Refining the exponential asymptotic expansion for the distribution function of a sum of a random number of nonnegative random variables

1997 ◽  
Vol 33 (1) ◽  
pp. 89-96 ◽  
Author(s):  
A. N. Nakonechnyi
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Random Number ◽  
Random Variables

Asymptotic approximations for coupon collectors

2009 ◽  
Vol 46 (1) ◽  
pp. 61-96
Author(s):  
Anna Pósfai ◽  
Sándor Csörgő
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Random Number ◽  
Limiting Distribution ◽  
Asymptotic Approximations ◽  
The One ◽  
First Time

A collector samples with replacement a set of n ≧ 2 distinct coupons until he has n − m , 0 ≦ m < n , distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if n → ∞ and m is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log n )/ n .

An expansion for the distribution function of a random sum

1973 ◽  
Vol 10 (4) ◽  
pp. 869-874 ◽  
Author(s):  
L. M. Marsh
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Special Form ◽  
Edgeworth Expansion ◽  
Random Number ◽  
Probability Generating Function ◽  
Random Variables ◽  
Fixed Number ◽  
Random Sum

The Edgeworth expansion gives an indication of the rate of convergence of the distribution function of the sum of a fixed number of random variables to the normal distribution. A similar expansion is given here for the distribution function of the sum of a random number N of random variables, when the probability generating function of N takes a special form.

An expansion for the distribution function of a random sum

1973 ◽  
Vol 10 (04) ◽  
pp. 869-874 ◽  
Author(s):  
L. M. Marsh
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Special Form ◽  
Edgeworth Expansion ◽  
Random Number ◽  
Probability Generating Function ◽  
Random Variables ◽  
Fixed Number ◽  
Random Sum

The Edgeworth expansion gives an indication of the rate of convergence of the distribution function of the sum of a fixed number of random variables to the normal distribution. A similar expansion is given here for the distribution function of the sum of a random number N of random variables, when the probability generating function of N takes a special form.

Binary decompositions of probability densities and random-bit simulation

2020 ◽  
Vol 26 (2) ◽  
pp. 163-169
Author(s):  
Vladimir Nekrutkin
Keyword(s):  
New York ◽  
Distribution Function ◽  
Random Number ◽  
Discrete Distribution ◽  
Random Number Generation ◽  
Academic Press ◽  
Bernoulli Trials ◽  
Algorithms And Complexity ◽  
Binary Decomposition ◽  
Probability Densities

AbstractThis paper is devoted to random-bit simulation of probability densities, supported on {[0,1]}. The term “random-bit” means that the source of randomness for simulation is a sequence of symmetrical Bernoulli trials. In contrast to the pioneer paper [D. E. Knuth and A. C. Yao, The complexity of nonuniform random number generation, Algorithms and Complexity, Academic Press, New York 1976, 357–428], the proposed method demands the knowledge of the probability density under simulation, and not the values of the corresponding distribution function. The method is based on the so-called binary decomposition of the density and comes down to simulation of a special discrete distribution to get several principal bits of output, while further bits of output are produced by “flipping a coin”. The complexity of the method is studied and several examples are presented.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

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.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution

scholarly journals A Shortcut to LAD Estimator Asymptotics

1991 ◽  
Vol 7 (4) ◽  
pp. 450-463 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Asymptotic Expansion ◽  
Regression Model ◽  
Taylor Series ◽  
Asymptotic Theory ◽  
Random Variables ◽  
Generalized Functions ◽  
Infinite Variance ◽  
Series Expansions ◽  
Linear Functionals ◽  
Taylor Series Expansions

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

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