II.—Asymptotic Renewal Theorems

1953 ◽  
Vol 64 (1) ◽  
pp. 9-48 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Distribution Function ◽  
Renewal Process ◽  
Age Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Renewal Theorem ◽  
Discrete Process ◽  
Main Emphasis ◽  
Image Position ◽  
Independent And Identically Distributed

SynopsisA sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete processwhere Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under whichThese new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (1) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X1,X2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X1,n,…,Xn,n be the sequence of order statistics of X1,…,Xn. For a sequence (cn)n≥1 of positive constants, the smallest fit off-line counting random variable is defined by Ne(cn) := max {j ≤ n : X1,n + … + Xj,n ≤ cn}. The asymptotic joint distributional comparison is given between the off-line count Ne(cn) and on-line counts Nnτ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑j≥1XτjI(τj≤n) ≤ cn. Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (cn)n≥1, we find sequences of positive constants (Bn)n≥1, (Δn)n≥1 and (Δ'n)n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

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.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (01) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X 1,X 2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X 1,n ,…,X n,n be the sequence of order statistics of X 1,…,X n . For a sequence (c n ) n≥1 of positive constants, the smallest fit off-line counting random variable is defined by N e (c n ) := max {j ≤ n : X 1,n + … + X j,n ≤ c n }. The asymptotic joint distributional comparison is given between the off-line count N e (c n ) and on-line counts N n τ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑ j≥1 X τ j I (τ j ≤n) ≤ c n . Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (c n ) n≥1, we find sequences of positive constants (B n ) n≥1, (Δ n ) n≥1 and (Δ' n ) n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

The Bahadur-Kiefer Representation ofLpRegression Estimators

1996 ◽  
Vol 12 (2) ◽  
pp. 257-283 ◽  
Author(s):  
Miguel A. Arcones
Keyword(s):  
Linear Regression ◽  
Regression Model ◽  
Positive Constant ◽  
Linear Regression Model ◽  
Random Variables ◽  
Random Variable ◽  
Regularity Conditions ◽  
Dimensional Parameter ◽  
Image Position ◽  
Independent And Identically Distributed

We consider the following linear regression model:whereare independent and identically distributed random variables, Yi, is real, Zihas values in Rm, Ui, is independent of Zi, and θ0is anm-dimensional parameter to be estimated. TheLpestimator of θ0is the value 6n such thatHere, we will give the exact Bahadur-Kiefer representation of θn, for each p ≥ 1. Explicitly, we will see that, under regularity conditions,whereandcis a positive constant, which depends onpand on the random variableX.

scholarly journals Exact and Limiting Probability Distributions of Some Smirnov Type Statistics

1965 ◽  
Vol 8 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Random Sample ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Probability Distributions ◽  
Continuous Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Exact Distributions ◽  
Image Position

Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variablesand and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
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.

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

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