This paper develops a likelihood for sequences of extremes when observations are dependent in time. The likelihood allows researchers to obtain more realistic standard errors of the generalized extreme‐value parameters. As a motivating example, annual minimum temperatures are examined from the Faraday/Vernadsky research station in Antarctica. Here, the year‐to‐year correlation in the series is about 0.46. Our likelihood allows the series to have temporal correlation but also keeps a generalized extreme‐value marginal distribution at each point in time. An analysis of the Faraday/Vernadsky annual minimum temperatures is conducted. While the standard error of the estimated trend increases when dependence is taken into account, it does not change the correlation‐ignored inference that annual minimum temperatures at the station are increasing.

We study a double-ended queue which consists of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system immediately. The matching follows first-come-first-serve principle. If a customer from one class cannot be matched immediately, he/she will stay in a queue and wait for the upcoming arrivals from the other class. Thus there cannot be non-zero numbers of customers from both classes simultaneously in the system. We also assume that each customer can leave the queue without being matched because of impatience. The arrival processes are assumed to be independent renewal processes, and the patience times for both classes are generally distributed. Under suitable heavy traffic conditions, assuming that the diffusion-scaled queue length process is stochastically bounded, we establish a simple asymptotic relationship between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time processes, and further show that the diffusion-scaled queue length process converges weakly to a diffusion process. We also provide a sufficient condition for the stochastic boundedness of the diffusion-scaled queue length process. At last, the explicit form of the stationary distribution of the limit diffusion process is derived.AMS 2010 subject classifications: Primary: 60F05, 60K25, 90B22; Secondary: 60K05.

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We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.