arXiv subjects
Publications and source records attributed to Edward Omey.
Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for $α\geq1$, $$ \int_0^x y^{α-1}(1-F(y))dy\textrm{ is SVF}\ \Longrightarrow\ \int_{[0,x]}y^αdF(y)\textrm{ is SVF, as } x\to\infty,$$ where $F(x)$ is a cumulative distribution function on $[0,\infty)$. Complementary results related to this transform and particular cases of this extended conjecture are discussed.
The paper deals with fluctuations of Kendall random walks, which are extremal Markov chains and iterated integral transforms with the Williamson kernel $Ψ(t) = \left(1-|t|^α\right)_+$, $α>0$. We obtain the joint distribution of the first ascending ladder epoch and height over any level $a \geq 0$ and distribution of maximum and minimum for these extremal Markovian sequences solving recursive integral equations. We show that distribution of the first crossing time of level $a \geq0$ is a mixture of geometric and negative binomial distributions. The Williamson transform is the main tool for considered problems connected with the Kendall convolution. All results are described by the Williamson transform of the unit step distribution of Kendall random walks. Using regular variation, we investigate the asymptotic properties of the maximum distribution.
The paper provides a new test of convergence and divergence of positive series. In particular, it extends the known test by Margaret Martin [%\emph{Bull. Amer. Math. Soc.} \textbf{47} (6), 452--457 (1941)]. \emph{Bull. Amer. Math. Soc.} \textbf{47}, 452--457 (1941)].
This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected to the Kendall convolution. In particular, based on its stochastic representation, we provide general formula for finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and connected continuous time stochastic process.
There are many tests for determining the convergence or divergence of series. The test of Raabe and the test of Betrand are relatively unknown and do not appear in most classical courses of analysis. Also, the link between these tests and regular variation is seldomly made. In this paper we offer a unified approach to some of the classical tests from a point of view of regular varying sequences.
In this paper, according to a certain criterion, we divide the exponential distribution class into three subclasses. One of them is closely related to the regular-variation-tailed distribution class, so it is called the semi-regular-variation-tailed distribution class. In the class, although all distributions are not convolution equivalent,they still have some good properties. We give the precise tail asymptotic expression of convolution of these distributions, and prove that the new class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.
Recently, new classes of positive and measurable functions, $\mathcal{M}(ρ)$ and $\mathcal{M}(\pm \infty)$, have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., 2015, 2016, 2017). Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type $\displaystyle \lim_{x\rightarrow \infty}\log U(x)/H(x)=ρ<\infty$ for a large class of normalizing functions $H$. It provides subclasses of $\mathcal{M}(0)$ and $\mathcal{M}(\pm\infty)$.
Let $f$ be a measurable, real function defined in a neighbourhood of infinity. The function $f$ is said to be of generalised regular variation if there exist functions $h \not\equiv 0$ and $g > 0$ such that $f(xt) - f(t) = h(x) g(t) + o(g(t))$ as $t \to \infty$ for all $x \in (0, \infty)$. Zooming in on the remainder term $o(g(t))$ leads eventually to a relation of the form $f(xt) - f(t) = h_1(x) g_1(t) + ... + h_n(x) g_n(t) + o(g_n(t))$, each $g_i$ being of smaller order than its predecessor $g_{i-1}$. The function $f$ is said to be generalised regularly varying of order $n$ with rate vector $\g = (g_1, >..., g_n)'$. Under general assumptions, $\g$ itself must be regularly varying in the sense that $\g(xt) = x^{\B} \g(t) + o(g_n(t))$ for some upper triangular matrix $\B \in \RR^{n \times n}$, and the vector of limit functions $\h = (h_1, >..., h_n)$ is of the form $\h(x) = \c \int_1^x u^\B u^{-1} \du$ for some row vector $\c \in \RR^{1 \times n}$. The usual results in the theory of regular variation such as uniform convergence and Potter bounds continue to hold. An interesting special case arises when all the rate functions $g_i$ are slowly varying, yielding $Π$-variation of order $n$, the canonical case being that $\B$ is equivalent to a single Jordan block with zero diagonal. The theory is applied to a long list of special functions.
Many statistics are based on functions of sample moments. Important examples are the sample variance $s_{n-1}^2$, the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central $t$-statistic $t(n)$. The definition of these quantities makes clear that the vector defined by (\sum_{i=1}^nX_i,\sum_{i=1}^nX_i^2) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector $(X,X^2)$ belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For simplicity, in restrict ourselves to positive random variables $X$.