Abstract It has been shown by Pronzato et al. (Bernoulli 23(4A):2617–2642, 2017; J Multivar Anal 168:276–289, 2018) that simplicial volumes formed by independent copies of random variables can be used to extend the definition of generalised variances. It is shown in this paper that exterior algebra is a natural environment in which to study these constructions. This is used to extend the formulation to covariances and correlations. The theory leads naturally to dispersion ordering, that is partial orderings in which one random variable is more disperse than another if one squared simplicial volume stochastically dominates the other.

Abstract We consider global optimization problems, where the feasible region $${\mathcal {X}}$$ X is a compact subset of $$\mathbb {R}^d$$ R d with $$d \ge 10$$ d ≥ 10 . For these problems, we demonstrate that the actual convergence of global random search algorithms is much slower than that given by the classical estimates, based on the asymptotic properties of random points, and that the usually recommended space exploration schemes are inefficient in the non-asymptotic regime. Moreover, we show that uniform sampling on entire $${\mathcal {X}}$$ X is much less efficient than uniform sampling on a suitable subset of $${\mathcal {X}}$$ X , and that the effect of replacement of random points by low-discrepancy sequences can be felt in small dimensions only.

Abstract Let $${\mathbb {Z}}_n = \{Z_1, \ldots , Z_n\}$$ Zn={Z1,…,Zn} be a design; that is, a collection ofnpoints $$Z_j \in [-1,1]^d$$ Zj∈[-1,1]d . We study the quality of quantisation of $$[-1,1]^d$$ [-1,1]d by the points of $${\mathbb {Z}}_n$$ Zn and the problem of quality of coverage of $$[-1,1]^d$$ [-1,1]d by $${{{\mathcal {B}}}}_d({\mathbb {Z}}_n,r)$$ Bd(Zn,r) , the union of balls centred at $$Z_j \in {\mathbb {Z}}_n$$ Zj∈Zn . We concentrate on the cases where the dimensiondis not small, $$d\ge 5$$ d≥5 , andnis not too large, $$n\le 2^d$$ n≤2d . We define the design $${{\mathbb {D}}_{n,\delta }}$$ Dn,δ as a $$2^{d-1}$$ 2d-1 design defined on vertices of the cube $$[-\delta ,\delta ]^d$$ [-δ,δ]d , $$0\le \delta \le 1$$ 0≤δ≤1 . For this design, we derive a closed-form expression for the quantisation error and very accurate approximations for the coverage area $${\text {vol}}{([-1,1]^d \cap {{{\mathcal {B}}}}_d({\mathbb {Z}}_n,r))}$$ vol([-1,1]d∩Bd(Zn,r)) . We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs $${{\mathbb {D}}_{n,\delta }}$$ Dn,δ .

Abstract We study properties of two probability distributions defined on the infinite set $$\{0,1,2, \ldots \}$$ {0,1,2,…} and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses $$1/\textcircled {1}$$ 1/1 to all points in the set $$ \{0,1,\ldots ,\textcircled {1}-1\}$$ {0,1,…,1-1} , where $$\textcircled {1}$$ 1 denotes the grossone. For this distribution, we study the problem of decomposing a random variable $$\xi $$ ξ with this distribution as a sum $$\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m$$ ξ=dξ1+⋯+ξm , where $$\xi _1 , \ldots , \xi _m$$ ξ1,…,ξm are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin $$(\textcircled {1},p)$$ (1,p) with $$p=c/\textcircled {1}^{\alpha }$$ p=c/1α with $$1/2<\alpha \le 1$$ 1/2<α≤1 . The accuracy of this approximation is assessed using a numerical study.

This article presents a new approach to blind equalization of an FIR channel. It is based on a bounded-error assumption and takes into account the fact that the input signal is in a finite alphabet. We show that even in the noisy case, identifiability can be guaranteed in finite time, provided that the support of the noise density is suitably bounded.