We present a short probabilistic proof of weak convergence result for the number cuts needed to isolate root random recursive tree. The is based on coupling related certain walk.

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent copies of an $\mathbb{R}^2$-valued random vector $(\xi, \eta)$ with arbitrarily dependent components. Put $T_n:= \xi_1+\ldots+\xi_{n-1} + \eta_n $ for $n\in\mathbb{N}$ and define $\tau(t) := \inf\{n\geq 1: T_n>t\}$ the first passage time into $(t,\infty)$, $N(t) :=\sum_{n\geq 1}1_{\{T_n\leq t\}}$ number visits to $(-\infty, t]$ $\rho(t):=\sup\{n\geq T_n \leq t\}$ associated last exit $t\in\mathbb{R}$. The standing assumption...

A classical fact of the theory almost periodic functions is existence their asymptotic distributions. In probabilistic terms, this means that if $f$ a Besicovitch function and $V$ random variable uniformly distributed on $[-1,1]$, then variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to proper non-degenerate variable. We prove functional extension result for processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ space functions, also sense weak convergence finite-dimensional further...

Buraczewski et al. (2023) proved a functional limit theorem (FLT) and law of the iterated logarithm (LIL) for random Dirichlet series ${\textstyle\sum _{k\ge 2}}\frac{{(\log k)^{\alpha }}}{{k^{1/2+s}}}{\eta _{k}}$ as $s\to 0+$, where $\alpha \gt -1/2$ ${\eta _{1}},{\eta _{2}},\dots $ are independent identically distributed variables with zero mean finite variance. A FLT LIL in boundary case =-1/2$. The is more demanding technically than -1/2$. _{p}}\frac{{\eta _{p}}}{{p^{1/2+s}}}$ sum taken...

Infinite sums of i.i.d. random variables discounted by a multiplicative walk are called perpetuities and have been studied many authors. The present paper provides log-type moment result for such under minimal conditions which is then utilized the study related moments a.s. limits certain martingales associated with supercritical branching walk. connection arises upon consideration size-biased version originally introduced Lyons. As by-product, necessary sufficient uniform integrability...

Let S 0 := and k ξ 1 + ··· for ∈ ℕ {1, 2, …}, where { : ℕ} are independent copies of a random variable with values in distribution p P{ = }, ℕ. We interpret the walk 0, 1, …} as particle jumping to right through integer positions. Fix n modify process by requiring that is bumped back its current state each time jump would bring larger than or equal . This constraint defines an increasing Markov chain R ( ) which never reaches call this barrier M denote number jumps paper focuses on...

Abstract Let X n be the number of cuts needed to isolate root in a random recursive tree with vertices. We provide weak convergence result for . The basic observation its proof is that probability distributions $\{X_n: n=2,3,\ldots\}$ are recursively defined by $X_n {{\rm d}\atop{=}} X_{n-D_n} + 1, \; n=2,3,\ldots, X_1=0$ , where D discrete variable ℙ ${\{ D_n = k \} {1 \over {k(k+1)}} {n {n-1}}, k=1,2,\ldots, n-1}$ which independent $(X_2,\ldots, X_n)$ This distributional recursion was not...

We consider an occupancy scheme in which “balls” are identified with n points sampled from the standard exponential distribution, while role of “boxes” is played by spacings induced independent random walk positive and nonlattice steps. discuss asymptotic behavior five quantities: index Kn* last occupied box, number Kn boxes, Kn, 0 empty boxes whose at most Kn*, Wn first box balls Zn box. It shown that limiting distribution properly scaled centered coincides renewals not exceeding logn. A...

We study the number of collisions, X n , an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts particles and is driven by rates determined a finite characteristic measure η(d x ) = −2 Λ(d ). Via coupling technique, we derive limiting laws using previous results on regenerative compositions derived from stick-breaking partitions unit interval. The possible include normal, stable index 1 ≤ α < 2, Mittag-Leffler distributions. apply, in particular, to case when η...

Let $(X_{1},\xi_{1}),(X_{2},\xi_{2}),\ldots$ be i.i.d. copies of a pair $(X,\xi)$ where $X$ is random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ positive variable. Define $S_{k}:=\xi_{1}+\cdots+\xi_{k}$, $k\in\mathbb{N}_{0}$ $Y(t):=\sum_{k\geq0}X_{k+1}(t-S_{k})\mathbf{1}_{\{S_{k}\leq t\}}$, $t\geq0$. We call $(Y(t))_{t\geq0}$ immigration at epochs renewal process. investigate weak convergence finite-dimensional distributions $(Y(ut))_{u>0}$ as $t\to\infty$. Under...

Consider a random trigonometric polynomial $X_n: \mathbb{R} \to $ of the form \[ X_n(t) = \sum _{k=1}^n \left ( \xi _k \sin (kt) + \eta \cos (kt)\right ), \] where $(\xi _1,\eta _1),(\xi _2,\eta _2),\ldots are independent identically distributed bivariate real vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in \mathbb{N} }$ be any sequence numbers. We prove that as $n\to \infty $, number zeros $X_n$ in interval $[s_n+a/n, s_n+ b/n]$ converges distribution to $[a,b]$...

The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies infinitely many boxes are produced by multiplicative walk, also known as residual allocation model or stick-breaking. We give an overview limit theorems concerning number occupied some balls out first $n$ thrown, and present new results empty within range.

We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, beta( , b )-coalescents > 1. show that some large-sample properties these can be derived by coupling coalescent an increasing Lévy process (subordinator), and exploiting parallels theory regenerative composition structures. In particular, we discuss limit distributions absorption time number collisions.