are envisageable. A Markov chain allowing uniform sampling of all possible solutions is given by Luby, Randall and Sinclair. In 2005, Morris and Sinclair, by using a flow argument, have shown that the mixing time of this Markov chain is $\mathcal{O}(n^{9/2+\epsilon})$, for any $\epsilon > 0$. By using a canonical path argument on the distributive lattice structure of the set of solutions, we obtain an improved bound, the mixing time is given as $\tau_{_{x}}(\epsilon) \leq n^{3} \ln (16 \epsilon^{-1})$.

[23]  arXiv:1803.06922 (cross-list from q-fin.RM) [pdf, ps, other]
Title: Approximation of Some Multivariate Risk Measures for Gaussian Risks
Authors: E. Hashorva
Subjects: Risk Management (q-fin.RM); Probability (math.PR)

Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate risks.

[24]  arXiv:1803.07005 (cross-list from math.AP) [pdf, ps, other]
Title: Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
Authors: Jonas M. Tölle
Comments: 25 pages, 53 references
Subjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA); Probability (math.PR)

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with $C^{1}$-vector field coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^o-formula.

Replacements for Tue, 20 Mar 18

[25]  arXiv:1505.01692 (replaced) [pdf, ps, other]
Title: Rough flows
Comments: v4, 55 pages; final version. The exposition has been polished to make the work easier to read
Subjects: Probability (math.PR); Classical Analysis and ODEs (math.CA)
[26]  arXiv:1611.04874 (replaced) [pdf, ps, other]
Title: The damped stochastic wave equation on p.c.f. fractals
Subjects: Probability (math.PR)
[27]  arXiv:1612.08485 (replaced) [pdf, ps, other]
Title: Limit theorems for random cubical homology
Comments: 19 pages, 14 figures
Subjects: Probability (math.PR); Algebraic Topology (math.AT)
[28]  arXiv:1703.09481 (replaced) [pdf, ps, other]
Title: Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions
Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
[29]  arXiv:1706.01015 (replaced) [pdf, other]
Title: The split-and-drift random graph, a null model for speciation
Comments: added Proposition 2.4 and formal proofs of Proposition 2.3 and 2.6
Subjects: Probability (math.PR); Combinatorics (math.CO); Populations and Evolution (q-bio.PE)
[30]  arXiv:1712.06841 (replaced) [pdf, other]
Title: Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces
Comments: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures
Subjects: Probability (math.PR)
[31]  arXiv:1802.08934 (replaced) [pdf, other]
Title: The Archimedean limit of random sorting networks
Authors: Duncan Dauvergne
Comments: 58 pages, 5 figures. Changes from v1: the proof of Lemma 7.2 has been lengthened for clarity; otherwise, only minor edits have been made
Subjects: Probability (math.PR); Combinatorics (math.CO)
[32]  arXiv:1803.05182 (replaced) [pdf, ps, other]
Title: Approximative Theorem of Incomplete Riemann-Stieltjes Sum of Stochastic Integral
Authors: Jingwei Liu
Comments: 4 figures
Subjects: Probability (math.PR)
[33]  arXiv:1706.09130 (replaced) [pdf, other]
Title: Potts models with a defect line
Comments: Revised version, with additional explanations and details. Should hopefully be easier to read
Subjects: Mathematical Physics (math-ph); Probability (math.PR)
[34]  arXiv:1711.05850 (replaced) [pdf, other]
Title: Local eigenvalue statistics of one-dimensional random non-selfadjoint pseudo-differential operators
Comments: 81 pages, 7 figures
Subjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP); Probability (math.PR)
[35]  arXiv:1803.04913 (replaced) [pdf, other]
Title: On non-commutativity in quantum theory (I): from classical to quantum probability
Authors: Luca Curcuraci
Comments: 19 pages, 1 figure
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR)
[36]  arXiv:1803.04916 (replaced) [pdf, other]
Title: On non-commutativity in quantum theory (II): toy models for non-commutative kinematics
Authors: Luca Curcuraci
Comments: 31 pages, 7 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR)
[37]  arXiv:1803.06299 (replaced) [pdf, other]
Title: On the existence of a scalar pressure field in the Bredinger problem
Authors: Aymeric Baradat
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC); Probability (math.PR)
[ total of 37 entries: 1-37 ]
[ showing up to 2000 entries per page: fewer | more ]

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