We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha$th power $(\alpha >1)$ of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of $2n-2$ new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower-tail large deviation principle (LDP) for $L = \sum_{i=1}^{n} ({S_i^2}/{i^2})$, where $\{S_n \colon n \geq 0\}$ is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$$(1/4<H\leqslant 1/2)$. Under Hörmander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

The purpose of this paper is to establish some limit theorems of delayed averages for countable nonhomogeneous Markov chains. The definition of the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for countable nonhomogeneous Markov chains is introduced first. Then a theorem about the generalized C-strong ergodicity and the generalized uniformly C-strong ergodicity for the nonhomogeneous Markov chains is established, and its applications to the information theory are given. Finally, the strong law of large numbers of delayed averages of bivariate functions for countable nonhomogeneous Markov chains is proved.

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 consider a model of branching Brownian motion in which the usual spatially homogeneous branching and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.

For a multivariate random walk with independent and identically distributed jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results of Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by Borovkov and Mogulskii from around 2000 with new auxiliary constructions, enabling us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a ‘corner’ at the ‘most probable hitting point’. We also discuss how our results can be extended to the case of more general target sets.

We show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.

In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$ together with a $T$-invariant measure whose base map $\unicode[STIX]{x1D703}$ satisfies certain topological and mixing conditions and the maps $T_{\unicode[STIX]{x1D714}}$ on the fibers are certain non-singular distance-expanding maps. Our results hold true when $\unicode[STIX]{x1D703}$ is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by $T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields162 (2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.

We consider a class of correlated Bernoulli variables, which have the following form: for some 0 < p < 1,

$$\begin{align}{P(X_{j+1}=1 \vert {\cal F}_{j})= (1-\theta_j)p+\theta_jS_j/j,}\end{align}$$

$S_n=\sum _{j=1}^nX_j$

${\cal F}_n=\sigma \{X_1,\ldots , X_n\}$

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

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.

Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

Let (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

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.