In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the firs...

Let n points be placed independently in d−dimensional space according to the density f(x) = Ade−λ‖x‖ α , λ > 0, x ∈ d, d ≥ 2. Let dn be the longest edge length for the nearest neighbor graph on these points. We show that (log n)1−1/αdn − bn converges weakly to the Gumbel distribution where bn ∼ log log n. We also show the strong law result, lim n→∞ (λ−1 log n)1−1/α dn √ log log n → d αλ , a.s. ...

We provide a new analysis framework for the adversarial multi-armed bandit problem. Using the notion of convex smoothing, we define a novel family of algorithms with minimax optimal regret guarantees. First, we show that regularization via the Tsallis entropy, which includes EXP3 as a special case, matches the O( √ NT ) minimax regret with a smaller constant factor. Second, we show that a wide ...

The authors present empirical distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of the halting time follow a distribution that, afte...

Abstract The Fairlie-Gumbel-Morgenstern (FGM) family has been investigated in detail for various continuous marginals such as Cauchy, normal, exponential, gamma, Weibull, lognormal and others. It has been a popular model for the bivariate distribution with mild dependence. However, bivariate FGM’s with continuous marginals on a bounded support discussed in the literature are only those with uni...

An interim drainage structure provides a short service during the construction months of the permanent drainage structure. This study presents a risk cost approach by which the design flood for an interim drainage structure can be derived from the capacity of the permanent structure, cost-capacity ratio, cost-damage ratio, and construction months during a year. To accommodate various constructi...

We investigate conditions for the existence of the limiting conditional distribution of a bivariate random vector when one component becomes large. We revisit the existing literature on the topic, and present some new sufficient conditions. We concentrate on the case where the conditioning variable belongs to the maximum domain of attraction of the Gumbel law, and we study geometric conditions ...

In a sample variance decomposition, with components functions of the sample's spacings, the largest component˜In is used in cluster detection. It is shown for normal samples that the asymptotic distribution of˜In is the Gumbel distribution. 1. Introduction. Clusters are nowadays data structures of considerable interest: Microarray data is used to attribute genes in clusters; gene expression is ...

Climate change poses a great risk to natural resources and agriculture in the Luvuvhu River Catchment. The catchment has experienced floods resulting from higher than normal rainfall associated with the Intertropical Convergence Zone and cyclones, which have caused enormous damage to property and impacted negatively on fauna and flora and human livelihoods. In order to understand the dynamics i...

Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certai...