The distribution theory for discrete-time renewal–reward processes with dependent rewards is developed through the derivation of double transforms. By dependent, we mean the more realistic setting in which the reward for an interarrival period is dependent on the duration of the associated interarrival time. The double transforms are the generating functions in time of the time-dependent reward probability-generating functions. Residue and saddlepoint approximations are used to invert such double transforms so that the reward distribution at arbitrary time n can be accurately approximated. In addition, double transforms are developed for the first-passage time distribution that the cumulative reward exceeds a fixed threshold amount. These distributions are accurately approximated by inverting the double transforms using residue and saddlepoint approximation methods. The residue methods also provide asymptotic expansions for moments and allow for the proof of central limit theorems related to these first passage times and reward amounts.

Novel prediction methods should always be compared to a baseline to determine their performance. Without this frame of reference, the performance score of a model is basically meaningless. What does it mean when a model achieves an F_1$F_1$ of 0.8 on a test set? A proper baseline is, therefore, required to evaluate the ‘goodness’ of a performance score. Comparing results with the latest state-of-the-art model is usually insightful. However, being state-of-the-art is dynamic, as newer models are continuously developed. Contrary to an advanced model, it is also possible to use a simple dummy classifier. However, the latter model could be beaten too easily, making the comparison less valuable. Furthermore, most existing baselines are stochastic and need to be computed repeatedly to get a reliable expected performance, which could be computationally expensive. We present a universal baseline method for all binary classification models, named the Dutch Draw (DD). This approach weighs simple classifiers and determines the best classifier to use as a baseline. Theoretically, we derive the DD baseline for many commonly used evaluation measures and show that in most situations it reduces to (almost) always predicting either zero or one. Summarizing, the DD baseline is general, as it is applicable to any binary classification problem; simple, as it can be quickly determined without training or parameter tuning; and informative, as insightful conclusions can be drawn from the results. The DD baseline serves two purposes. First, it is a robust and universal baseline that enables comparisons across research papers. Second, it provides a sanity check during the prediction model’s development process. When a model does not outperform the DD baseline, it is a major warning sign.

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.

We derive large-sample and other limiting distributions of components of the allele frequency spectrum vector, \mathbf{M}_n$\mathbf{M}_n$, joint with the number of alleles, K_n$K_n$, from a sample of n genes. Models analysed include those constructed from gamma and \alpha$\alpha$-stable subordinators by Kingman (thus including the Ewens model), the two-parameter extension by Pitman and Yor, and a two-parameter version constructed by omitting large jumps from an \alpha$\alpha$-stable subordinator. In each case the limiting distribution of a finite number of components of \mathbf{M}_n$\mathbf{M}_n$ is derived, joint with K_n$K_n$. New results include that in the Poisson–Dirichlet case, \mathbf{M}_n$\mathbf{M}_n$ and K_n$K_n$ are asymptotically independent after centering and norming for K_n$K_n$, and it is notable, especially for statistical applications, that in other cases the limiting distribution of a finite number of components of \mathbf{M}_n$\mathbf{M}_n$, after centering and an unusual n^{\alpha/2}$n^{\alpha/2}$ norming, conditional on that of K_n$K_n$, is normal.

A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

In this note we provide an upper bound for the difference between the value function of a distributionally robust Markov decision problem and the value function of a non-robust Markov decision problem, where the ambiguity set of probability kernels of the distributionally robust Markov decision process is described by a Wasserstein ball around some reference kernel whereas the non-robust Markov decision process behaves according to a fixed probability kernel contained in the ambiguity set. Our derived upper bound for the difference between the value functions is dimension-free and depends linearly on the radius of the Wasserstein ball.

We consider an \mathrm{M}/\mathrm{G}/\infty$\mathrm{M}/\mathrm{G}/\infty$ queue with infinite expected service time. We then provide the transience/recurrence classification of the states (the system is said to be at state n if there are n customers being served), observing also that here (unlike irreducible Markov chains, for example) it is possible for recurrent and transient states to coexist. We also prove a lower bound on the growth speed in the transient case.

We consider continuous-state branching processes (CB processes) which become extinct almost surely. First, we tackle the problem of describing the stationary measures on (0,+\infty)$(0,+\infty)$ for such CB processes. We give a representation of the stationary measure in terms of scale functions of related Lévy processes. Then we prove that the stationary measure can be obtained from the vague limit of the potential measure, and, in the critical case, can also be obtained from the vague limit of a normalized transition probability. Next, we prove some limit theorems for the CB process conditioned on extinction in a near future and on extinction at a fixed time. We obtain non-degenerate limit distributions which are of the size-biased type of the stationary measure in the critical case and of the Yaglom distribution in the subcritical case. Finally we explore some further properties of the limit distributions.

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model contains a crossing of an n\times n$n\times n$ square, how wide is this crossing? The answer depends on whether we consider the critical, sub-, or super-critical regime, and is different for the occupied and vacant sets.

We study a version of the stochastic control problem of minimizing the sum of running and controlling costs, where control opportunities are restricted to independent Poisson arrival times. Under a general setting driven by a general Lévy process, we show the optimality of a periodic barrier strategy, which moves the process upward to the barrier whenever it is observed to be below it. The convergence of the optimal solutions to those in the continuous-observation case is also shown.

We consider a robust optimal investment–reinsurance problem to minimize the goal-reaching probability that the value of the wealth process reaches a low barrier before a high goal for an ambiguity-averse insurer. The insurer invests its surplus in a constrained financial market, where the proportion of borrowed amount to the current wealth level is no more than a given constant, and short-selling is prohibited. We assume that the insurer purchases per-claim reinsurance to transfer its risk exposure to a reinsurer whose premium is computed according to the mean–variance premium principle. Using the stochastic control approach based on the Hamilton–Jacobi–Bellman equation, we derive robust optimal investment–reinsurance strategies and the corresponding value functions. We conclude that the behavior of borrowing typically occurs with a lower wealth level. Finally, numerical examples are given to illustrate our results.

The embedding problem of Markov chains examines whether a stochastic matrix\mathbf{P} $\mathbf{P}$ can arise as the transition matrix from time 0 to time 1 of a continuous-time Markov chain. When the chain is homogeneous, it checks if \mathbf{P}=\exp{\mathbf{Q}}$\mathbf{P}=\exp{\mathbf{Q}}$ for a rate matrix \mathbf{Q}$\mathbf{Q}$ with zero row sums and non-negative off-diagonal elements, called a Markov generator. It is known that a Markov generator may not always exist or be unique. This paper addresses finding \mathbf{Q}$\mathbf{Q}$, assuming that the process has at most one jump per unit time interval, and focuses on the problem of aligning the conditional one-jump transition matrix from time 0 to time 1 with \mathbf{P}$\mathbf{P}$. We derive a formula for this matrix in terms of \mathbf{Q}$\mathbf{Q}$ and establish that for any \mathbf{P}$\mathbf{P}$ with non-zero diagonal entries, a unique \mathbf{Q}$\mathbf{Q}$, called the {\unicode{x1D7D9}}${\unicode{x1D7D9}}$-generator, exists. We compare the {\unicode{x1D7D9}}${\unicode{x1D7D9}}$-generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of \mathbf{P}$\mathbf{P}$ in some practical cases.

We study a signaling game between an employer and a potential employee, where the employee has private information regarding their production capacity. At the initial stage, the employee communicates a salary claim, after which the true production capacity is gradually revealed to the employer as the unknown drift of a Brownian motion representing the revenues generated by the employee. Subsequently, the employer has the possibility to choose a time to fire the employee in case the estimated production capacity falls short of the salary. In this setup, we use filtering and optimal stopping theory to derive an equilibrium in which the employee provides a randomized salary claim and the employer uses a threshold strategy in terms of the conditional probability for the high production capacity. The analysis is robust in the sense that various extensions of the basic model can be solved using the same methodology, including cases with positive firing costs, incomplete information about an individual’s own type, as well as an additional interview phase.

We propose a novel and unified sampling scheme, called the accelerated group sequential sampling scheme, which incorporates four different types of sampling scheme: (i) the classic Anscombe–Chow–Robbins purely sequential sampling scheme; (ii) the accelerated sequential sampling scheme; (iii) the relatively new k-at-a-time group sequential sampling scheme; and (iv) the new k-at-a-time accelerated group sequential sampling scheme. The first-order and second-order properties of this unified sequential sampling scheme are fully investigated with two illustrations on minimum risk point estimation for the mean of a normal distribution and on bounded variance point estimation for the location parameter of a negative exponential distribution. We also provide extensive Monte Carlo simulation studies and real data analyses for each illustration.

We introduce gradient flow aggregation, a random growth model. Given existing particles \{x_1,\ldots,x_n\} \subset \mathbb{R}^2$\{x_1,\ldots,x_n\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at \infty$\infty$ and flows in direction of the vector field \nabla E$\nabla E$ where E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}$E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}$, 0 < \alpha < \infty$0 < \alpha < \infty$. The case \alpha = 0$\alpha = 0$ refers to the logarithmic energy {-}\sum\log\|x-x_i\|${-}\sum\log\|x-x_i\|$. Particles stop once they are at distance 1 from one of the existing particles, at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten’s method, can be used to deduce sub-ballistic growth for 0 \leq \alpha < 1$0 \leq \alpha < 1$, \text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}$\text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}$. This is optimal when \alpha = 0$\alpha = 0$. The case \alpha = 0$\alpha = 0$ leads to a ‘round’ full-dimensional tree. The larger the value of \alpha$\alpha$, the sparser the tree. Some instances of the higher-dimensional setting are also discussed.

We investigate the tail behavior of the first-passage time for Sinai’s random walk in a random environment. Our method relies on the connection between Sinai’s walk and branching processes with immigration in a random environment, and the analysis on some important quantities of these branching processes such as extinction time, maximum population, and total population.

This paper studies a novel Brownian functional defined as the supremum of a weighted average of the running Brownian range and its running reversal from extrema on the unit interval. We derive the Laplace transform for the squared reciprocal of this functional, which leads to explicit moment expressions that are new to the literature. We show that the proposed Brownian functional can be used to estimate the spot volatility of financial returns based on high-frequency price observations.

We consider the constrained-degree percolation model in a random environment (CDPRE) on the square lattice. In this model, each vertex v has an independent random constraint \kappa_v$\kappa_v$ which takes the value j\in \{0,1,2,3\}$j\in \{0,1,2,3\}$ with probability \rho_j$\rho_j$. The dynamics is as follows: at time t=0$t=0$ all edges are closed; each edge e attempts to open at a random time U(e)\sim \mathrm{U}(0,1]$U(e)\sim \mathrm{U}(0,1]$, independently of all the other edges. It succeeds if at time U(e) both its end vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, this result is meaningful only if the supremum of all values of t for which the expected size of the open cluster of the origin is finite is larger than \frac12$\frac12$. We prove this last fact by showing a sharp phase transition for an intermediate model.

Early investigation of Pólya urns considered drawing balls one at a time. In the last two decades, several authors have considered multiple drawing in each step, but mostly for schemes involving two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a ‘core’ square matrix. We examine cases where the urn is irreducible and demonstrate its relationship to matrix irreducibility for its core matrix, with examples provided. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small, critical, and large index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit with a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

The problem of reservation in a large distributed system is analyzed via a new mathematical model. The target application is car-sharing systems. This model is motivated by the large station-based car-sharing system in France called Autolib’. This system can be described as a closed stochastic network where the nodes are the stations and the customers are the cars. The user can reserve a car and a parking space. We study the evolution of the system when the reservation of parking spaces and cars is effective for all users. The asymptotic behavior of the underlying stochastic network is given when the number N of stations and the fleet size M increase at the same rate. The analysis involves a Markov process on a state space with dimension of order N^2$N^2$. It is quite remarkable that the state process describing the evolution of the stations, whose dimension is of order N, converges in distribution, although not Markov, to a non-homogeneous Markov process. We prove this mean-field convergence. We also prove, using combinatorial arguments, that the mean-field limit has a unique equilibrium measure when the time between reserving and picking up the car is sufficiently small. This result extends the case where only the parking space can be reserved.