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We study random convex cones defined as positive hulls of d -dimensional random walks and bridges. We compute expectations of various geometric functionals of these cones such as the number of k -dimensional faces and the sums of conic quermassintegrals of their k -dimensional faces. These expectations are expressed in terms of Stirling numbers of both kinds and their B -analogues. [ABSTRACT FROM AUTHOR]

Let (B (t)) t ∈ Θ with Θ = Z or Θ = R be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in span ¯ { B (s) , s ≤ t } providing the best possible mean square approximation to the variable B (τ) with τ > t. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process B by a stationary differentiable process X taking into account the kinetic energy that X spends in its approximation efforts. [ABSTRACT FROM AUTHOR]

We prove weak convergence on the Skorokhod space of Galton–Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes (1979). [ABSTRACT FROM AUTHOR]

We revisit conservative/dissipative and positive/null decompositions of stationary max-stable processes. Originally, both decompositions were defined in an abstract way based on the underlying non-singular flow representation. We provide simple criteria which allow to tell whether a given spectral function belongs to the conservative/dissipative or positive/null part of the de Haan spectral representation. Specifically, we prove that a spectral function is null-recurrent iff it converges to 0 in the Cesàro sense. For processes with locally bounded sample paths we show that a spectral function is dissipative iff it converges to 0 . Surprisingly, for such processes a spectral function is integrable a.s. iff it converges to 0 a.s. Based on these results, we provide new criteria for ergodicity, mixing, and existence of a mixed moving maximum representation of a stationary max-stable process in terms of its spectral functions. In particular, we study a decomposition of max-stable processes which characterizes the mixing property. [ABSTRACT FROM AUTHOR]

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of an inverse stable subordinator, also known as a Mittag-Leffler process, and a power function. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean ‘inter-shot’ distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times. [ABSTRACT FROM AUTHOR]

We study stationary max-stable processes { η ( t ) : t ∈ R } admitting a representation of the form η ( t ) = max i ∈ N ( U i + Y i ( t ) ) , where ∑ i = 1 ∞ δ U i is a Poisson point process on R with intensity e − u d u , and Y 1 , Y 2 , … are i.i.d. copies of a process { Y ( t ) : t ∈ R } obtained by running a Lévy process for positive t and a dual Lévy process for negative t . We give a general construction of such Lévy–Brown–Resnick processes, where the restrictions of Y to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form M n ( t ) = max i = 1 , … , n ξ i ( s n + t ) , where ξ 1 , ξ 2 , … are i.i.d. Lévy processes and s n is a sequence such that s n ∼ c log n with c > 0 . Also, we consider maxima of the form max i = 1 , … , n Z i ( t / log n ) , where Z 1 , Z 2 , … are i.i.d. Ornstein–Uhlenbeck processes driven by an α -stable noise with skewness parameter β = − 1 . After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick (1977) to the totally skewed α -stable case. [ABSTRACT FROM AUTHOR]

Abstract: Let be independent identically distributed (i.i.d.) random variables with , . Suppose that for all and some . Let and . We are interested in the limiting distribution of the multiscale scan statistic We prove that for an appropriate normalizing sequence , the random variable converges to the Gumbel extreme-value law . The behavior of depends strongly on the distribution of the ’s. We distinguish between four cases. In the superlogarithmic case we assume that for every . In this case, we show that the main contribution to comes from the intervals having length of order , , where and is the order of the first non-vanishing cumulant of (not counting the variance). In the logarithmic case we assume that the function attains its maximum at some unique point . In this case, we show that the main contribution to comes from the intervals of length , , where . In the sublogarithmic case we assume that the tail of is heavier than , for some . In this case, the main contribution to comes from the intervals of length and in fact, under regularity assumptions, from the intervals of length 1. In the remaining, fourth case, the ’s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length , . We argue that our results cover most interesting distributions with light tails. The proofs are based on the precise asymptotic estimates for large and moderate deviation probabilities for sums of i.i.d. random variables due to Cramér, Bahadur, Ranga Rao, Petrov and others, and a careful extreme value analysis of the random field of standardized increments by the double sum method. [Copyright &y& Elsevier]