We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

We consider an infinitely divisible random field in ℝd given as an integral of a kernel function with respect to a Lévy basis. Under mild regularity conditions, we derive central limit theorems for the moment estimators of the mean and the variogram of the field.

We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability ${\textstyle{1 \over 2}}$. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having n vertices of degree 3 and α(n) vertices of degree 4, where α(n)/n → 0 as n → ∞, with independent and identically distributed standard exponential edge lengths. Depending on whether $\alpha(n) \gg \sqrt{n}$ or $\alpha(n)=O(\sqrt{n})$, we prove that, as n → ∞, these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. We use our analysis of this model to make progress towards a conjecture of Aronson, Frieze and Pittel (1998) concerning the number of vertices which remain unmatched when we use the Karp–Sipser algorithm to find a matching on the Erdős–Rényi random graph.

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

As an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.

We consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp (f). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance. The setting and proof of the extended Laplace principle are based on the weak convergence approach to large deviations by Dupuis and Ellis (2011).

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

We discuss a rich family of directed series–parallel (SP) graphs grown by the simultaneous random series or parallel development of multiple edges. The family portrays a spectrum that spans a wide range of SP graphs: from simple models, where only as few as one edge is chosen for evolution at each discrete point in time, to complex hierarchical lattice networks grown by a take-all strategy, where all the edges in the existing network are developed.

The family of SP graphs we discuss is grown from an initial seed graph with τ0 edges under an arbitrary building sequence, $\{k_{n}\}_{n=1}^{\infty}$, of nonnegative integers (with $k_n \le \tau _0 + \sum\nolimits_{i = 1}^n {k_i} $, for arbitrary τ0 ≥ 1), that specifies the number of edges subjected to evolution at time n. We study the average north polar degree and show that we can go beyond averages to strong laws. We also find the exact average number of critical edges. The asymptotics of the critical edges are facilitated under the regularity condition that $k_n/\sum\nolimits_{i = 1}^n {k_i} $ converges to a constant (as n → ∞), a natural condition easily met by practical strategies, such as single-edge evolution and take-all choice, and much in between.

$\mathbb{Z}^{d}$-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

Consider a supercritical Crump‒Jagers process in which all births are at integer times (the lattice case). Let μ̂(z) be the generating function of the intensity of the offspring process, and consider the complex roots of μ̂(z)=1. The root of smallest absolute value is e-α=1∕m, where α>0 is the Malthusian parameter; let γ* be the root of second smallest absolute value. Subject to some technical conditions, the second-order fluctuations of the age distribution exhibit one of three types of behaviour: (i) when γ*>e-α∕2=m-1∕2, they are asymptotically normal; (ii) when γ*=e-α∕2, they are still asymptotically normal, but with a larger variance; and (iii) when γ*<e-α∕2, the fluctuations are in general oscillatory and (degenerate cases excluded) do not converge in distribution. This trichotomy is similar to what has been observed in related situations, such as some other branching processes and for Pólya urns. The results lead to a symbolic calculus describing the limits. The asymptotic results also apply to the total of other (random) characteristics of the population.

We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the logarithm of the block-counting process by means of a drifted subordinator.

We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.

The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

The effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certain random initial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.