The range of a simple random walk on ℤ

1996 ◽  
Vol 28 (4) ◽  
pp. 1014-1033 ◽  
Author(s):  
P. Vallois
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Simple Random Walk ◽  
The Difference ◽  
First Time

Let θ (a) be the first time when the range (Rn; n ≧ 0) is equal to a, Rn being equal to the difference of the maximum and the minimum, taken at time n, of a simple random walk on ℤ. We compute the g.f. of θ (a); this allows us to compute the distributions of θ (a) and Rn. We also investigate the asymptotic behaviour of θ (n), n going to infinity.

The range of a simple random walk on ℤ

1996 ◽  
Vol 28 (04) ◽  
pp. 1014-1033 ◽  
Author(s):  
P. Vallois
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Simple Random Walk ◽  
The Difference ◽  
First Time

Letθ(a) be the first time when the range (Rn;n≧ 0) is equal toa, Rnbeing equal to the difference of the maximum and the minimum, taken at timen, of a simple random walk on ℤ. We compute the g.f. ofθ(a); this allows us to compute the distributions ofθ(a) andRn.We also investigate the asymptotic behaviour ofθ(n),ngoing to infinity.

scholarly journals A simple random walk model explains the disruption process of hierarchical, Eccentric three-body systems

2020 ◽  
Vol 498 (1) ◽  
pp. 665-673
Author(s):  
Jonathan Mushkin ◽  
Boaz Katz
Keyword(s):  
Random Walk ◽  
Initial Conditions ◽  
Simple Random Walk ◽  
Survival Times ◽  
Step Size ◽  
Orbital Parameters ◽  
Unstable Set ◽  
Three Body ◽  
First Time ◽  
Comparable Mass

ABSTRACT We study the disruption process of hierarchical three-body systems with bodies of comparable mass. Such systems have long survival times that vary by orders of magnitude depending on the initial conditions. By comparing with three-body numerical integrations, we show that the evolution and disruption of such systems can be statistically described as a simple random walk process in the outer orbit’s energy, where the energy exchange per pericenter passage (step size) is calculated from the initial conditions. In our derivation of the step size, we use previous analytic results for parabolic encounters, and average over the (Kozai–Lidov) oscillations in orbital parameters, which are faster then the energy diffusion time-scale. While similar random walk models were studied before, this work differs in two manners: (a) this is the first time that the Kozai–Lidov averaged step size is derived from first principles and demonstrated to reproduce the statistical evolution of numerical ensembles without fitting parameters, and (b) it provides a characteristic lifetime, instead of answering the binary question (stable/unstable), set by case-specific criteria.

scholarly journals Diameter and Stationary Distribution of Random $r$-Out Digraphs

10.37236/9485 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Louigi Addario-Berry ◽  
Borja Balle ◽  
Guillem Perarnau
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Stationary Distribution ◽  
High Probability ◽  
Branching Processes ◽  
Simple Random Walk ◽  
Undirected Graphs

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $\eta_r>0$ such that $\mathrm{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. The constant $\eta_r$ is related to branching processes and also appears in other models of random undirected graphs. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\pi_{\max}$ and $\pi_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\pi_{\max} = n^{-1+o(1)}$ and $\pi_{\min}=n^{-(1+\eta_r)+o(1)}$. Our proof shows that the vertices with $\pi(v)$ near to $\pi_{\min}$ lie at the top of "narrow, slippery tower"; such vertices are also responsible for increasing the diameter from $(1+o(1))\log_r n$ to $(1+\eta_r+o(1))\log_r{n}$.

A simple random walk and an associated asymptotic behaviour of the Bessel functions

1962 ◽  
Vol 58 (4) ◽  
pp. 708-709 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Bessel Functions ◽  
Simple Random Walk ◽  
Image Position

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .

scholarly journals Risk theory and Wiener processes

1972 ◽  
Vol 7 (1) ◽  
pp. 96-99 ◽  
Author(s):  
H. Bohman
Keyword(s):  
Random Walk ◽  
Risk Process ◽  
Risk Theory ◽  
Practical Applications ◽  
The Difference ◽  
The Individual ◽  
Image Position ◽  
The Right ◽  
Risk Processes ◽  
First Time

We will in this paper consider the risk process from the point of view of random walk in one dimension. The particle starts out at the origin. Each claim is equivalent to a step in the random walk. The length of the step is equal to the amount of the claim minus the amount of the premium which has been obtained since the preceding claim. If the difference is positive the particle advances to the right and if the difference is negative to the left. At distance U to the right from the origin there is a barrier. The problem is to find the distribution function of X, the time it takes the particle to cross the barrier for the first time.In most practical applications of risk theory U is large in comparison to the individual steps of the particle. We will in this paper assume that U is large in comparison to the individual steps and draw certain conclusions about the risk processes from this assumption.The individual steps of the particle have a certain distribution. The corresponding characteristic function is ϕ. For reasons which will be seen later we will consider ϕ to be a function of it = θ instead of t. This means thatThe mean value and the standard deviation of each step is equal to m and σ respectively. We now writeWe now define two random variables X and Y.X = time to cross the barrier for the first timeY = X σ2/U2.

Strategies for Oral Delivery of Metal-Saturated Lactoferrin

2019 ◽  
Vol 20 (11) ◽  
pp. 1046-1051 ◽  
Author(s):  
Przemysław Gajda-Morszewski ◽  
Klaudyna Śpiewak-Wojtyła ◽  
Maria Oszajca ◽  
Małgorzata Brindell
Keyword(s):  
Biological Activity ◽  
Oral Delivery ◽  
Therapeutic Potential ◽  
Structure And Function ◽  
The Difference ◽  
And Function ◽  
First Time

Lactoferrin was isolated and purified for the first time over 50-years ago. Since then, extensive studies on the structure and function of this protein have been performed and the research is still being continued. In this mini-review we focus on presenting recent scientific efforts towards the elucidation of the role and therapeutic potential of lactoferrin saturated with iron(III) or manganese(III) ions. The difference in biological activity of metal-saturated lactoferrin vs. the unmetalated one is emphasized. The strategies for oral delivery of lactoferrin, are also reviewed, with particular attention to the metalated protein.

Nokhri, Ger, and the Art of Separation in the Hebrew Bible

2018 ◽  
Author(s):  
Adi Ophir ◽  
Ishay Rosen-Zvi
Keyword(s):  
Hebrew Bible ◽  
Biblical Literature ◽  
The Difference ◽  
First Time

This chapter traces the developments of various terms denoting “others” in biblical literature. In much of the biblical corpus, Israel is still one goy among many, and the difference between it and its Others is neither binary nor stable. After a brief analysis of the dynamics of familial and ethnwic separations in Genesis and Exodus, this chapter concentrates on the priestly and Deuteronomistic modes of separating peoples, examines the novelty and limitedness of the Deuteronomistic legislation, where the nokhri (stranger) is systematically contrasted for the first time with the Israelite (referred to as “your brother”), and follows the various modes of separations and their rationales.

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

Species diversity and abundance of Dacus (Diptera: Tephritidae) in five ecosystems of Penang, West Malaysia

1982 ◽  
Vol 72 (4) ◽  
pp. 709-716 ◽  
Author(s):  
Tan Keng-Hong ◽  
Lee Soo-Lam
Keyword(s):  
Species Diversity ◽  
Capsicum Annuum ◽  
Fruits And Vegetables ◽  
Methyl Eugenol ◽  
Dacus Dorsalis ◽  
The Difference ◽  
Vegetable Farm ◽  
First Time

AbstractDacus dorsalis Hend. infested eleven, D. cucurbitae Coq. five and D. umbrosus F. two of the eighteen common fruits and vegetables grown in Penang, West Malaysia. D. tau (Wlk.) infested bacang (Mangifera foetida), D. caudatus F. chilli (Capsicum annuum) and D. frauenfeldi Schin. water guava (Eugenia javanica), together with D. dorsalis. Pomelo (Citrus grandis) was found infested for the first time by D. cucurbitae. No flies were trapped using Capilure and trimedlure as baits. Cue-lure attracted D. caudatus, D. cucurbitae, D. frauenfeldi, D. occipitalis (Bez.) and D. tau. Methyl eugenol attracted D. dorsalis and D. umbrosus. Dorsalure was less attractive to D. caudatus and D. dorsalis than cue-lure and methyl eugenol, respectively, but it was equally attractive to D. frauenfeldi as cue-lure. Using traps baited with cue-lure or methyl eugenol in five ecosystems, the highest numbers of males of D. dorsalis, D. umbrosus, D. frauenfeldi and D. caudatus trapped were from a village, on a vegetable farm for D. cucurbitae, and D. occipitalis was only caught in a forest. Analysis showed that for each species of Dacus the difference between ecosystems was highly significant. The few examples caught in grassland were probably migrants.

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