We consider random rectangles in $\mathbb{R}^2$ that are distributed according to a Poisson random measure, i.e. independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields, and give statistical properties of these limits.

We study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$. For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example.

The accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

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.

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.

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.

Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability ϕ(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

In this paper we consider the optimal scaling of high-dimensional random walk Metropolis algorithms for densities differentiable in the L p mean but which may be irregular at some points (such as the Laplace density, for example) and/or supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to ∞. As the log-density might be nondifferentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of Roberts et al. (1997). This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

The contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

We derive some limit theorems associated with the Ewens sampling formula when its parameter is increasing together with a sample size. Moreover, the limit results are applied in order to investigate asymptotic properties of the maximum likelihood estimator.

We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

An urn contains black and red balls. Let Z n be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball b n is drawn. If b n is black and Z n-1<U, then b n is replaced together with a random number B n of black balls. If b n is red and Z n-1>L, then b n is replaced together with a random number R n of red balls. Otherwise, no additional balls are added, and b n alone is replaced. In this paper we assume that R n =B n . Then, under mild conditions, it is shown that Z n →a.s. Z for some random variable Z, and D n ≔√n(Z n -Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(D n ∈⋅|𝒢n )→w 𝒩(0,σ2) a.s., where ℙ(D n ∈⋅|𝒢n ) is a regular version of the conditional distribution of D n given the past 𝒢n . Thus, in particular, one obtains D n →𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.