We study convergence rates, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact subset of $\mathbb{R}^d$. We propose two different approaches for this study. The first is based on the notion of a minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index, which is much used in extreme value theory. The second approach is based on a $\beta$-mixing condition together with a local-type dependence assumption. More precisely, all our results concern stationary $\beta$-mixing sequences satisfying a tail condition, known as the (a, b)-standard assumption, together with a local-type dependence condition or stationary sequences satisfying the (a, b)-standard assumption and having a positive minimal index. We prove that the optimal rates of the independent and identically distributed setting can be reached. We apply our results to stationary Markov chains on a ball, or to a class of Markov chains on a circle or on a torus. We study with simulations the particular examples of a Möbius Markov chain on the unit circle and of a Markov chain on the unit square wrapped on a torus.

We develop explicit bounds for the tail of the distribution of the all-time supremum of a random walk with negative drift, where the increments have a truncated heavy-tailed distribution. As an application, we consider a ruin problem in the presence of reinsurance.

This work concerns stochastic differential equations with jumps. We prove convergence for solutions to a sequence of (possibly degenerate) stochastic differential equations with jumps when the coefficients converge in some appropriate sense. Then some special cases are analyzed and some concrete and verifiable conditions are given.

We investigate geometric properties of invariant spatio-temporal random fields $X\colon\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space $\mathbb M^d$ in any dimension $d\ge 2$, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as $T\to +\infty$) of the average on [0,T] of the volume of the excursion set on the manifold, i.e. of $\lbrace X(\cdot, t)\ge u\rbrace$ (for any $u >0$). The Fourier components of X may have short or long memory in time, i.e. integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in Marinucci et al. (2021) and allows us to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as $T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field X.

We prove an ergodic theorem for Markov chains indexed by the Ulam–Harris–Neveu tree over large subsets with arbitrary shape under two assumptions: (i) with high probability, two vertices in the large subset are far from each other, and (ii) with high probability, those two vertices have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that these assumptions are satisfied for some usual trees. Finally, with Markov chain Monte Carlo considerations in mind, we prove that when the underlying Markov chain is stationary and reversible, the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes. In doing so, we prove that the Hosoya–Wiener polynomial is minimized over $[{-}1,1]$ by the line graph among trees of a given size.

This paper focuses on the averaging principle concerning the fast–slow McKean–Vlasov stochastic differential equations driven by mixed fractional Brownian motion with Hurst parameter $\tfrac{1}{2} < H < 1$. The integral associated with Brownian motion is the standard Itô integral, while the integral with respect to fractional Brownian motion is a generalized Riemann–Stieltjes integral. Under the non-Lipschitz condition and certain appropriate assumptions regarding the coefficients, we initially establish the existence and uniqueness theorem for the fast–slow McKean–Vlasov stochastic differential equation driven by mixed fractional Brownian motion. Subsequently, we demonstrate the averaging principle of the fast–slow McKean–Vlasov stochastic differential equations, signifying that the slow stochastic differential equation converges to the associated averaged equation in terms of mean-square convergence.

In this paper we derive cumulant bounds for subgraph counts and power-weighted edge lengths in a class of spatial random networks known as weight-dependent random connection models. These bounds give rise to different probabilistic results, from which we mainly focus on moderate deviations of the respective statistics, but also show a concentration inequality and a normal approximation result. This involves dealing with long-range spatial correlations induced by the profile function and the weight distribution. We start by deriving the bounds for the classical case of a Poisson vertex set, and then provide extensions to α-determinantal processes.

We define a random graph obtained by connecting each point of $\mathbb{Z}^d$ independently and uniformly to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed k-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between the vertices in at least one, respectively precisely two, directions. For these graphs we study the question of percolation, i.e. the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected k-neighbor graph never percolates, while the directed k-neighbor graph percolates whenever $k \geq d+1$, $k \geq 3$, and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected 2-neighbor graph percolates for $d=2$, the undirected 3-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

We develop general conditions for weak convergence of adaptive Markov chain Monte Carlo processes and this is shown to imply a weak law of large numbers for bounded Lipschitz continuous functions. This allows an estimation theory for adaptive Markov chain Monte Carlo where previously developed theory in total variation may fail or be difficult to establish. Extensions of weak convergence to general Wasserstein distances are established, along with a weak law of large numbers for possibly unbounded Lipschitz functions. Applications are applied to autoregressive processes in various settings, unadjusted Langevin processes, and adaptive Metropolis–Hastings.

Consider nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,\ldots$ of a random variable X with a probability density function f. If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of $R=(X_{N+1},X_{N+2},\ldots)$ conditioned on $S=(X_1,\ldots,X_N)$ does not depend on f. If also the lengths of the intervals exhibit a Markovian structure, $R\mid S$ becomes a Markov chain of a certain order $s\ge0$. If $s=0$ then $X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time $\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.

We introduce a financial market model featuring a risky asset whose price follows a sticky geometric Brownian motion and a riskless asset that grows with a constant interest rate $r\in \mathbb R$. We prove that this model satisfies no arbitrage and no free lunch with vanishing risk only when $r=0$. Under this condition, we derive the corresponding arbitrage-free pricing equation, assess the replicability, and give a representation of the replication strategy. We then show that all locally bounded replicable payoffs for the standard Black–Scholes model are also replicable for the sticky model. Last, we evaluate via numerical experiments the impact of hedging in discrete time and of misrepresenting price stickiness.

We focus on obtaining Block–Savits type characterizations for different ageing classes as well as some important renewal classes by using the Laplace transform. We also introduce a novel approach, based on the equilibrium distribution, to handle situations where the techniques of Block and Savits (1980) either fail or involve tedious calculations. Our approach in conjunction with the theory of total positivity yields Vinogradov’s (1973) result for the increasing failure rate class when the distribution function is continuous. We also present simple but elegant proofs for Block and Savits’ results for the decreasing mean residual life, new better than used in expectation, and harmonic new better than used in expectation classes as applications of our approach. We address several other related issues that are germane to our problem. Finally, we conclude with a short discussion on the issue of convolutions.

We introduce a comprehensive method for establishing stochastic orders among order statistics in the independent and identically distributed case. This approach relies on the assumption that the underlying distribution is linked to a reference distribution through a transform order. Notably, this method exhibits broad applicability, particularly since several well-known nonparametric distribution families can be defined using relevant transform orders, including the convex and the star transform orders. Moreover, for convex-ordered families, we show that an application of Jensen’s inequality gives bounds for the probability that a random variable exceeds the expected value of its corresponding order statistic.

We consider a superprocess $\{X_t\colon t\geq 0\}$ in a random environment described by a Gaussian field $\{W(t,x)\colon t\geq 0,x\in \mathbb{R}^d\}$. First, we set up a representation of $\mathbb{E}[\langle g, X_t\rangle\mathrm{e}^{-\langle \,f,X_t\rangle }\mid\sigma(W)\vee\sigma(X_r,0\leq r\leq s)]$ for $0\leq s < t$ and some functions f,g, which generalizes the result in Mytnik and Xiong (2007, Theorem 2.15). Next, we give a uniform upper bound for the conditional log-Laplace equation with unbounded initial values. We then use this to establish the corresponding conditional entrance law. Finally, the excursion representation of $\{X_t\colon t\geq 0\}$ is given.

Gaussian random polytopes have received a lot of attention, especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less-studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated with Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to $C(\alpha)^{d+o(d)}$, where $C(\alpha)$ is some constant which depends on the constant of proportionality $\alpha$. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When the number of points is 2d, we determine the constant C such that the expected number of pairs of estranged facets is equal to $C^{d+o(d)}$.

Considering a double-indexed array $(Y_{n,i:\,n\ge 1,i\ge 1})$ of non-negative regularly varying random variables, we study the random-length weighted sums and maxima from its ‘row’ sequences. These sums and maxima may have the same tail and extremal indices (Markovich and Rodionov 2020). The main constraints of the latter results are that there exists a unique series in a scheme of series with the minimum tail index and the tail of the term number is lighter than the tail of the terms. Here, a bounded random number of series are allowed to have the minimum tail index and the tail of the term number may be heavier than the tail of the terms. We derive the tail and extremal indices of the weighted non-stationary random-length sequences under a broader set of conditions than in Markovich and Rodionov (2020). We provide examples of random sequences for which the assumptions are valid. Perspectives in adopting the results in different application areas are formulated.

We establish a sample path moderate deviation principle for the integrated shot noise process with Poisson arrivals and non-stationary noises. As in Pang and Taqqu (2019), we assume that the noise is conditionally independent given the arrival times, and the distribution of each noise depends on its arrival time. As applications, we derive moderate deviation principles for the workload process and the running maximum process for a stochastic fluid queue with the integrated shot noise process as the input; we also show that a steady-state distribution exists and derive the exact tail asymptotics.

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPSs), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of ‘sufficientness’ postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observations. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a ‘pure’ sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

This paper is concerned with the growth rate of susceptible–infectious–recovered epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterised by the presence of cliques (fully connected subgraphs). We study epidemics on random intersection graphs with a mixed Poisson degree distribution and show that in the limit of large population sizes the number of infected individuals grows exponentially during the early phase of the epidemic, as is generally the case for epidemics on asymptotically unclustered networks. The Malthusian parameter is shown to satisfy a variant of the classical Euler–Lotka equation. To obtain these results we construct a coupling of the epidemic process and a continuous-time multitype branching process, where the type of an individual is (essentially) given by the length of its infectious period. Asymptotic results are then obtained via an embedded single-type Crump–Mode–Jagers branching process.