Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.

Download Full-text

Negative binomial sums of random variables and discounted reward processes

Journal of Applied Probability ◽

10.1239/jap/1032265207 ◽

1998 ◽

Vol 35 (3) ◽

pp. 589-599

Author(s):

William L. Cooper

Keyword(s):

Differential Equation ◽

Negative Binomial ◽

Infinite Horizon ◽

Random Variables ◽

Random Variable ◽

Random Time ◽

Sums Of Random Variables ◽

Total Reward ◽

Binomial Random Variable ◽

Binomial Sums

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

Download Full-text

Negative binomial sums of random variables and discounted reward processes

Journal of Applied Probability ◽

10.1017/s0021900200016247 ◽

1998 ◽

Vol 35 (03) ◽

pp. 589-599

Author(s):

William L. Cooper

Keyword(s):

Differential Equation ◽

Negative Binomial ◽

Infinite Horizon ◽

Random Variables ◽

Random Variable ◽

Random Time ◽

Sums Of Random Variables ◽

Total Reward ◽

Binomial Random Variable ◽

Binomial Sums

Download Full-text

Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽

10.3390/math9090981 ◽

2021 ◽

Vol 9 (9) ◽

pp. 981

Author(s):

Patricia Ortega-Jiménez ◽

Miguel A. Sordo ◽

Alfonso Suárez-Llorens

Keyword(s):

Financial Risk ◽

Sufficient Conditions ◽

Random Variables ◽

Stochastic Ordering ◽

Random Variable ◽

Distribution Functions ◽

Stochastic Comparisons ◽

Stochastic Orderings ◽

Absolute Value ◽

The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

Download Full-text

On Distribution Characteristics of a Fuzzy Random Variable

Austrian Journal of Statistics ◽

10.17713/ajs.v47i2.581 ◽

2018 ◽

Vol 47 (2) ◽

pp. 53-67 ◽

Cited By ~ 2

Author(s):

Jalal Chachi

Keyword(s):

Probability Theory ◽

Distribution Function ◽

Cumulative Distribution Function ◽

Random Variables ◽

Fuzzy Random Variable ◽

Random Variable ◽

Cumulative Distribution ◽

Distribution Characteristics ◽

Fuzzy Random Variables ◽

Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

Download Full-text

Embedding sequences of successive maxima in extremal processes, with applications

Journal of Applied Probability ◽

10.2307/3214208 ◽

1987 ◽

Vol 24 (4) ◽

pp. 827-837 ◽

Cited By ~ 27

Author(s):

Rocco Ballerini ◽

Sidney I. Resnick

Keyword(s):

Population Size ◽

Limit Theorems ◽

Random Variables ◽

Record Values ◽

Record Times ◽

Extremal Processes ◽

Sample Record

Consequences of embedding sequences {Mn} in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Download Full-text

A refinement of normal approximation to Poisson binomial

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.717 ◽

2005 ◽

Vol 2005 (5) ◽

pp. 717-728 ◽

Cited By ~ 4

Author(s):

K. Neammanee

Keyword(s):

Normal Distribution ◽

Normal Approximation ◽

Random Variables ◽

Random Variable ◽

Standard Normal Distribution ◽

Bernoulli Random Variables ◽

Binomial Random Variable ◽

Standard Normal ◽

Correction Terms ◽

Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

Download Full-text

Asymptotic Law of thejth Records in the Bivariate Exponential Case

Journal of Mathematics ◽

10.1155/2014/458914 ◽

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.

Download Full-text

Extremal processes and random measures

Journal of Applied Probability ◽

10.2307/3212445 ◽

1975 ◽

Vol 12 (2) ◽

pp. 316-323 ◽

Cited By ~ 9

Author(s):

R. W. Shorrock

Keyword(s):

Random Measure ◽

Record Values ◽

Poisson Processes ◽

Two Dimensional ◽

Random Measures ◽

Record Times ◽

Upper Record Values ◽

Extremal Processes ◽

Upper Record ◽

Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

Download Full-text

Record values and inter-record times

Journal of Applied Probability ◽

10.1017/s002190020011842x ◽

1973 ◽

Vol 10 (03) ◽

pp. 543-555 ◽

Cited By ~ 11

Author(s):

R. W. Shorrock

Keyword(s):

Road Traffic ◽

Stochastic Order ◽

Jump Process ◽

Record Values ◽

Asymptotic Results ◽

Markov Jump ◽

Record Times ◽

Continuous State ◽

Upper Record Values ◽

Upper Record

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process. The possible relevance of this result to single lane road traffic flow is indicated.

Download Full-text

Embedding sequences of successive maxima in extremal processes, with applications

Journal of Applied Probability ◽

10.1017/s0021900200116717 ◽

1987 ◽

Vol 24 (04) ◽

pp. 827-837 ◽

Cited By ~ 5

Author(s):

Rocco Ballerini ◽

Sidney I. Resnick

Keyword(s):

Population Size ◽

Limit Theorems ◽

Random Variables ◽

Record Values ◽

Record Times ◽

Extremal Processes ◽

Sample Record

Consequences of embedding sequences {Mn } in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Download Full-text