arXiv subjects

Martynas Manstavičius

Publications and source records attributed to Martynas Manstavičius.

2 recordsLinked to original sources

arXiv2017

Randomly stopped maximum and maximum of sums with consistently varying distributions

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables, and $η$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=ξ_1+ξ_2+\cdots+ξ_n$ for $n\geqslant1$. We consider conditions for random variables $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution functions of the random maximum $ξ_{(η)}:=\max\{0,ξ_1,ξ_2,\ldots,ξ_η\}$ and of the random maximum of sums $S_{(η)}:=\max\{S_0,S_1,S_2,\ldots,S_η\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{ξ_1,ξ_2,\ldots\}$ are not necessarily identically distributed.

math.PR↗

arXiv2006

A note about Khoshnevisan--Xiao conjecture

Khoshnevisan and Xiao showed in [Ann. Probab. 33 (2005) 841--878] that the statement about almost surely vanishing Bessel--Riesz capacity of the image of a Borel set $G\subset\mathbb{R}_+$ under a symmetric Lévy process $X$ in $\mathbb{R}^d$ is equivalent to the vanishing of a deterministic $f$-capacity for a particular function $f$ defined in terms of the characteristic exponent of $X$. The authors conjectured that a similar statement is true for all Lévy processes in $\mathbb{R}^d$. We show that the conjecture is true provided we extend the definition of $f$ and require certain integrability conditions which cannot be avoided in general.

math.PR↗