Optimality of myopic stopping times for geometric discounting

1988 ◽  
Vol 25 (2) ◽  
pp. 437-443 ◽  
Author(s):  
Bert Fristedt ◽  
Donald A. Berry
Keyword(s):  
Beta Distribution ◽  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Discount Factor ◽  
Stopping Times ◽  
Bernoulli Random Variables

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

Optimality of myopic stopping times for geometric discounting

1988 ◽  
Vol 25 (02) ◽  
pp. 437-443 ◽  
Author(s):  
Bert Fristedt ◽  
Donald A. Berry
Keyword(s):  
Beta Distribution ◽  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Discount Factor ◽  
Stopping Times ◽  
Bernoulli Random Variables

Consider a sequence of conditionally independent Bernoulli random variables taking on the values 1 and − 1. The objective is to stop the sequence in order to maximize the discounted sum. Suppose the Bernoulli parameter has a beta distribution with integral parameters. It is optimal to stop when the conditional expectation of the next random variable is negative provided the discount factor is less than or equal to . Moreover, is best possible. The case where the parameters of the beta distribution are arbitrary positive numbers is also treated.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
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.

On an Integral Equation for Discounted Compound – Annuity Distributions

1989 ◽  
Vol 19 (2) ◽  
pp. 191-198 ◽  
Author(s):  
Colin M. Ramsay
Keyword(s):  
Integral Equation ◽  
Random Variables ◽  
Random Variable ◽  
Discrete Distributions ◽  
Discount Factor ◽  
Present Value ◽  
New Class ◽  
Image Position ◽  
Independent And Identically Distributed

AbstractWe consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let Xk denote the total claims generated in the kth year, k ≥ 1. The Xk's are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as where υ is the discount factor. An integral equation similar to that given by Panjer (1981) is developed for the pdf of SN(υ). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion:where an is the present value of an n-year immediate annuity.

On the stationary distribution of some extremal Markovian sequences

1990 ◽  
Vol 27 (2) ◽  
pp. 291-302 ◽  
Author(s):  
M. T. Alpuim ◽  
E. Athayde
Keyword(s):  
Stationary Distribution ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Stationary Distributions

This paper is concerned with the Markovian sequence Xn = Zn max{Xn–1, Yn},n ≧ 1, where X0 is any random variable, {Zn} and {Yn} are independent sequences of i.i.d. random variables both independent of X0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

On the stationary distribution of some extremal Markovian sequences

1990 ◽  
Vol 27 (02) ◽  
pp. 291-302 ◽  
Author(s):  
M. T. Alpuim ◽  
E. Athayde
Keyword(s):  
Stationary Distribution ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Stationary Distributions

This paper is concerned with the Markovian sequence Xn = Zn max{Xn– 1, Yn },n ≧ 1, where X 0 is any random variable, {Zn } and {Yn } are independent sequences of i.i.d. random variables both independent of X 0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn 's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

Random Variables and Probability

2021 ◽  
pp. 109-124
Author(s):  
Timothy E. Essington
Keyword(s):  
Parameter Estimation ◽  
Model Selection ◽  
Beta Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Density Functions ◽  
The Core ◽  
Discrete Random Variables ◽  
Probability Mass ◽  
T Distribution

The chapter “Random Variables and Probability” serves as both a review and a reference on probability. The random variable is the core concept in understanding probability, parameter estimation, and model selection. This chapter reviews the basic idea of a random variable and discusses the two main kinds of random variables: discrete random variables and continuous random variables. It covers the distinction between discrete and continuous random variables and outlines the most common probability mass or density functions used in ecology. Advanced sections cover distributions such as the gamma distribution, Student’s t-distribution, the beta distribution, the beta-binomial distribution, and zero-inflated models.

scholarly journals On Generalized Slash Distributions: Representation by Hypergeometric Functions

Stats ◽  
2019 ◽  
Vol 2 (3) ◽  
pp. 371-387
Author(s):  
Peter Zörnig
Keyword(s):  
Beta Distribution ◽  
Hypergeometric Functions ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Continuous Random Variable ◽  
Generalized Hypergeometric Functions ◽  
Slash Distribution

The popular concept of slash distribution is generalized by considering the quotient Z = X/Y of independent random variables X and Y, where X is any continuous random variable and Y has a general beta distribution. The density of Z can usually be expressed by means of generalized hypergeometric functions. We study the distribution of Z for various parent distributions of X and indicate a possible application in finance.

Some applications of the Stein-Chen method for proving Poisson convergence

1989 ◽  
Vol 21 (1) ◽  
pp. 74-90 ◽  
Author(s):  
A. D. Barbour ◽  
Lars Holst
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Upper Bounds ◽  
Poisson Random Variable ◽  
Bernoulli Random Variables ◽  
Capture Recapture ◽  
Variational Distance

Let W be a sum of Bernoulli random variables and Uλ a Poisson random variable having the same mean λ = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and Uλ are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and Pólya sampling, of capture-recapture, of spacings and of matching and ménage.

The qq-bit (I): Central limits with left q-Jordan–Wigner embeddings, monotone interacting Fock space, Azema random variable, probabilistic meaning of q

2018 ◽  
Vol 21 (04) ◽  
pp. 1850030 ◽  
Author(s):  
Luigi Accardi ◽  
Yun-Gang Lu
Keyword(s):  
Tensor Product ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Random Variable ◽  
Asymmetry Index ◽  
Trivial Extension ◽  
Bernoulli Random Variables ◽  
Boson Fock Space ◽  
Interacting Fock Space

The [Formula: see text]-bit is the [Formula: see text]-deformation of the [Formula: see text]-bit. It arises canonically from the quantum decomposition of Bernoulli random variables and the [Formula: see text]-parameter has a natural probabilistic and physical interpretation as asymmetry index of the given random variable. The connection between a new type of [Formula: see text]-deformation (generalizing the Hudson–Parthasarathy bosonization technique and different from the usual one) and the Azema martingale was established by Parthasarathy. Inspired by this result, Schürmann first introduced left and right [Formula: see text]-JW-embeddings of [Formula: see text] ([Formula: see text] complex matrices) into the infinite tensor product [Formula: see text], proved central limit theorems (CLT) based on these embeddings in the context of ∗-bi-algebras and constructed a general theory of [Formula: see text]-Levy processes on ∗-bi-algebras. For [Formula: see text], left [Formula: see text]-JW-embeddings define the Jordan–Wigner transformation, used to construct a tensor representation of the Fermi anti-commutation relations (bosonization). For [Formula: see text], they reduce to the usual tensor embeddings that were at the basis of the first quantum CLT due to von Waldenfels. The present paper is the first of a series of four in which we study these theorems in the tensor product context. We prove convergence of the CLT for all [Formula: see text]. The moments of the limit random variable coincide with those found by Parthasarathy in the case [Formula: see text]. We prove that the space where the limit random variable is represented is not the Boson Fock space, as in Parthasarathy, but the monotone Fock space in the case [Formula: see text] and a non-trivial deformation of it for [Formula: see text]. The main analytical tool in the proof is a non-trivial extension of a recently proved multi-dimensional, higher order Cesaro-type theorem. The present paper deals with the standard CLT, i.e. the limit is a single random variable. Paper1 deals with the functional extension of this CLT, leading to a process. In paper2 the left [Formula: see text]-JW–embeddings are replaced by symmetric [Formula: see text]-embeddings. The radical differences between the results of the present paper and those of2 raise the problem to characterize those CLT for which the limit space provides the canonical decomposition of all the underlying classical random variables (see the Introduction, Lemma 4.5 and Sec. 5 of the present paper for the origin of this problem). This problem is solved in the paper3 for CLT associated to states satisfying a generalized Fock property. The states considered in this series have this property.

scholarly journals On the continuity of the vector valued and set valued conditional expectations

1989 ◽  
Vol 12 (3) ◽  
pp. 477-486 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Conditional Expectation ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Conditional Expectations ◽  
Vector Valued

In this paper we study the dependence of the vector valued conditional expectation (for both single valued and set valued random variables), on the σ–field and random variable that determine it. So we prove that it is continuous for theL1(X)convergence of the sub–σ–fields and of the random variables. We also present a sufficient condition for theL1(X)–convergence of the sub–σ–fields. Then we extend the work to the set valued conditional expectation using the Kuratowski–Mosco (K–M) convergence and the convergence in the Δ–metric. We also prove a property of the set valued conditional expectation.

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