An extension of a lemma of Wald

1966 ◽  
Vol 3 (1) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y1, y2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

An extension of a lemma of Wald

1966 ◽  
Vol 3 (01) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y 1, y 2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

scholarly journals On a transformation of the weighted compound Poisson process

1971 ◽  
Vol 6 (1) ◽  
pp. 42-46 ◽  
Author(s):  
Hans Bühlmann ◽  
Roberto Buzzi
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Characteristic Functions ◽  
Compound Poisson ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Poisson Variable ◽  
Image Position ◽  
Independent Identically Distributed

We are using the following terminology—essentially following Feller:a) Compound Poisson VariableThis is a random variable where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variablehence(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functionsb) Weighted Compound Poisson VariableThis is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)henceor in the language of characteristic functionsLet [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads thenIt is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (04) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y 1, Y2 , · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the X il…i m .

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (02) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X 1, · ··, Xn ) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c &gt; 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α&gt; 0, then for fixed c the expected number, En (c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (4) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y1, Y2, · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the Xil…im.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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 Improving the Asmussen–Kroese-Type Simulation Estimators

2012 ◽  
Vol 49 (4) ◽  
pp. 1188-1193 ◽  
Author(s):  
Samim Ghamami ◽  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Nonnegative Integer ◽  
Rare Event ◽  
Random Variables ◽  
Random Variable ◽  
Heavy Tailed ◽  
Independent Identically Distributed

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

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