Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. In the context-specific case, where conditional independence is queried relative to a specific value of the conditioning variables, we introduce the notion of a source DAG to disclose the valid conditional independence relations. In the context-free case, we characterize conditional independence through a modified separation concept, ∗-separation, combined with a tropical eigenvalue condition. We also introduce the notion of an impact graph, which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry.

In this work, we introduce a generalization of the classical Pólya urn scheme (Ann. Inst. Henri Poincaré 1 (1930) 117–161) with colors indexed by a Polish space, say, S. The urns are defined as finite measures on S endowed with the Borel σ-algebra, say, $\mathcal{S}$. The generalization is an extension of a model introduced earlier by Blackwell and MacQueen (Ann. Statist. 1 (1973) 353–355). We present a novel approach of representing the observed sequence of colors from such a scheme in terms an associated branching Markov chain on the random recursive tree. The work presents fairly general asymptotic results for this new generalized urn models. As special cases, we show that the results on classical urns, as well as, some of the results proved recently for infinite color urn models in (Bernoulli 23 (2017) 3243–3267; Statist. Probab. Lett. 92 (2014) 232–240), can easily be derived using the general asymptotic. We also demonstrate some newer results for infinite color urns.

We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semilinear PDE with superlinear gradient, monotone generator and singular terminal value. We also establish an asymptotic analysis of the robust model for small amounts of uncertainty and analyze the effect of robustness on optimal trading strategies and liquidation costs. In particular, in our model ambiguity aversion is observationally equivalent to increased risk aversion. This suggests that ambiguity aversion increases liquidation rates.

Over the last 25 years, techniques based on drift and minorization (d&m) have been mainstays in the convergence analysis of MCMC algorithms. However, results presented herein suggest that d&m may be less useful in the emerging area of convergence complexity analysis, which is the study of how the convergence behavior of Monte Carlo Markov chains scales with sample size, n, and/or number of covariates, p. The problem appears to be that minorization can become a serious liability as dimension increases. Alternative methods of constructing convergence rate bounds (with respect to total variation distance) that do not require minorization are investigated. Based on Wasserstein distances and random mappings, these methods can produce bounds that are substantially more robust to increasing dimension than those based on d&m. The Wasserstein-based bounds are used to develop strong convergence complexity results for MCMC algorithms used in Bayesian probit regression and random effects models in the challenging asymptotic regime where n and p are both large.

The Markov dynamics of interlaced particle arrays, introduced by A. Borodin and P. L. Ferrari in (Comm. Math. Phys. 325 (2014) 603–684), is a classical example of $(2+1)$-dimensional random growth model belonging to the so-called Anisotropic KPZ universality class. In (Comm. Pure Appl. Math. 72 (2018) 620–666), a hydrodynamic limit—the convergence of the height profile, after space/time rescaling, to the solution of a deterministic Hamilton–Jacobi PDE with nonconvex Hamiltonian—was proven when either the initial profile is convex, or for small times, before the solution develops shocks. In the present work, we give a simpler proof, that works for all times and for all initial profiles for which the limit equation makes sense. In particular, the convexity assumption is dropped. The main new idea is a new viewpoint about “finite speed of propagation” that allows to bypass the need of a priori control of the interface gradients, or equivalently of inter-particle distances.

We consider regular variation for marked point processes with independent heavy-tailed marks and prove a single large point heuristic: the limit measure is concentrated on the cone of point measures with one single point. We then investigate successive hidden regular variation removing the cone of point measures with at most k points, $\mathit{k}\ge 1$, and prove a multiple large point phenomenon: the limit measure is concentrated on the cone of point measures with $\mathit{k}+1$ points. We show how these results imply hidden regular variation in Skorokhod space of the associated risk process, in connection with the single/multiple large point heuristic from (Ann. Probab. 47 (2019) 3551–3605). Finally, we provide an application to risk theory in a reinsurance model where the k largest claims are covered and we study the asymptotic behavior of the residual risk.

We define a new class of Ξ-coalescents characterized by a possibly infinite measure over the nonnegative integers. We call them symmetric coalescents since they are the unique family of exchangeable coalescents satisfying a symmetry property on their coagulation rates: they are invariant under any transformation that consists of moving one element from one block to another without changing the total number of blocks. We illustrate the diversity of behaviors of this family of processes by introducing and studying a one parameter subclass, the $(\mathit{\beta },\mathit{S})$-coalescents. We also embed this family in a larger class of Ξ-coalescents arising as the limit genealogies of Wright–Fisher models with bottlenecks. Some convergence results rely on a new Skorokhod type metric, that induces the Meyer–Zheng topology, which allows us to study the scaling limit of non-Markovian processes using standard techniques.

In the present paper, we show that (under some minor technical assumption) Complex Gaussian multiplicative chaos defined as the complex exponential of a log-correlated Gaussian field can be obtained by taking the limit of the exponential of the field convoluted with a smoothing kernel. We consider two types of chaos: ${\mathit{e}}^{\mathit{\gamma }\mathit{X}}$ for a log correlated field X and $\mathit{\gamma }=\mathit{\alpha }+\mathit{i}\mathit{\beta }$, $\mathit{\alpha },\mathit{\beta }\in \mathbb{R}$ and ${\mathit{e}}^{\mathit{\alpha }\mathit{X}+\mathit{i}\mathit{\beta }\mathit{Y}}$ for X and Y two independent fields with $\mathit{\alpha },\mathit{\beta }\in \mathbb{R}$. Our result is valid in the range

$${\mathcal{P}}_{\mathrm{sub}}:=\{{\mathit{\alpha }}^2+{\mathit{\beta }}^2<\mathit{d}\}\cup \{|\mathit{\alpha }|\in (\sqrt{\mathit{d}/2},\sqrt{2\mathit{d}})\text{and}|\mathit{\beta }|<\sqrt{2\mathit{d}}-|\mathit{\alpha }|\},$$

which, up to boundary, is conjectured to be optimal.

We study a single-period optimal transport problem on ${\mathbb{R}}^2$ with a covariance-type cost function $\mathit{c}(\mathit{x},\mathit{y})=({\mathit{x}}_1-{\mathit{y}}_1)({\mathit{x}}_2-{\mathit{y}}_2)$ and a backward martingale constraint. We show that a transport plan γ is optimal if and only if there is a maximal monotone set G that supports the x-marginal of γ and such that $\mathit{c}(\mathit{x},\mathit{y})={min}_{\mathit{z}\in \mathit{G}}\mathit{c}(\mathit{z},\mathit{y})$ for every $(\mathit{x},\mathit{y})\in supp\mathit{\gamma }$. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from (Rev. Econ. Stud. 61 (1994) 131–152).

Given a p-coin that lands heads with unknown probability p, we wish to produce an $\mathit{f}(\mathit{p})$-coin for a given function $\mathit{f}:(0,1)\to (0,1)$. This problem is commonly known as the Bernoulli factory and results on its solvability and complexity have been obtained in (ACM Trans. Model. Comput. Simul. 4 (1994) 213–219; Ann. Appl. Probab. 15 (2005) 93–115). Nevertheless, generic ways to design a practical Bernoulli factory for a given function f exist only in a few special cases. We present a constructive way to build an efficient Bernoulli factory when $\mathit{f}(\mathit{p})$ is a rational function with coefficients in $\mathbb{R}$. Moreover, we extend the Bernoulli factory problem to a more general setting where we have access to an m-sided die and we wish to roll a v-sided one; that is, we consider rational functions between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains—a task that we show can be achieved using perfect simulation techniques with the original m-sided die as the only source of randomness. In the Bernoulli factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in p. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.

The stochastic Kuramoto model defined on a sequence of graphs is analyzed: the emphasis is posed on the relationship between the mean field limit, the connectivity of the underlying graph and the long time behavior. We give an explicit deterministic condition on the sequence of graphs such that, for any finite time and any initial condition, even dependent on the network, the empirical measure of the system stays close to the solution of the McKean–Vlasov equation associated to the classical mean field limit. Under this condition, we study the long time behavior in the subcritical and in the supercritical regime: in both regimes, the empirical measure stays close to the (possibly degenerate) manifold of stable stationary solutions, up to times which can diverge as fast as the exponential of the size of the system, before large deviation phenomena take over. The condition on the sequence of graphs is derived by means of Grothendieck’s inequality and expressed through a concentration in ${\ell }_{\infty }\to {\ell }_1$ norm. It is shown to be satisfied by a large class of graphs, random and deterministic, provided that the average number of neighbors per site diverges, as the size of the system tends to infinity.

We study the outliers for two models which have an interesting connection. On the one hand, we study a specific class of planar Coulomb gases which are determinantal. It corresponds to the case where the confining potential is the logarithmic potential of a radial probability measure. On the other hand, we study the zeros of random polynomials that appear to be closely related to the first model. Their behavior far from the origin is shown to depend only on the decaying properties of the probability measure generating the potential. A similar feature is observed for their behavior near the origin. Furthermore, in some cases, the appearance of outliers is observed, and the zeros of random polynomials and the Coulomb gases are seen to exhibit exactly the same behavior, which is related to the unweighted Bergman kernel.

This paper is concerned with a stochastic linear-quadratic optimal control problem with regime switching, random coefficients and cone control constraint. The randomness of the coefficients comes from two aspects: the Brownian motion and the Markov chain. Using Itô’s lemma for Markov chain, we obtain the optimal state feedback control and optimal cost value explicitly via two new systems of extended stochastic Riccati equations (ESREs). We prove the existence and uniqueness of the two ESREs using tools including multidimensional comparison theorem, truncation function technique, log transformation and the John–Nirenberg inequality. These results are then applied to study mean-variance portfolio selection problems with and without short-selling prohibition with random parameters depending on both the Brownian motion and the Markov chain. Finally, the efficient portfolios and efficient frontiers are presented in closed forms.

In this paper we consider the posterior consistency of Bayesian inference procedures when the family of models consists of appropriate stochastic processes. Specifically, we suppose that one observes an unknown ergodic process and one has access to a family of models consisting of dependent processes. In this context, we consider Gibbs posterior inference, which is a loss-based generalization of standard Bayesian inference. Our main results characterize the asymptotic behavior of the Gibbs posterior distributions on the space of models. Furthermore, we show that in the case of properly specified models our convergence results may be used to establish posterior consistency. Our model processes are defined via the thermodynamic formalism for dynamical systems, and they allow for a large degree of dependence, including both Markov chains of unbounded orders and processes that are not Markov of any order. This work establishes close connections between Gibbs posterior inference and the thermodynamic formalism for dynamical systems, which we hope will lead to new questions and results in both nonparametric Bayesian analysis and the thermodynamic formalism.

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability $1-\mathit{q}$ and the opposite value with probability q, where $\mathit{q}\in [0,1]$. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of q for which the optimal reconstruction method has a probability of error bounded away from $1/2$. We also show that the probability of error is bounded by a constant times q. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g., Plug–Pichler—stochastic programming, Hellwig—game theory, Aldous—stability of optimal stopping, Hoover–Keisler—model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality.

A natural compatible metric for the adapted weak topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug–Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etcetera in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.

We consider sequences of U-processes based on symmetric kernels of a fixed order, that possibly depend on the sample size. Our main contribution is the derivation of a set of analytic sufficient conditions, under which the aforementioned U-processes weakly converge to a linear combination of time-changed independent Brownian motions. In view of the underlying symmetric structure, the involved time-changes and weights remarkably depend only on the order of the U-statistic, and have consequently a universal nature. Checking these sufficient conditions requires calculations that have roughly the same complexity as those involved in the computation of fourth moments and cumulants. As such, when applied to the degenerate case, our findings are infinite-dimensional extensions of the central limit theorems (CLTs) proved in de Jong (J. Multivariate Anal. 34 (1990) 275–289) and Döbler and Peccati (Electron. J. Probab. 22 (2017) Paper No. 2). As important tools in our analysis, we exploit the multidimensional central limit theorems established in Döbler and Peccati (Electron. J. Probab. 24 (2019) Paper No. 5) together with upper bounds on absolute moments of degenerate U-statistics by Ibragimov and Sharakhmetov (Studia Sci. Math. Hungar. 39 (2002) 251–275), and also prove some novel multiplication formulae for degenerate symmetric U-statistics—allowing for different sample sizes—that are of independent interest. We provide applications to random geometric graphs and to a class of U-statistics of order two, whose Gaussian fluctuations have been recently studied by Robins et al. (Stochastic Process. Appl. 126 (2016) 3733–3759), in connection with quadratic estimators in nonparametric models. In particular, our application to random graphs yields a class of new functional central limit theorems for subgraph counting statistics, extending previous findings in the literature. Finally, some connections with invariance principles in changepoint analysis are established.

We extend Stein’s celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size n and degree d, where we obtain the optimal convergence rate $\sqrt{{\mathit{n}}^3/\mathit{d}}$ under only moment assumptions, and to degenerate U-statistics and Poisson functionals, where we strengthen a few of the fourth moment bounds in the literature on the Wasserstein distance.

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a Λ-coalescent, and fragmentation dislocates at a finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters ${\mathit{\theta }}_{\star }\le {\mathit{\theta }}^{\star }\in [0,\infty ]$, so that if ${\mathit{\theta }}^{\star }<1$, the process comes down from infinity and if ${\mathit{\theta }}_{\star }>1$, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters ${\mathit{\theta }}^{\star }$, ${\mathit{\theta }}_{\star }$ coincide and are explicit.

We consider a mean-field system of path-dependent stochastic interacting diffusions in random media over a finite time window. The interaction term is given as a function of the empirical measure and is allowed to be nonlinear and path dependent. We prove that the sequence of empirical measures of the full trajectories satisfies a large deviation principle with explicit rate function. The minimizer of the rate function is characterized as the path-dependent McKean–Vlasov diffusion associated to the system. As corollary, we obtain a strong law of large numbers for the sequence of empirical measures. The proof is based on a decoupling technique by associating to the system a convenient family of product measures. To illustrate, we apply our results for the delayed stochastic Kuramoto model and for a SDE version of Galves–Löcherbach model.

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans ${\mathit{\pi }}^1,{\mathit{\pi }}^2,\dots$ converges weakly to a transport plan π, then π is also optimal (between its marginals).

Alfonsi, Corbetta and Jourdain (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 1706–1729) asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since martingale optimal transport plans are not characterized by a “monotonicity”-property of their supports.

In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet (In Séminaire de Probabilités XLVIII (2016) 13–32 Springer). An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.