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Let (a_k)_{k\in \mathbb N} be an increasing sequence of positive integers satisfying the Hadamard gap condition a_{k+1}/a_k> q >1 for all k\in \mathbb N, and let \begin{equation*} S_n(\omega) = \sum _{k=1}^n \cos (2\pi a_k \omega), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then S_n is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space \Omega = [0,1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (S_n)_{n\in \mathbb {N}} has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition a_{k+1}/a_k \to \infty, we prove that the sequence (S_n/n)_{n \in \mathbb {N}} does indeed satisfy a large deviation principle with speed n and the same rate function \widetilde {I} as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when a_k= q^k for some q\in \{2,3,\ldots \}, (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle (with speed n) and a rate function I_q that is different from \widetilde {I}, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of I_q. In addition, we also prove that I_q converges pointwise to \widetilde I as q\to \infty. Furthermore, we construct a random perturbation (a_k)_{k \in \mathbb {N}} of the sequence (2^k)_{k \in \mathbb {N}} for which a_{k+1}/a_k \to 2 as k\to \infty, but for which at the same time (S_n/n)_{n \in \mathbb {N}} satisfies a large deviation principle with the same rate function \widetilde {I} as in the independent case, which is surprisingly different from the rate function I_2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a_k)_{k\in \mathbb N}. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems. [ABSTRACT FROM AUTHOR]

Let X1, . . ., Xn be a standard normal sample in Rd. We compute exactly the expected volume of the Gaussian polytope conv [X1, . . ., Xn], the symmetric Gaussian polytope conv [±X1, . . .,±Xn], and the Gaussian zonotope [0,X1] + · · · + [0,Xn] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson’s formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including conv [l1X1, . . ., lnXn] and conv[±l1X1, . . .,±lnXn], where l1, . . ., ln ≥ 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Δn−1 to the expected maximum of independent standard Gaussian random variables ξ1, . . ., ξn given that the maximum has multiplicity k. Namely, we show that Vk(Δn−1) = (2π)k/2/k! · lim ε↓0 ε1−kE[max{ξ1, . . ., ξn}1{ξ(n) −ξ(n−k+1) ≤ε}], where ξ(1) ≤ ·· · ≤ ξ(n) denote the order statistics. A similar result holds for the cross-polytope if we replace ξ1, . . ., ξn with their absolute values. [ABSTRACT FROM AUTHOR]