Let $(A_i)_{i \geq 0}$ be a finite-state irreducible aperiodic Markov chain and f a lattice score function such that the average score is negative and positive scores are possible. Define $S_0\coloneqq 0$ and $S_k\coloneqq \sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first excursion above 0, and $M_n\coloneqq \max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$ the local score, first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of $S^+$ and derive a new approximation for the tail behaviour of $Q_1$ , together with an asymptotic equivalence for the distribution of $M_n$ . Computational methods are explicitly presented in a simple application case. The new approximations are compared with those proposed by Karlin and Dembo (1992) in order to evaluate improvements, both in the simple application case and on the real data examples considered by Karlin and Altschul (1990).

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Hawkes processes have been widely used in many areas, but their probability properties can be quite difficult. In this paper an elementary approach is presented to obtain moments of Hawkes processes and/or the intensity of a number of marked Hawkes processes, in which the detailed outline is given step by step; it works not only for all Markovian Hawkes processes but also for some non-Markovian Hawkes processes. The approach is simpler and more convenient than usual methods such as the Dynkin formula and martingale methods. The method is applied to one-dimensional Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, inhomogeneous Poisson processes, and non-Markovian cases. Several results are obtained which may be useful in studying Hawkes processes and other counting processes. Our proposed method is an extension of the Dynkin formula, which is simple and easy to use.

We study the problem of stopping a Brownian bridge X in order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.

Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

We explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-($2\ell+1$) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.

In the first part of this paper we study approximations of trajectories of piecewise deterministic processes (PDPs) when the flow is not given explicitly by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for piecewise deterministic Markov processes (PDMPs). These estimates are the building blocks of the multilevel Monte Carlo (MLMC) method, which we study in the second part. The coupling required by the MLMC is based on the thinning procedure. In the third part we apply these results to a two-dimensional Morris–Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms classical Monte Carlo.

This article studies the expected occupancy probabilities on an alphabet. Unlike the standard situation, where observations are assumed to be independent and identically distributed, we assume that they follow a regime-switching Markov chain. For this model, we (1) give finite sample bounds on the expected occupancy probabilities, and (2) provide detailed asymptotics in the case where the underlying distribution is regularly varying. We find that in the regularly varying case the finite sample bounds are rate optimal and have, up to a constant, the same rate of decay as the asymptotic result.

Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

In this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.

Let X = (Xt)t≥0 be a stochastic process issued from $x \in \mathbb{R}$ that admits a marginal stationary measure v, i.e. vPt f = vf for all t ≥ 0, where $\textbf{P}_t\,f(x)= \mathbb{E}_x[f(\textbf{X}_t)]$. In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections.” Also, update first published online date, if available. into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L2 (v)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt)t ≥ 0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov (to appear, Mem. Amer. Math. Soc., 179p), and then time-change by subordinators and their inverses.

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by

$$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

$\unicode[STIX]{x1D6EF}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$\unicode[STIX]{x1D6E5}_{s}$

$Q_{s}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$w_{\unicode[STIX]{x1D6FF}}$

${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$Q_{s}$

$Q_{s}$

${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$

$J_{s}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and the box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad and the Assouad dimension. For common models of random fractal sets, we obtain a dichotomy of its behaviour by finding a threshold function where the quasi-Assouad behaviour transitions to the Assouad dimension. This threshold can be considered a phase transition, and we compute the threshold for the Gromov boundary of Galton–Watson trees and one-variable random self-similar and self-affine constructions. We describe how the stochastically self-similar model can be derived from the Galton–Watson tree result.

We consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling.

For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases.

We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution.

We also mention several open questions.

We study the distribution of the negative Wiener–Hopf factor for a class of two-sided jump Lévy processes whose positive jumps have a rational Laplace transform. The positive Wiener–Hopf factor for this class of processes was studied by Lewis and Mordecki (2008). Here we obtain a formula for the Laplace transform of the negative Wiener–Hopf factor, as well as an explicit expression for its probability density in terms of sums of convolutions of known functions. Under additional regularity conditions on the Lévy measure of the studied processes, we also provide asymptotic results as $u\to-\infty$ for the distribution function F(u) of the negative Wiener–Hopf factor. We illustrate our results in some particular examples.

In a recent article, Döring et al. (2018) conditioned a stable process with two-sided jumps to avoid an interval. As usual, the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found by Döring et al. does not exist or is not invariant, but nonetheless we will characterise the conditioned process as a Markov process.

Parrondo’s coin-tossing games were introduced as a toy model of the flashing Brownian ratchet in statistical physics but have emerged as a paradigm for a much broader phenomenon that occurs if there is a reversal in direction in some system parameter when two similar dynamics are combined. Our focus here, however, is on the original Parrondo games, usually labeled A and B. We show that if the parameters of the games are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games A and B are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but can also be arbitrarily close to 1 (i.e. 100%).

We find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.