Consider the perpetuity equation $X \stackrel{\mathcal {D}} {=} A X + B$, where $(A,B)$ and $X$ on right-hand side are independent. The Kesten–Grincevičius–Goldie theorem states that if $\mathbf{E} A^\kappa = 1$, \log _+ < \infty $, |B|^\kappa then $\mathbf{P} \{ > x \} \sim c x^{-\kappa }$. Assume |B|^\nu $ for some $\nu \kappa consider two cases (i) $; (ii) A^t all $t $. We show under appropriate additional assumptions $A$ asymptotic } \ell (x) holds, $\ell is a nonconstant slowly varying...

In this paper we generalize some of the classical results Rényi and Sulanke (1963), (1964) in context spindle convexity. A planar convex disc S is if it intersection congruent closed circular discs. The finitely many discs called a polygon. We prove asymptotic formulae for expectation number vertices, missed area, perimeter difference uniform random polygons contained sufficiently smooth disc.

Let $V_{t}$ be a driftless subordinator, and let denote $m_{t}^{(1)} \geq m_{t}^{(2)} \geq\ldots$ its jump sequence on interval $[0,t]$. Put $V_{t}^{(k)} = V_{t} - m_{t}^{(1)} \ldots- m_{t}^{(k)}$ for the $k$-trimmed subordinator. In this note we characterize under what conditions limiting distribution of ratios / m_{t}^{(k+1)}$ $m_{t}^{(k+1)} exist, as $t \downarrow0$ or \to\infty$.

Abstract We extend Goldie's implicit renewal theorem to the arithmetic case, which allows us determine tail behavior of solution various random fixed point equations. It turns out that and nonarithmetic cases are very different. Under appropriate conditions we obtain X equations = D AX + B ∨ is ℓ( x ) q ( -κ , where a logarithmically periodic function e h ), &gt; 0, with being span distribution log A ℓ slowly varying function. In particular, not necessarily regularly varying. use theoretic...

We examine the almost-sure asymptotics of solution to stochastic heat equation driven by a Lévy space-time white noise. When spatial point is fixed and time tends infinity, we show that develops unusually high peaks over short intervals, even in case additive noise, which leads breakdown an intuitively expected strong law large numbers. More precisely, if normalize increasing nonnegative function, either obtain convergence $0$, or limit superior and/or inferior will be infinite. A detailed...

We prove a Tauberian theorem for the Laplace-Stieltjes transform, Karamata-type theorem, and monotone density in framework of regularly log-periodic functions.We provide several applications these results: example, we that tail nonnegative random variable is if only same holds its Laplace transform at 0, determine exact behavior fixed points certain smoothing transforms.

We investigate the moment asymptotics of solution to stochastic heat equation driven by a $(d+1)$-dimensional L\'evy space--time white noise. Unlike case Gaussian noise, typically has no finite moments order $1+2/d$ or higher. Intermittency $p$, that is, exponential growth $p$th as time tends infinity, is established in dimension $d=1$ for all values $p\in(1,3)$, and higher dimensions some $p\in(1,1+2/d)$. The proof relies on new lower bound integrals against compensated Poisson measures....

We consider the self-normalized sums $T_{n}=\sum_{i=1}^{n}X_{i}Y_{i}/\sum_{i=1}^{n}Y_{i}$, where ${Y_{i} : i\geq 1}$ are non-negative i.i.d. random variables, and ${X_{i} 1} $ independent of i \geq 1}$. The main result paper is that each subsequential limit law T_n$ continuous for any non-degenerate $X_1$ with finite expectation, if only $Y_1$ in centered Feller class.

Dedicated to the memory of Sándor CsörgőWe show somewhat unexpectedly that whenever a general Bernstein-type maximal inequality holds for partial sums sequence random variables, form is also valid.

Multitype inhomogeneous Galton--Watson processes with immigration are investigated, where the offspring mean matrix slowly converges to a critical matrix. Under general conditions we obtain limit distribution for process, coordinates of vector not necessarily independent.

The European Centre for Disease Prevention and Control called the attention in March 2012 to risk of measles Ukraine among visitors UEFA Football Championship. Large populations supporters travelled various locations Poland Ukraine, depending on schedule Euro outcome games, possibly carrying disease from one location another. In present study, we propose a novel two-phase multitype branching process model with immigration describe major epidemic connection large-scale sports-related mass...