The random record model

1977 ◽  
Vol 356 (1687) ◽  
pp. 529-547 ◽  
Keyword(s):  
Point Process ◽  
Large Time ◽  
Asymptotic Properties ◽  
Fixed Interval ◽  
Independent Point ◽  
Time Intervals ◽  
Record Times ◽  
Index Numbers ◽  
Independent And Identically Distributed

This paper contains a detailed study of the random record model, which is the process of successive upper records in a sequence of independent and identically distributed variates occurring at the points of an independent point process. Asymptotic properties of the record and inter-record times, and of the number of records in a fixed interval, are derived, for both large time intervals and large record index numbers. A generalization which permits some dependence between the variates and the point process is also discussed, with emphasis on explicit results.

The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1

1973 ◽  
Vol 59 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Harvey Segur
Keyword(s):  
Water Waves ◽  
Large Time ◽  
Vries Equation ◽  
Asymptotic Properties ◽  
Series Representation ◽  
Convergent Series ◽  
Asymptotic Results ◽  
De Vries ◽  
Method Of Solution

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (1) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

scholarly journals Turning the body into a clock: Accurate timing is facilitated by simple stereotyped interactions with the environment

2020 ◽  
Vol 117 (23) ◽  
pp. 13084-13093 ◽  
Author(s):  
Mostafa Safaie ◽  
Maria-Teresa Jurado-Parras ◽  
Stefania Sarno ◽  
Jordane Louis ◽  
Corane Karoutchi ◽  
...  
Keyword(s):  
Fixed Interval ◽  
The Body ◽  
Trial And Error ◽  
Time Intervals ◽  
Timing Accuracy ◽  
Comparable Level ◽  
Regular Time ◽  
Stereotyped Behaviors ◽  
Moving Direction ◽  
Freely Moving

How animals adapt their behavior according to regular time intervals between events is not well understood, especially when intervals last several seconds. One possibility is that animals use disembodied internal neuronal representations of time to decide when to initiate a given action at the end of an interval. However, animals rarely remain immobile during time intervals but tend to perform stereotyped behaviors, raising the possibility that motor routines improve timing accuracy. To test this possibility, we used a task in which rats, freely moving on a motorized treadmill, could obtain a reward if they approached it after a fixed interval. Most animals took advantage of the treadmill length and its moving direction to develop, by trial-and-error, the same motor routine whose execution resulted in the precise timing of their reward approaches. Noticeably, when proficient animals did not follow this routine, their temporal accuracy decreased. Then, naïve animals were trained in modified versions of the task designed to prevent the development of this routine. Compared to rats trained in the first protocol, these animals didn’t reach a comparable level of timing accuracy. Altogether, our results indicate that timing accuracy in rats is improved when the environment affords cues that animals can incorporate into motor routines.

A point-process approach to almost-sure behaviour of record values and order statistics

1996 ◽  
Vol 28 (02) ◽  
pp. 426-462 ◽  
Author(s):  
Charles M. Goldie ◽  
Ross A. Maller
Keyword(s):  
Order Statistics ◽  
Point Process ◽  
Relative Stability ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Record Values ◽  
Process Approach ◽  
Integral Conditions ◽  
Comprehensive Investigation ◽  
Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima(the rth largest among a sample ofni.i.d. random variables), partial record valuesand differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties asa.s. for some finite constantc, ora.s. for some constantcin [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail ofF,which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (3) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (01) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

scholarly journals Asymptotic Law of thejth Records in the Bivariate Exponential Case

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Grine Azedine
Keyword(s):  
Exponential Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Asymptotic Independence ◽  
Record Times ◽  
Bivariate Exponential ◽  
Exponential Case ◽  
Independent And Identically Distributed

We consider a sequence(Xi,Yi)1⩽i⩽nof independent and identically distributed random variables with joint cumulative distribution H(x,y), which has exponential marginalsF(x)andG(y)with parameterλ=1. We also assume thatXi(ω)≠Yi(ω),∀i∈N, andω∈Ω. We denoteRk(j)k⩾1andSk(j)k⩾1by the sequences of thejth records in the sequences(Xi)1⩽i⩽n,(Yi)1⩽i⩽n, respectively. The main result of of the paper is to prove the asymptotic independence ofRk(j)k⩾1andSk(j)k⩾1using the property of stopping time of thejth record times and that of the exponential distribution.

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