arXiv subjects

Ildar Ibragimov

Publications and source records attributed to Ildar Ibragimov.

3 recordsLinked to original sources

arXiv2016

Some extensions of linear approximation and prediction problems for stationary processes

Let $(B(t))_{t\in Θ}$ with $Θ={\mathbb Z}$ or $Θ={\mathbb R}$ be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in $\overline{span}\{B(s),s\le t\}$ 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.

math.PR↗

arXiv2011

On random surface area

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \emph{non-random} smooth field.

math.PR↗

arXiv2011

On distribution of zeros of random polynomials in complex plane

Let $G_n(z)=ξ_0+ξ_1z+...+ξ_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2π]$ asymptotically as $n\to\infty$. We also prove that the condition $\E\ln(1+|ξ_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.

math.PR↗