The tail behaviour of a distribution function and its connection to the growth of its entire characteristic function

1978 ◽  
Vol 81 (1) ◽  
pp. 217-231 ◽  
Author(s):  
Monika Dewess
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Tail Behaviour ◽  
Entire Characteristic Function

A model for a system subject to random shocks

1993 ◽  
Vol 30 (4) ◽  
pp. 979-984 ◽  
Author(s):  
Eui Yong Lee ◽  
Jiyeon Lee
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Stochastic Model ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
Random Shocks ◽  
Random Amount ◽  
Explicit Expressions

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

scholarly journals From Characteristic Function to Distribution Function: A Simple Framework for the Theory

1991 ◽  
Vol 7 (4) ◽  
pp. 519-529 ◽  
Author(s):  
N.G. Shephard
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Unified Framework ◽  
New Approach ◽  
Inversion Theorem

A unified framework is established for the study of the computation of the distribution function from the characteristic function. A new approach to the proof of Gurland's and Gil-Pelaez's univariate inversion theorem is suggested. A multivariate inversion theorem is then derived using this technique.

scholarly journals Approximations to Risk Theory's F(x, t) by Means of the Gamma Distribution

1977 ◽  
Vol 9 (1-2) ◽  
pp. 213-218 ◽  
Author(s):  
Hilary L. Seal
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Gamma Distribution ◽  
Gamma Function ◽  
Function Approximation ◽  
Good Accuracy ◽  
Electronic Computer ◽  
Pocket Computer ◽  
Power Approximation ◽  
Aggregate Claims

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

ON THE TWO-POINT CORRELATION FUNCTION FOR DISPERSIONS OF NONOVERLAPPING SPHERES

1998 ◽  
Vol 08 (02) ◽  
pp. 359-377 ◽  
Author(s):  
KONSTANTIN Z. MARKOV ◽  
JOHN R. WILLIS
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Characteristic Function ◽  
Radial Distribution Function ◽  
Radial Distribution ◽  
Integral Formula ◽  
Probability Densities ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Point Level

Random dispersions of spheres are useful and appropriate models for a wide class of particulate random materials. They can be described in two equivalent and alternative ways — either by the multipoint moments of the characteristic function of the region, occupied by the spheres, or by the probability densities of the spheres' centers. On the "two-point" level, a simple and convenient integral formula is derived which interconnects the radial distribution function of the spheres with the two-point correlation of the said characteristic function. As one of the possible applications of the formula, the behavior of the correlation function near the origin is studied in more detail and related to the behavior of the radial distribution function at the "touching" separation of the spheres.

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