We consider the problem of estimating fractional processes based on noisy high-frequency data. Generalizing the idea of pre-averaging to a fractional setting, we exhibit a sequence of consistent estimators for the unknown parameters of interest by proving a law of large numbers for associated variation functionals. In contrast to the semimartingale setting, the optimal window size for pre-averaging depends on the unknown roughness parameter of the underlying process. We evaluate the performance of our estimators in a simulation study and use them to empirically verify Kolmogorov’s $2/3$-law in turbulence data contaminated by instrument noise.

Let $\{X_{i}\}_{i\geq1}$ be a sequence of independent and identically distributed random variables and $T\in\{1,2,\ldots\}$ a stopping time associated with this sequence. In this paper, the distribution of the minimum observation, $\min\{X_{1},X_{2},\ldots,X_{T}\}$, until the stopping time T is provided by proposing a methodology based on an appropriate change of the initial probability measure of the probability space to a truncated (shifted) one on the $X_{i}$. As an application of the aforementioned general result, the random variables $X_{1},X_{2},\ldots$ are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process $\{Y_{t},t\geq0\}$, while the stopping time T is set to be the number of summands until the sum of the $X_{i}$ exceeds t for the first time, i.e. $T=Y_{t}+1$. Under this setup, the distribution of the minimal spacing, $D_{t}=\min\{X_{1},X_{2},\ldots,X_{Y_{t}+1}\}$, that starts in the interval [0, t] is investigated and a stochastic ordering relation for $D_{t}$ is obtained. In addition, bounds for the tail probability of $D_{t}$ are provided when the interarrival times have the increasing failure rate / decreasing failure rate property. In the special case of a Poisson process, an exact formula, as well as closed-form bounds and an asymptotic result, are derived for the tail probability of $D_{t}$. Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of $D_{t}$ are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.

We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of $({a_T-1})/{T{\textrm{e}}^{b_TTx}}$, the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of $1/T^2$, the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.

We consider an inhomogeneous Erdős–Rényi random graph ensemble with exponentially decaying random disconnection probabilities determined by an independent and identically distributed field of variables with heavy tails and infinite mean associated with the vertices of the graph. This model was recently investigated in the physics literature (Garuccio, Lalli, and Garlaschelli 2023) as a scale-invariant random graph within the context of network renormalization. From a mathematical perspective, the model fits in the class of scale-free inhomogeneous random graphs whose asymptotic geometrical features have recently attracted interest. While for this type of graph several results are known when the underlying vertex variables have finite mean and variance, here instead we consider the case of one-sided stable variables with necessarily infinite mean. To simplify our analysis, we assume that the variables are sampled from a Pareto distribution with parameter $\alpha\in(0,1)$. We start by characterizing the asymptotic distributions of the typical degrees and some related observables. In particular, we show that the degree of a vertex converges in distribution, after proper scaling, to a mixed Poisson law. We then show that correlations among degrees of different vertices are asymptotically non-vanishing, but at the same time a form of asymptotic tail independence is found when looking at the behavior of the joint Laplace transform around zero. Moreover, we present some findings concerning the asymptotic density of wedges and triangles, and show a cross-over for the existence of dust (i.e. disconnected vertices).

In this paper we investigate large-scale linear systems driven by a fractional Brownian motion (fBm) with Hurst parameter $H\in [1/2, 1)$. We interpret these equations either in the sense of Young ($H>1/2$) or Stratonovich ($H=1/2$). In particular, fractional Young differential equations are well suited to modeling real-world phenomena as they capture memory effects, unlike other frameworks. Although it is very complex to solve them in high dimensions, model reduction schemes for Young or Stratonovich settings have not yet been much studied. To address this gap, we analyze important features of fundamental solutions associated with the underlying systems. We prove a weak type of semigroup property which is the foundation of studying system Gramians. From the Gramians introduced, a dominant subspace can be identified, which is shown in this paper as well. The difficulty for fractional drivers with $H>1/2$ is that there is no link between the corresponding Gramians and algebraic equations, making the computation very difficult. Therefore we further propose empirical Gramians that can be learned from simulation data. Subsequently, we introduce projection-based reduced-order models using the dominant subspace information. We point out that such projections are not always optimal for Stratonovich equations, as stability might not be preserved and since the error might be larger than expected. Therefore an improved reduced-order model is proposed for $H=1/2$. We validate our techniques conducting numerical experiments on some large-scale stochastic differential equations driven by fBm resulting from spatial discretizations of fractional stochastic PDEs. Overall, our study provides useful insights into the applicability and effectiveness of reduced-order methods for stochastic systems with fractional noise, which can potentially aid in the development of more efficient computational strategies for practical applications.

We study backward stochastic difference equations (BS$\Delta$Es) driven by a d-dimensional stochastic process on a lattice, whose increments take only $d+1$ possible values that generate the lattice. Interpreting the driving process as a d-dimensional asset price process, we provide applications to an optimal investment problem and to a market equilibrium analysis, where utility functionals are defined via BS$\Delta$Es.

We study sequential optimal stopping with partial reversibility. The optimal stopping problem is subject to implementation delay, which is random and exponentially distributed. Once the stopping decision is made, the decision maker can, by incurring a cost, call the decision off and restart the stopping problem. The optimization criterion is to maximize the expected present value of the total payoff. We characterize the value function in terms of a Bellman principle for a wide class of payoff functions and potentially multidimensional strong Markov dynamics. We also analyse the case of linear diffusion dynamics and characterize the value function and the optimal decision rule for a wide class of payoff functions.

The generalised random graph contains n vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments, and study the point process $\mathcal{C}_n$ on $\{3,4,\ldots\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set A, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets A at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and longest cycles when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung–Lu model and the Norros–Reittu model.

We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter $\alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the sparse regime $\alpha>\tfrac{1}{2}$, the rank at which these two rankings cease to coincide is $n^{1/(1+8\alpha)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $\alpha = \tfrac{1}{2}$, which corresponds to the connectivity threshold of the model. For $\alpha < \tfrac{1}{2}$, the maximum degree is of order $n - O(n^{\alpha + 1/2})$, whereas for $\alpha \geq \tfrac{1}{2}$, the maximum degree is of order $n^{1/(2\alpha)}$. In the $\alpha < \tfrac{1}{2}$ and $\alpha > \tfrac{1}{2}$ cases, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $\alpha < \tfrac{1}{2}$ and Fréchet for $\alpha > \tfrac{1}{2}$). This refines previous estimates on the maximum degree for $\alpha > \tfrac{1}{2}$ and extends the study of large degrees to the dense regime $\alpha \leq \tfrac{1}{2}$.

We derive the exact asymptotics of $\mathbb{P} {\{\sup\nolimits_{\boldsymbol{t}\in {\mathcal{A}}}X(\boldsymbol{t})>u \}} \textrm{ as}\ u\to\infty,$ for a centered Gaussian field $X({\boldsymbol{t}}),\ {\boldsymbol{t}}\in \mathcal{A}\subset\mathbb{R}^n$, $n>1$ with continuous sample paths almost surely, for which $\arg \max_{\boldsymbol{t}\in {\mathcal{A}}} {\mathrm{Var}}(X(\boldsymbol{t}))$ is a Jordan set with a finite and positive Lebesgue measure of dimension $k\le n$ and its dependence structure is not necessarily locally stationary. Our findings are applied to derive the asymptotics of tail probabilities related to performance tables and chi processes, particularly when the covariance structure is not locally stationary.

We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^{d}$ for every activity $\lambda < ({e}^{L} \hat{C}_{\phi})^{-1}$, where L is the local stability constant and $\hat{C}_{\phi} \,:\!=\, \sup_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - {e}^{-\left\lvert \phi(x, y) \right\rvert} \mathrm{d} y$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle–Penrose bound by a factor of at least ${e}$, where the improvements become larger as the negative parts of the potential become more prominent (i.e. for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (2004 Random Structures & Algorithms 24, 461–479): We show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth–death dynamics) that converges rapidly to the finite-volume Gibbs point process while the effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.

A non-uniqueness phase for infinite clusters is proven for a class of marked random connection models (RCMs) on the d-dimensional hyperbolic space, ${\mathbb{H}^d}$, in a high volume-scaling regime. The approach taken in this paper utilizes the spherical transform on ${\mathbb{H}^d}$ to diagonalize convolution by the adjacency function and the two-point function and bound their $L^2\to L^2$ operator norms. Under some circumstances, this spherical transform approach also provides bounds on the triangle diagram that allows for a derivation of certain mean-field critical exponents. In particular, the results are applied to some Boolean and weight-dependent hyperbolic RCMs. While most of the paper is concerned with the high volume-scaling regime, the existence of the non-uniqueness phase is also proven without this scaling for some RCMs whose resulting graphs are almost surely not locally finite.

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.

In this paper, we investigate a class of McKean–Vlasov stochastic differential equations (SDEs) with Lévy-type perturbations. We first establish the existence and uniqueness theorem for the solutions of the McKean–Vlasov SDEs by utilizing an Eulerlike approximation. Then, under suitable conditions, we demonstrate that the solutions of the McKean–Vlasov SDEs can be approximated by the solutions of the associated averaged McKean–Vlasov SDEs in the sense of mean square convergence. In contrast to existing work, a novel feature of this study is the use of a much weaker condition, locally Lipschitz continuity in the state variables, allowing for possibly superlinearly growing drift, while maintaining linearly growing diffusion and jump coefficients. Therefore, our results apply to a broader class of McKean–Vlasov SDEs.

This paper investigates the asymptotic properties of parameter estimation for the Ewens–Pitman partition with parameters $0\lt\alpha\lt1$ and $\theta\gt-\alpha$. Specifically, we show that the maximum-likelihood estimator (MLE) of $\alpha$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $\alpha$. Our proof relies on a stable martingale central limit theorem, which is of independent interest.

Asymptotic properties of random graph sequences, like the occurrence of a giant component or full connectivity in Erdös–Rényi graphs, are usually derived with very specific choices for the defining parameters. The question arises as to what extent those parameter choices may be perturbed without losing the asymptotic property. For two sequences of graph distributions, asymptotic equivalence (convergence in total variation) and contiguity have been considered by Janson (2010) and others; here we use so-called remote contiguity to show that connectivity properties are preserved in more heavily perturbed Erdös–Rényi graphs. The techniques we demonstrate here with random graphs also extend to general asymptotic properties, e.g. in more complex large-graph limits, scaling limits, large-sample limits, etc.

The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenization problems for this type of functional.

Exchangeable partitions of the integers and their corresponding mass partitions on $\mathcal{P}_{\infty} = \{\mathbf{s} = (s_{1},s_{2},\ldots)\colon s_{1} \geq s_{2} \geq \cdots \geq 0$ and $\sum_{k=1}^{\infty}s_{k} = 1\}$ play a vital role in combinatorial stochastic processes and their applications. In this work, we continue our focus on the class of Gibbs partitions of the integers and the corresponding stable Poisson–Kingman-distributed mass partitions generated by the normalized jumps of a stable subordinator with an index $\alpha \in (0,1)$, subject to further mixing. This remarkable class of infinitely exchangeable random partitions is characterized by probabilities that have Gibbs (product) form. These partitions have practical applications in combinatorial stochastic processes, random tree/graph growth models, and Bayesian statistics. The most notable class consists of random partitions generated from the two-parameter Poisson–Dirichlet distribution $\mathrm{PD}(\alpha,\theta)$. While the utility of Gibbs partitions is recognized, there is limited understanding of the broader class. Here, as a continuation of our work, we address this gap by extending the dual coagulation/fragmentation results of Pitman (1999), developed for the Poisson–Dirichlet family, to all Gibbs models and their corresponding Poisson–Kingman mass partitions, creating nested families of Gibbs partitions and mass partitions. We focus primarily on fragmentation operations, identifying which classes correspond to these operations and providing significant calculations for the resulting Gibbs partitions. Furthermore, for completion, we provide definitive results for dual coagulation operations using dependent processes. We demonstrate the applicability of our results by establishing new findings for Brownian excursion partitions, Mittag-Leffler, and size-biased generalized gamma models.

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

The continuous random energy model (CREM) was introduced by Bovier and Kurkova in 2004 as a toy model of disordered systems. Among other things, their work indicates that there exists a critical point $\beta_\mathrm{c}$ such that the partition function exhibits a phase transition. The present work focuses on the high-temperature regime where $\beta<\beta_\mathrm{c}$. We show that, for all $\beta<\beta_\mathrm{c}$ and for all $s>0$, the negative s moment of the CREM partition function is comparable with the expectation of the CREM partition function to the power of $-s$, up to constants that are independent of N.