Spatial quantiles are among the most successful concepts of multivariate quantiles. In particular, they are essentially the only quantiles that can be computed in high dimensions. There has been an intense research activity to study spatial quantiles in the last two decades, yet surprisingly little is known about their robustness properties. In the present work, we carefully study the breakdown point of spatial quantiles. We offer three approaches, that show diverse distinctive advantages. The first approach is a constructive one: it is conceptually simple and allows us to derive the finite-sample breakdown point of spatial quantiles. While the second approach is not constructive and does not identify the global breakdown point of spatial quantiles, it provides an upper bound on the breakdown point under contamination in any fixed direction. It also allows us to determine the breakdown point of spatial Lp-quantiles for any p > 1. Last but not least, the third approach characterizes precisely when breakdown occurs under any given contamination scheme, hence provides the breakdown points associated with very diverse contamination scenarios. Quite nicely, this last approach further covers cases where the contamination and/or the ground probability measures are continuous distributions. An intriguing corollary of our results states that, in high dimensions, the "practical" breakdown point exceeds the theoretical one. Throughout, our theoretical results are illustrated through numerical exercises. Part of our results cover infinite-dimensional Hilbert spaces as well.
We consider the fractal Sierpi\'{n}ski gasket or carpet graph in dimension $d\geq 2,$ denoted by $G$. At time $0$, we place a Poisson point process of particles onto the graph and let them perform independent simple random walks, which in this setting exhibit sub-diffusive behaviour. We generalise the concept of particle process dependent Lipschitz percolation to the (coarse graining of the) space-time graph $G\times \mathbb{R}$, where the opened/closed state of space-time cells is measurable with respect to the particle process inside the cell. We then provide an application of this generalised framework and prove the following: if particles can spread an infection when they share a site of $G$, and if they recover independently at some rate $\gamma>0$, then if $\gamma$ is sufficiently small, the infection started with a single infected particle survives indefinitely with positive probability.
We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a c & agrave;dl & agrave;g path and an occupation measure associated with this path. Using this result, we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for c & agrave;dl & agrave;g semimartingales, in particular for alpha-stable processes, and Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher-order variation but no local times.
We establish an explicit characterisation of Lévy measures on both L^p-spaces and UMD Banach spaces. In the case of L^p-spaces, Lévy measures are characterised by an integrability condition, which directly generalises the known description of Lévy measures on sequence spaces. The latter has been the only known description of Lévy measures on infinite dimensional Banach spaces that are not Hilbert. Lévy measures on UMD Banach spaces are characterised by the finiteness of the expectation of a random γ-radonifying norm. Although this description is more abstract, it reduces to simple integrability conditions in the case of L^p-spaces.
We consider the Kardar-Parisi-Zhang (KPZ) fixed point H(x,τ) with the narrow-wedge initial condition and investigate the distribution of H(x,τ) conditioned on a large height at an earlier space-time point H(x',τ'). As H(x',τ') tends to infinity, we prove that the conditional one-point distribution of H(x,τ) in the regime τ>τ' converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution and that the next two lower-order error terms can be expressed as derivatives of the Tracy-Widom distribution. The lowe order expansion here is analogue to the Edgeworth expansion in the central limit theorem. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in where they study the finite-dimensional distribution of H(x,τ) conditioned on a large height at a later space-time point H(x',τ'). They show, with the narrow-wedge initial condition, that the conditional random field H(x,τ) in the regime τ<τ' converges to the minimum of two independent Brownian bridges modified by linear drifts as H(x',τ') goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes τ>τ' and τ<τ' when H(x',τ') goes to infinity.
We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and P\'olya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\mathbb R^d$ for $d\in\{1,2,3\}$, and we bring evidence for a similar result for $d\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments.
We study strong approximation of scalar additive noise driven stochastic differential equations (SDEs) at time point $1$ in the case that the drift coefficient is bounded and has Sobolev regularity $s\in(0,1)$. Recently, it has been shown in [arXiv:2101.12185v2 (2022)] that for such SDEs the equidistant Euler approximation achieves an $L^2$-error rate of at least $(1+s)/2$, up to an arbitrary small $\varepsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In the present article we prove a matching lower error bound for $s\in(1/2,1)$. More precisely we show that, for every $s\in(1/2,1)$, the $L^2$-error rate $(1+s)/2$ can, up to a logarithmic term, not be improved in general by no numerical method based on finitely many evaluations of $W$ at fixed time points. Up to now, this result was known in the literature only for the cases $s=1/2-$ and $s=1-$. For the proof we employ the coupling of noise technique recently introduced in [arXiv:2010.00915 (2020)] to bound the $L^2$-error of an arbitrary approximation from below by the $L^2$-distance of two occupation time functionals provided by a specifically chosen drift coefficient with Sobolev regularity $s$ and two solutions of the corresponding SDE with coupled driving Brownian motions. For the analysis of the latter distance we employ a transformation of the original SDE to overcome the problem of correlated increments of the difference of the two coupled solutions, occupation time estimates to cope with the lack of regularity of the chosen drift coefficient around the point $0$ and scaling properties of the drift coefficient.
We study averaging for Stochastic Differential Equations (SDEs) and Poisson equations. We succeed in obtaining a uniform in time (UiT) averaging result, with a rate, for fully coupled SDE models with super-linearly growing coefficients. This is the main result of this paper and is, to the best of our knowledge, the first UiT multiscale result with a rate. Very few UiT averaging results exist in the literature, and they almost exclusively apply to multiscale systems of Ordinary Differential Equations. Among these few, none of those we are aware of comes with a rate of convergence. The UiT nature of this result and the rate of convergence given by the main theorem, make it important as theoretical underpinning for a range of applications, such as applications to statistical methodology, molecular dynamics etc. Key to obtaining both our UiT averaging result and to enable dealing with the super-linear growth of the coefficients is conquering exponential decay in time of the space-derivatives of appropriate Markov semigroups. We refer to this property as being Strongly Exponentially Stable (SES). The analytic approach to proving averaging results we take requires studying a family of Poisson problems associated with the generator of the (fast component of the) SDE dynamics. The study of Poisson equations in non-compact state space is notoriously difficult, with current literature mostly covering the case when the coefficients of the Partial Differential Equation (PDE) are either bounded or satisfy linear growth assumptions. In this paper we treat Poisson equations on non-compact state spaces for coefficients that can grow super-linearly. We demonstrate how SES can be employed not only to prove the UiT result for the slow-fast system but also to overcome some of the technical hurdles in the analysis of Poisson problems, which is of independent interest as well.
We prove that every (possibly infinite) graph of degree at most d has a 4-dependent random proper 4d(d+1)/2-coloring, and one can construct it as a finitary factor of iid. For unimodular transitive (or unimodular random) graphs we construct an automorphisminvariant (respectively, unimodular) 2-dependent coloring by 3d(d+1)/2 colors. In particular, there exist random proper colorings for & Zopf;d and for the regular tree that are 2-dependent and automorphism-invariant, or 4-dependent and finitary factor of iid.
For finite interacting particle systems with strong repulsing-attracting or general interactions, we prove global weak well-posedness almost up to the critical threshold of the strengths of attracting interactions (independent of the number of particles), and establish other regularity results, such as a heat kernel bound in the regions where strongly attracting particles are close to each other. Our main analytic instruments are a variant of De Giorgi's method in L^p and an abstract desingularization theorem.
We study the gap processes in a degenerate system of three particles interacting through their ranks. We obtain the Laplace transform of the invariant measure of these gaps, and an explicit expression for the corresponding invariant density. To derive these results, we start from the basic adjoint relationship characterizing the invariant measure, and apply a combination of two approaches: first, the invariance methodology of W. Tutte, thanks to which we compute the Laplace transform in closed form; second, a recursive compensation approach which leads to the density of the invariant measure as an infinite convolution of exponential functions. As in the case of Brownian motion with reflection or killing at the endpoints of an interval, certain Jacobi theta functions play a crucial role in our computations.
We develop the study of a particular class of biorthogonal measures, encompassing at the same time several random matrix models and partition functions of polymers. This general framework allows us to characterize the partition functions of the Log Gamma polymer and the mixed polymer in terms of explicit biorthogonal measures, as it was previously done for the homogeneous O'Connell-Yor polymer by Imamura and Sasamoto. In addition, we show that the biorthogonal measures associated to these three polymer models (Log Gamma, O'Connell-Yor, and mixed polymer) converge to random matrix eigenvalue distributions in small temperature limits. We also clarify the connection between different Fredholm determinant representations and explain how our results might be useful for asymptotic analysis and large deviation estimates.
We condition a Brownian motion on having an atypically small L_2-norm on a long time interval. The obtained limiting process is a non-stationary Ornstein-Uhlenbeck process.
For any convex function F of n-dimensional Gaussian vector g with Ee(lambda F(g))< infinity for any lambda > 0, we show that lambda(-1) ln Ee(lambda F(g)) is convex in lambda is an element of R. Based on this convexity, we draw three major consequences. The first recovers a version of the Paouris-Valettas lower deviation inequality for Gaussian convex functions with an improved exponent. The second establishes a quantitative bound for the Dotsenko-Franz-Mezard conjecture arising from the study of the Sherrington-Kirkpatrick (SK) mean-field spin glass model, which states that in the absence of external field, the annealed free energy of negative replica is asymptotically equal to the free energy. The last further establishes the differentiability for this annealed free energy with respect to the negative replica variable at any temperature and external field.
For any edge weight distribution, we consider the uniform spanning tree (UST) on finite graphs with i.i.d. random edge weights. We show that, for bounded degree expander graphs and finite boxes of ${\mathbb Z}^d$, the diameter of the UST is of order $n^{1/2+o(1)}$ with high probability, where $n$ is the number of vertices.
We prove the continuity of the extremal decomposition measure of the free state of low temperature Potts models, and more generally of ferromagnetic finite-spin models, on a regular tree, including general clock models. The decomposition is supported on uncountably many inhomogeneous extremal states, that we call glassy states. The method of proof provides explicit concentration bounds on branch overlaps, which play the role of an order parameter for typical extremals. The result extends to the counterpart of the free state (called central state) in a wide range of models which have no symmetry, allowing also the presence of sufficiently small field terms. Our work shows in particular that the decomposition of central states into uncountably many glassy states in finite-spin models on trees at low temperature is a generic phenomenon, and does not rely on symmetries of the Hamiltonian.
We present a new proof of $|x|^{-(d-2)}$ decay of critical two-point functions for spread-out statistical mechanical models on $\mathbb{Z}^d$ above the upper critical dimension, based on the lace expansion and assuming appropriate diagrammatic estimates. Applications include spread-out models of the Ising model and self-avoiding walk in dimensions $d>4$, and spread-out percolation for $d>6$. The proof is based on an extension of the new Gaussian deconvolution theorem we obtained in a recent paper. It provides a technically simpler and conceptually more transparent approach than the method of Hara, van der Hofstad and Slade (2003).
Building on Talagrand's proof of the Hoffmann-Jørgensen inequality for L_p spaces and its version for the exponential Orlicz spaces we provide a full characterization of Orlicz functions Ψ for which an analogous inequality holds in the Orlicz space L_Ψ(F), where F is an arbitrary Banach space. As an application we present a characterization of Talagrand-type concentration inequality for suprema of empirical processes with envelope in L_Ψ (equivalently for sums of independent F-valued random variables in L_Ψ(F)). This result generalizes in particular an inequality by the first-named author concerning exponentially integrable summands and a recent inequality due to Chamakh-Gobet-Liu on summands with β-heavy tails. Another corollary concerns concentration for convex functions of independent, unbounded random variables, generalizing recent results due to Klochkov-Zhivotovskiy and Sambale. We also obtain a corollary concerning boundedness in L_Ψ(F) of partial sums of a series of independent random variables, generalizing the original result by Hoffmann-Jørgensen.
