The aim of this paper is to present some ideas how to relax the notion of the optimal solution of the stochastic optimization problem. In the deterministic case, $\epsilon$-minimal solutions and level-minimal solutions are considered as desired relaxations. We call them approximative solutions and we introduce some possibilities how to combine them with randomness. Relations among random versions of approximative solutions and their consistency are presented in this paper. No measurability is assumed, therefore,...

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T(X₁,...,Xₙ)=n^{-1}{\sum }_{1\le iLet{q}_i\left(i\ge 1\right)beeigenvaluesoftheHilbert-Schmidtoperatorassociatedwiththekernelh(x,y),andq₁bethelargestinabsolutevalueone.Weprovethat}$Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12)$,$where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and ${\beta }^{\text{'}}:=E\left|h(X,X₁)\right|³+{E\left|h(X,X₁)\right|}^{18/5}+E\left|g\left(X\right)\right|³+{E\left|g\left(X\right)\right|}^{18/5}+1<\infty$.

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $𝛬$-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty$. Some of our results hold in the case of a general $𝛬$-coalescent...

We consider the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions. A diagnostic procedure for assessing homogeneity of the conditional densities is also proposed.

Limiting laws, as t→∞, for brownian motion penalised by the longest length of excursions up to t, or up to the last zero before t, or again, up to the first zero after t, are shown to exist, and are characterized.

Let $(X_n,n\ge 1),({\tilde{X}}_n,n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${ℝ}^k$ and $S_n=X_1+\cdots +X_n$, ${\tilde{S}}_n={\tilde{X}}_1+\cdots +{\tilde{X}}_n$, $n\ge 1$. Assuming that $E{X}_1=E{\tilde{X}}_1$, $E|X_1{|}^2<\infty$, $E|{\tilde{X}}_1{|}^{k+2}<\infty$ and the existence of a density of ${\tilde{X}}_1$ satisfying the certain conditions we prove the following inequalities: $$v(S_n,{\tilde{S}}_n)\le cmax\{v(X_1,{\tilde{X}}_1),{\zeta }_2(X_1,{\tilde{X}}_1)\},n=1,2,\cdots ,$$ where $v$ and ${\zeta }_2$ are the total variation and Zolotarev’s metrics, respectively.

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise $\varphi$-mixing random variables, and the Baum-Katz-type result for arrays of rowwise $\varphi$-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\varphi$-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).

In this paper we prove two convergence theorems for set-valued conditional expectations. The first is a set-valued generalization of Levy’s martingale convergence theorem, while the second involves a nonmonotone sequence of sub $\sigma$-fields.

Branching process approximation to the initial stages of an epidemic process has been used since the 1950's as a technique for providing stochastic counterparts to deterministic epidemic threshold theorems. One way of describing the approximation is to construct both branching and epidemic processes on the same probability space, in such a way that their paths coincide for as long as possible. In this paper, it is shown, in the context of a Markovian model of parasitic infection, that coincidence...