We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

Extended gamma processes have been seen as a flexible extension of standard gamma processes in the recent reliability literature, for the purpose of cumulative deterioration modeling. The probabilistic properties of the standard gamma process have been well explored since the 1970s, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and ageing properties for the corresponding level-crossing times are now well understood. The aim of this paper is to explore similar properties for extended gamma processes and see which ones can be broadened to this new context. As a by-product, new stochastic comparisons for convolutions of gamma random variables are also obtained.

We study the tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Comparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all $p\geq1$. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

The objective of this study was to analyse the dynamics of spatial dispersion of the coronavirus disease 2019 (COVID-19) in Brazil by correlating them to socioeconomic indicators. This is an ecological study of COVID-19 cases and deaths between 26 February and 31 July 2020. All Brazilian counties were used as units of analysis. The incidence, mortality, Bayesian incidence and mortality rates, global and local Moran indices were calculated. A geographic weighted regression analysis was conducted to assess the relationship between incidence and mortality due to COVID-19 and socioeconomic indicators (independent variables). There were confirmed 2 662 485 cases of COVID-19 reported in Brazil from February to July 2020 with higher rates of incidence in the north and northeast. The Moran global index of incidence rate (0.50, P = 0.01) and mortality (0.45 with P = 0.01) indicate a positive spatial autocorrelation with high standards in the north, northeast and in the largest urban centres between cities in the southeast region. In the same period, there were 92 475 deaths from COVID-19, with higher mortality rates in the northern states of Brazil, mainly Amazonas, Pará and Amapá. The results show that there is a geospatial correlation of COVID-19 in large urban centres and regions with the lowest human development index in the country. In the geographic weighted regression, it was possible to identify that the percentage of people living in residences with density higher than 2 per dormitory, the municipality human development index (MHDI) and the social vulnerability index were the indicators that most contributed to explaining incidence, social development index and the municipality human development index contributed the most to the mortality model. We hope that the findings will contribute to reorienting public health responses to combat COVID-19 in Brazil, the new epicentre of the disease in South America, as well as in other countries that have similar epidemiological and health characteristics to those in Brazil.

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes $S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Motivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

We consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.

We study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at the jump times of an independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as an impulse-type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and illustrate the results with different well-known cost and diffusion structures.

We give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

We are interested in the property of coming down from infinity of continuous-state branching processes with competition in a Lévy environment. We first study the event of extinction for such a family of processes under Grey’s condition. Moreover, if we add an integrability condition on the competition mechanism then the process comes down from infinity regardless of the long-time behaviour of the environment.

A fever clinic within a hospital plays a vital role in pandemic control because it serves as an outpost for pandemic discovery, monitoring and handling. As the outbreak of coronavirus disease 2019 (COVID-19) in Wuhan was gradually brought under control, the fever clinic in the West Campus of Wuhan Union Hospital introduced a new model for construction and management of temporary mobile isolation wards. A traditional battlefield hospital model was combined with pandemic control regulations, to build a complex of mobile isolation wards that used adaptive design and construction for medical operational, medical waste management and water drainage systems. The mobile isolation wards allowed for the sharing of medical resources with the fever clinic. This increased the capacity and efficiency of receiving, screening, triaging and isolation and observation of patients with fever. The innovative mobile isolation wards also controlled new sudden outbreaks of COVID-19. We document the adaptive design and construction model of the novel complex of mobile isolation wards and explain its characteristics, functions and use.

As most children infected with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) present with mild symptoms or they are asymptomatic, the optimal strategy for molecular testing it is not well defined. The aim of the study was to determine the extent and aetiology of molecular testing for SARS-CoV-2 in Greek paediatric departments during the first phase of the pandemic and identify possible differences in incidence, depending on the age group and geographical area. We conducted a nationwide study of molecular testing for SARS-CoV-2 of children in paediatric departments between March and June 2020. A total of 65 paediatric departments participated in the study, representing 4901 children who were tested for SARS-CoV-2 and 90 (1.8%) were positive. Most paediatric cases were associated with topical outbreaks. Adolescents 11–16 years had the highest positivity rate (3.6%) followed by children 6–10 years (1.9%). However, since the testing rate significantly differed between age groups, the modified incidence of SARS-CoV-2 infection per age group was highest in infants <1 year (19.25/105 population). Most children tested presented with fever (70.9%), respiratory (50.1%) or gastrointestinal symptoms (28.1%). Significant differences were detected between public and private hospitals regarding the positivity rate (2.34% vs. 0.39%, P-value <0.001). Significant variation in SARS-CoV-2 molecular testing positivity rate and incidence between age groups indicate discrepancies in risk factors among different age groups that shall be considered when ordering molecular testing.

The Ethiopian government has several initiatives to expand and intensify the dairy industry; however, the risk of bovine tuberculosis (bTB) spread is a challenge. To assess the rate of expansion and risk factors for transmission of bTB within-herds, we carried out a repeated cross-sectional survey at two time points, 2016/17 and 2018, in three regional cities, namely, Gondar, Hawassa and Mekelle, representing the emerging dairy belts of Ethiopia. The total number of herds involved was 128, comprising an average of 2303 cattle in each round. The Single Intradermal Comparative Cervical Tuberculin (SICCT) test was used to identify reactor status and data on herd-level risk factors were collected using a structured questionnaire. In the first survey, the apparent prevalence of bTB, as measured by the SICCT test, was 4.5% (95% CI 3.7–5.4%) at the individual animal-level and 24% (95% CI 17.5–32%) at the herd-level. There was no statistically significant change in the overall apparent prevalence or regional distribution at the second survey, consistent with the infection being endemic. The incidence rate was estimated at 3.6 (95% CI 2.8–4.5) and 6.6 (95% CI 3.0–12.6) cases/100 cattle (or herd)-years at the animal- and herd-levels, respectively. Risk factors significantly associated with the within-herd transmission of bTB were age group and within-herd apparent prevalence at the start of the observation period. We noted that farmers voluntarily took steps to remove reactor cattle from their herds as a consequence of the information shared after the first survey. Removal of reactors between surveys was associated with a reduced risk of transmission within these herds. However, with no regulatory barriers to the sale of reactor animals, such actions could potentially lead to further spread between herds. We therefore advocate the importance of setting up regulations and then establishing a systematic bTB surveillance programme to monitor the impact prior to implementing any control measures in Ethiopia.

Researchers often trim observations with small values of the denominator A when they estimate moments of the form $\mathbb {E}[B/A]$. Large trimming is common in practice to reduce variance, but it incurs a large bias. This paper provides a novel method of correcting the large trimming bias. If a researcher is willing to assume that the joint distribution between A and B is smooth, then the trimming bias may be estimated well. Along with the proposed bias correction method, we also develop an inference method. Practical advantages of the proposed method are demonstrated through simulation studies, where the data generating process entails a heavy-tailed distribution of $B/A$. Applying the proposed method to the Compustat database, we analyze the history of external financial dependence of U.S. manufacturing firms for years 2000–2010.

Under the classical long-span asymptotic framework, we develop a class of generalized laplace (GL) inference methods for the change-point dates in a linear time series regression model with multiple structural changes analyzed in, e.g., Bai and Perron (1998, Econometrica 66, 47–78). The GL estimator is defined by an integration rather than optimization-based method and relies on the LS criterion function. It is interpreted as a classical (non-Bayesian) estimator, and the inference methods proposed retain a frequentist interpretation. This approach provides a better approximation about the uncertainty in the data of the change-points relative to existing methods. On the theoretical side, depending on some input (smoothing) parameter, the class of GL estimators exhibits a dual limiting distribution, namely the classical shrinkage asymptotic distribution or a Bayes-type asymptotic distribution. We propose an inference method based on highest density regions using the latter distribution. We show that it has attractive theoretical properties not shared by the other popular alternatives, i.e., it is bet-proof. Simulations confirm that these theoretical properties translate to good finite-sample performance.