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We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph G(N, p). We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime β > 1 and the case when an external magnetic field is present. In both cases, we assume that p = p(N) satisfies p³N → ∞. [ABSTRACT FROM AUTHOR]

The β-Delaunay tessellation in ℝd-1 is a generalization of the classical Poisson-Delaunay tessellation. As a first result of this paper we show that the shape of a weighted typical cell of a βDelaunay tessellation, conditioned on having large volume, is close to the shape of a regular simplex inℝd-1. This generalizes earlier results of Hug and Schneider about the typical (non-weighted) Poisson-Delaunay simplex. Second, the asymptotic behaviour of the volume of weighted typical cells in high-dimensional β-Delaunay tessellation is analysed, as d → ∞. In particular, various high dimensional limit theorems, such as quantitative central limit theorems as well as moderate and large deviation principles, are derived. [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 introduce and study the r -Lah distribution whose definition involves r -Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements. [ABSTRACT FROM AUTHOR]

Let r = r(n) be a sequence of integers such that r ≤ n and let X1, ..., Xr+1 be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on ℝn. Limit theorems for the log-volume and the volume of the random convex hull of X1, ..., Xr+1 are established in high dimensions, that is, as r and n tend to infinity simultaneously. This includes Berry-Esseen-type central limit theorems, log-normal limit theorems, and moderate and large deviations. Also different types of mod-φ convergence are derived. The results heavily depend on the asymptotic growth of r relative to n. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if r = o(n) (respectively, r ~ αn for some 0 < α < 1). [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]

Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k -dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k -dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξ k 's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types A n − 1 and B n . This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. [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]

A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volume of some infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions S1, S2, C1, C2 studied by Biane, Pitman, Yor [Bull. AMS 38 (2001)]. We show that the k-th intrinsic volume of the set of all functions on [0, 1] which have Lipschitz constant bounded by 1 and which vanish at 0 (respectively, which have vanishing integral) is given by ... This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001); Elect. Comm. Probab. 8 (2003)], who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the d-dimensional Brownian motion which is due to Eldan [Elect. J. Probab. 19 (2014)]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process. [ABSTRACT FROM AUTHOR]

In the classical leader election procedure all players toss coins independently and those who get tails leave the game, while those who get heads move to the next round where the procedure is repeated. We investigate a generalizion of this procedure in which the labels (positions) of the players who remain in the game are determined using an integer-valued random walk. We study the asymptotics of some relevant quantities for this model such as: the positions of the persons who remained after n rounds; the total number of rounds until all the persons among 1, 2,..., M leave the game; and the number of players among 1, 2,..., M who survived the first n rounds. Our results lead to some interesting connection with Galton-Watson branching processes and with the solutions of certain stochasticfixed point equations arising in the context of the stability of point processes under thinning. We describe the set of solutions to these equations and thus provide a characterization of one-dimensional point processes that are stable with respect to thinning by integer-valued random walks. [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]