Exponential tightness of a family of Skorohod integrals is studied in this paper. We first provide a counterexample to illustrate that in general the exponential tightness with speed ε similar to Itô integral does not hold, even for any speed ${\mathit{\epsilon }}^{\mathit{\alpha }}$ with $\mathit{\alpha }>0$. Then, some characterizations of this subject are given. Application is also provided to illustrate our results.

We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization $n^{1∕6}$). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.

In this note, we establish that the stationary distribution of a possibly non-equilibrium Langevin diffusion converges, as the damping parameter goes to infinity (or equivalently in the Smoluchowski-Kramers vanishing mass limit), toward a tensor product of the stationary distribution of the corresponding overdamped process and of a Gaussian distribution.

In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of Erdős–Rényi random graphs $G(n,p_n)$, where $p_n=n^{-\mathit{\alpha }}$ for $0<\mathit{\alpha }<1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-1 neighborhoods, G is exactly reconstructable for $0<\mathit{\alpha }<\frac{1}{3}$, but not reconstructable for $\frac{1}{2}<\mathit{\alpha }<1$. Given the collection of distance-2 neighborhoods, G is exactly reconstructable for $\mathit{\alpha }\in \left(0,\frac{1}{2}\right)\cup \left(\frac{1}{2},\frac{3}{5}\right)$, but not reconstructable for $\frac{3}{4}<\mathit{\alpha }<1$.

In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the n-th generation, which may be different from the driving increment distribution. This model was first introduced by Bandyopadhyay and Ghosh [2] and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Under very minimal assumptions, we derive the large deviation principle (LDP) for the right-most position of a particle in generation n. As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by Gantert and Höfelsauer [7].

The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time 0. The particle moves according to a Brownian motion with drift $\mathrm{\mu }=2$ and diffusion coefficient ${\mathit{\sigma }}^2=2$, until an independent exponential time of parameter 1. At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to ∞. The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.

Colour an element of ${\mathbb{Z}}^d$ white if its coordinates are coprime and black otherwise. What does this colouring look like when seen from a “uniformly chosen” random point of ${\mathbb{Z}}^d$? More generally, label every element of ${\mathbb{Z}}^d$ by its greatest common divisor: what do the labels look like around a “uniform” random point of ${\mathbb{Z}}^d$? We answer these questions and generalisations of them, which includes a result a “local/graphon” convergence. We also investigate the percolative properties of the colouring under study.

In this paper, we extend upon a result by Mueller and Tribe regarding Funaki’s model of a random string. Specifically, we examine the rate of escape of this model in dimensions $d\ge 7$. We also provide a bound for the rate of approach to the origin in dimension $d=6$.

We analyse the eigenvectors of the adjacency matrix of the Erdős-Rényi graph on N vertices with edge probability $\frac{d}{N}$. We determine the full region of delocalization by determining the critical values of $\frac{d}{logN}$ down to which delocalization persists: for $\frac{d}{logN}>\frac{1}{log4-1}$ all eigenvectors are completely delocalized, and for $\frac{d}{logN}>1$ all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [1, 3] that localized eigenvectors exist in the corresponding spectral regions.

We obtain a sharp estimate of the speed of convergence in the Boolean central limit theorem for measures with finite sixth moment. The main tool is a quantitative version of the Stieltjes-Perron inversion formula.

In this note, using the Kendall-Cranston coupling, we study on Kähler (resp. quaternion Kähler) manifolds first eigenvalue estimates in terms of dimension, diameter, and lower bounds on the holomorphic (resp. quaternionic) sectional curvature.

Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on H remains somewhat mysterious. We study the case when a positive weight w is put on the edges of the H-copies in G, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase transition. For large enough w we show that the $\mathsf{FSF}$ is connected, while for a wide family of H and T, the $\mathsf{FSF}$ is disconnected when w is small (relying on [4]). Finally, we prove that when H is the graph of one edge, then for any w, the $\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

Let $M_n$ be a random $n\times n$ matrix with i.i.d. $Bernoulli(1∕2)$ entries. We show that for fixed $k\ge 1$,

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}{log}_2\mathbb{P}[corank{M}_n\ge k]=-k.$$

In this paper we study the self-intersection of paths solving elliptic stochastic differential equations driven by fractional Brownian motion. We show that such a path has no self-intersection – except for paths forming a set of zero $(r,q)$-capacity in the sample space – provided the dimension d of the space and the Hurst parameter H satisfy the inequality $d>rq+2∕H$. This inequality is sharp in the case of brownian motion and fractional brownian motion according to existing results. Various results exist for the critical case where $d=rq+4$ for Brownian motion.

It is well known that given two probability measures μ and ν on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

We point out an error in “The remainder in the renewal theorem”, and show that the result is essentially correct in two important special cases.

This paper concerns a variational representation formula for Wiener functionals. Let $B={\{B_t\}}_{t\ge 0}$ be a standard d-dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$ of B up to time 1, the expectation $\mathbb{E}\left[e^{F(B)}\right]$ admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also $F(B)$ to be a functional of B over the whole time interval, we prove that the Boué–Dupuis formula holds true provided that both $e^{F(B)}$ and $F(B)$ are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein–Uhlenbeck semigroup in ${\mathbb{R}}^d$, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d-dimensional Gaussian space.

In this paper, we prove that the price of a defaultable bond, under a Vasicek short rate dynamics coupled with a Cox-Ingersoll-Ross default intensity model, is a real analytic function, in a neighborhood of the origin, of the correlation parameter between the Brownian motions driving the processes, used to express the dependence between the short rate and the default intensity of the bond issuer. Employing conditioning and a change of numéraire technique, we obtain a manageable representation of the bond price in this non-affine model which allows us to control its derivatives and assess the convergence of the series. By truncating the expansion at the second order, a quadratic approximation formula for the price is then provided. Finally, practical applications of the result are highlighted by performing a numerical comparison with alternative pricing methodologies.

We consider an iterated Kolmogorov diffusion $X_t$ of step n. The small ball problem for $X_t$ is solved by means of the Gaussian correlation inequality. We also prove Chung’s laws of iterated logarithm for $X_t$ both at time zero and infinity.

The geometric sum plays a significant role in risk theory and reliability theory [Kalashnikov (1997)] and a prototypical example of the geometric sum is Rényi’s theorem [Rényi (1956)] saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum [Sugakova (1995), Kalashnikov (1997), Peköz & Röllin (2011)] but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein’s method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.

We study the weak limit of the arboreal gas along any exhaustion of a regular tree with wired boundary conditions. We prove that this limit exists, does not depend on the choice of exhaustion, and undergoes a phase transition. Below and at criticality, we prove the model is equivalent to bond percolation. Above criticality, we characterise the model as the superposition of critical bond percolation and a random collection of infinite one-ended paths. This provides a simple example of an arboreal gas model that continues to exhibit critical-like behaviour throughout its supercritical phase.

We withdraw [1, Theorem 1].

Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in general frameworks such as the one of Polish spaces. However, for the martingale transport problem several works based on different strategies established stability results for $\mathbb{R}$only. We show that the restriction to dimension $d=1$ is not accidental by presenting a sequence of marginal distributions on ${\mathbb{R}}^2$ for which the martingale optimal transport problem is neither stable w.r.t. the value nor the set of minimizers. Our construction adapts to any dimension $d\ge 2$. For $d\ge 2$ it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in $d=1$.

We investigate the compact interface property in a recently introduced variant of the stochastic heat equation that incorporates dormancy, or equivalently seed banks. There individuals can enter a dormant state during which they are no longer subject to spatial dispersal and genetic drift. This models a state of low metabolic activity as found in microbial species. Mathematically, one obtains a memory effect since mass accumulated by the active population will be retained for all times in the seed bank. This raises the question whether the introduction of a seed bank into the system leads to a qualitatively different behaviour of a possible interface. Here, we aim to show that nevertheless in the stochastic heat equation with seed bank compact interfaces are retained through all times in both the active and dormant population. We use duality and a comparison argument with partial functional differential equations to tackle technical difficulties that emerge due to the lack of the martingale property of our solutions which was crucial in the classical non seed bank case.

By investigating McKean-Vlasov SDEs, the order preservation and positive correlation are characterized for nonlinear Fokker-Planck equations. The main results recover the corresponding criteria on these properties established in [3, 5] for diffusion processes or linear Fokker-Planck equations.

We show that for lazy simple random walks on finite spherically symmetric trees, the ratio of the mixing time and the relaxation time is bounded by a universal constant. Consequently, lazy simple random walks on any sequence of finite spherically symmetric trees do not exhibit pre-cutoff; this conclusion also holds for continuous-time simple random walks. This answers a question recently proposed by Gantert, Nestoridi, and Schmid. We also show that for lazy simple random walks on finite spherically symmetric trees, hitting times of vertices are (uniformly) non concentrated. Finally, we study the stability of our results under rough isometries.

Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the “the least favorable perturbation” ${\hat{Y}}_{\mathit{\epsilon }}$, which maximizes the prediction error $E{(X-E(X|X+Y))}^2$ in the class of Y with $var(Y)\le \mathit{\epsilon }$. We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse.

Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-λ Poisson process while being chased and consumed by blue particles according to a rate-1 Poisson process. Given a growing sequence of finite graphs, the critical rate ${\mathit{\lambda }}_c$ is the largest value of λ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on ${\mathit{\lambda }}_c$ for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.

It has been discovered that the Kadomtsev-Petviashvili (KP) equation governs the distribution of the fluctuation of many random growth models. In particular, the Tracy-Widom distributions appear as special self-similar solutions of the KP equation. We prove that the anti-derivative of the Baik-Rains distribution, which governs the fluctuation of the models in the KPZ universality class starting with stationary initial data, satisfies the KP equation. The result is first derived formally by taking a limit of the generating function of the KPZ equation, which satisfies the KP equation. Then we prove it directly using the explicit Painlevé II formulation of the Baik-Rains distribution.

Let ${\{{\mathrm{\eta }}_i(t),t\in [0,1]\}}_{i=1}^k$ be independent copies of $\mathrm{\eta }=\{\mathrm{\eta }(t),t\in [0,1]\}$, a mean zero continuous Gaussian process. Let

$Y_k:=Y_k(t)={\sum }_{i=1}^k{\mathrm{\eta }}_i^2(t),t\in [0,1].$

This paper shows how exact local (at 0) and uniform moduli of continuity (on [0,1]) of $Y_k$ can be obtained from the exact local and uniform moduli of continuity of η.

The results in this paper provide new information on asymptotic properties of classical models: the neutral Kingman coalescent under a general finite-alleles, parent-dependent mutation mechanism, and its generalisation, the ancestral selection graph. Several relevant quantities related to these fundamental models are not explicitly known when mutations are parent dependent. Examples include the probability that a sample taken from a population has a certain type configuration, and the transition probabilities of their block counting jump chains. In this paper, asymptotic results are derived for these quantities, as the sample size goes to infinity. It is shown that the sampling probabilities decay polynomially in the sample size with multiplying constant depending on the stationary density of the Wright-Fisher diffusion and that the transition probabilities converge to the limit of frequencies of types in the sample.

In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee [14].

González Cázares and Ivanovs (2021) suggested a new method for “recovering” the Brownian motion component from the trajectory of a Lévy process that required sampling from an independent Brownian motion process. We show that such a procedure works equally well without any additional source of randomness if one uses normal quantiles instead of the ordered increments of the auxiliary Brownian motion process.

Reinforced processes are known to provide a stochastic representation for the quasi-stationary distribution of a given killed Markov process – describing the killed Markov process at fixed time instants. In this paper we shall adapt the construction to provide a pathwise description. We also obtain a stochastic approximation for the quasi-limiting distributions of reducible killed Markov processes as a corollary.

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\ge 2$. We show that the eigenvalues stay distinct for all times, and that the process Λ of eigenvalues is a kinetic diffusion (i.e. the pair $(\mathrm{\Lambda },\dot{\mathrm{\Lambda }})$ of Λ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that Λ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.

We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble ${\sum }_{i\ne j}f(L_N({\mathit{\theta }}_i-{\mathit{\theta }}_j))$ for arbitrary $\mathit{\beta }>0$ in the mesoscopic regime $L_N=\mathcal{O}(N^{2∕3-\mathit{ϵ}})$. In addition, we obtain similar results for bipartite statistics.

We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.

When the value L of the directed landscape at a point $(\mathbf{p};\mathbf{q})$ is sufficiently large, the geodesic from p to q is rigid and its location fluctuates of order $L^{-1∕4}$ around its expectation. We further show that at a midpoint of the geodesic, the location of the geodesic and the value of the directed landscape after appropriate scaling converge to two independent Gaussians.

We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.

We study noise sensitivity of the consensus opinion of the voter model on finite graphs, with respect to noise affecting the initial opinions and noise affecting the dynamics. We prove that the final opinion is stable with respect to small perturbations of the initial configuration, and is sensitive to perturbations of the dynamics governing the evolution of the process. Our proofs rely on the duality relationship between the voter model and coalescing random walks, and on a precise description of this evolution when we have coupled dynamics.

The first aim of this paper is to wonder to what extent we can generalize the central limit theorem of Gordin [6] under the so-called ${\mathbb{L}}^1$-projective criteria to ergodic stationary random fields when completely commuting filtrations are considered. Surprisingly it appears that this result cannot be extended to its full generality and that an additional condition is needed.

We consider the elephant random walk with general step distribution. We calculate the first four moments of the limiting distribution of the position rescaled by $n^{\mathit{\alpha }}$ in the superdiffusive regime where α is the memory parameter. This extends the results obtained by Bercu in [Ber17].

In this paper we investigate some geometric functionals for band-limited Gaussian and isotropic spherical random fields in dimension 2. In particular, we focus on the area of excursion sets, providing its behavior in the high energy limit. Our results are based on Wiener chaos expansion for non linear transform of Gaussian fields and on an explicit derivation on the high-frequency limit of the covariance function of the field. As a simple corollary we establish also the Central Limit Theorem for the excursion area.

Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.

We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem – without the need of renormalization – in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.

We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.

In this paper we prove the weak convergence, in the high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions in a subset of the high-temperature phase. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.

Consider a càdlàg local martingale M with square brackets $[M]$. In this paper, we provide upper and lower bounds for expectations of the type $\mathbb{E}{\left[M\right]}_{\mathit{\tau }}^{q∕2}$, for any stopping time τ and $q\ge 2$, in terms of predictable processes. This result can be thought of as a Burkholder-Davis-Gundy type inequality in the sense that it can be used to relate the expectation of the running maximum $|M^{\ast }{|}^q$ to the expectation of the dual previsible projections of the relevant powers of the associated jumps of M. The case for a class of moderate functions is also discussed.

A weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in [7].

The paper contains the study of sharp extensions of weak-type estimates for a martingale maximal function. Given $1<p<\mathrm{\infty }$ and a pair $(x,y)$ of nonnegative numbers satisfying $x^p\le y$, we identify the optimal upper bounds for $\Vert |{sup}_n{f}_n|{\Vert }_{p,\mathrm{\infty }}$, for nonnegative martingales $f={(f_n)}_{n\ge 0}$ satisfying $\Vert f{\Vert }_1=x$ and $\Vert f{\Vert }_p^p=y$.

Consider a 2-dimensional soft random geometric graph $G(\mathit{\lambda },s,\mathit{\varphi })$, obtained by placing a Poisson($\mathit{\lambda }{s}^2$) number of vertices uniformly at random in a square of side s, with edges placed between each pair $x,y$ of vertices with probability $\mathit{\varphi }(\Vert x-y\Vert )$, where $\mathit{\varphi }:{\mathbb{R}}_{+}\to [0,1]$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(\mathit{\lambda },s,\mathit{\varphi })$ in the large-s limit with $(\mathit{\lambda },\mathit{\varphi })$ fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value ${\mathit{\lambda }}_c(\mathit{\varphi })$.

This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.

Occupancy processes are a broad class of discrete time Markov chains on ${\{0,1\}}^n$ encompassing models from ecology and epidemiology. This model is compared to a collection of n independent Markov chains on $\{0,1\}$, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non–nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of “weak records” (or “ladder points”) in two kinds of one–dimensional non–nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases.

Let $T_D$ denote the first exit time of a Brownian motion from a domain D in ${\mathbb{R}}^n$. Given domains $U,W\subseteq {\mathbb{R}}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from U than W, meaning $\mathbf{P}(T_U<t)>\mathbf{P}(T_W<t)$ for t small. We show that the primary factor in the probability of fast exits from domains is the proximity of the closest regular part of the boundary to the origin. We also prove a result on the complementary question of longs stays, meaning $\mathbf{P}(T_U>t)>\mathbf{P}(T_W>t)$ for t large. This result, which applies only in two dimensions, shows that the unit disk $\mathbb{D}$ has the lowest probability of long stays amongst all Schlicht domains.

The halfspace depth is a prominent tool of nonparametric inference for multivariate data. We consider it in the general context of finite Borel measures μ on ${\mathbb{R}}^d$. The halfspace depth of a point $x\in {\mathbb{R}}^d$ is defined as the infimum of the μ-masses of halfspaces that contain x. We say that a measure μ has a simple (halfspace) depth if the set of all attained halfspace depth values of μ on ${\mathbb{R}}^d$ is finite. We give a complete description of measures with simple depths by showing that the halfspace depth of μ is simple if and only if μ is atomic with finitely many atoms. This result completely resolves the halfspace depth characterization problem for the particular situation of simple halfspace depths and datasets. We also discuss the cardinality of the set of the attained halfspace depth values.

Consider the set of Borel probability measures on ${\mathbf{R}}^k$ and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme value distributions is dense and of the first Baire category. In addition, the analogue result holds in the context of free probability theory.

It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors. This procedure is known as matrix scaling, and has found numerous applications in operations research, economics, image processing, and machine learning. In this work, we establish the stability of matrix scaling to random bounded perturbations. Specifically, letting $\tilde{A}\in {\mathbb{R}}^{M\times N}$ be a positive and bounded random matrix whose entries assume a certain type of independence, we provide a concentration inequality for the scaling factors of $\tilde{A}$ around those of $A=\mathbb{E}[\tilde{A}]$. This result is employed to study the convergence rate of the scaling factors of $\tilde{A}$ to those of A, as well as the concentration of the scaled version of $\tilde{A}$ around the scaled version of A in operator norm, as $M,N\to \mathrm{\infty }$. We demonstrate our results in several simulations.

The Stein couplings of Chen and Roellin [6] vastly expanded the range of applications for which coupling constructions in Stein’s method for normal approximation could be applied, and subsumed both Stein’s classical exchangeable pair, as well as the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.

We show that for low enough temperatures, but still above the AT line, the Jacobian of the TAP equations for the SK model has a macroscopic fraction of eigenvalues outside the unit interval. This provides a simple explanation for the numerical instability of the fixed points, which thus occurs already in high temperature. The insight leads to some algorithmic considerations on the low temperature regime.

We say that a random integer variable X is monotone if the modulus of the characteristic function of X is decreasing on $[0,\mathrm{\pi }]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the trick of exponential tilting, we obtain sharp estimates for the point probabilities in the tail under a slightly stronger assumption on the random integer variables which we call strong monotonicity.

In this note, we consider the parabolic Anderson model on ${\mathbb{R}}_{+}\times \mathbb{R}$, driven by a Gaussian noise which is fractional in time with index $H_0>1∕2$ and fractional in space with index $0<H<1∕2$ such that $H_0+H>3∕4$. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p-th moments with $p\ge 2$.