scholarly journals Two-dimensional symmetric stable distributions and Their projections

2005 ◽  
Vol 180 ◽  
pp. 135-149
Author(s):  
Katsuya Kojo
Keyword(s):  
Stable Distribution ◽  
Stable Distributions ◽  
Negative Answer ◽  
Affirmative Answer ◽  
Two Dimensional ◽  
Symmetric Stable Distribution

AbstractWe study the problem whether a given 2-dimensional symmetric stable distribution with index α (0 < α ≤ 1) is determined by its 1-dimensional projections in some specified directions. We give some conditions for the affirmative answer and for the negative answer.

scholarly journals An Universal, Simple, Circular Statistics-Based Estimator of α for Symmetric Stable Family

2019 ◽  
Vol 12 (4) ◽  
pp. 171
Author(s):  
Ashis SenGupta ◽  
Moumita Roy
Keyword(s):  
Entire Range ◽  
Goodness Of Fit ◽  
Stable Distribution ◽  
Real Life ◽  
Circular Statistics ◽  
Directional Statistics ◽  
Symmetric Stable Distribution ◽  
Stable Family ◽  
Family Of Distributions ◽  
Better Than

The aim of this article is to obtain a simple and efficient estimator of the index parameter of symmetric stable distribution that holds universally, i.e., over the entire range of the parameter. We appeal to directional statistics on the classical result on wrapping of a distribution in obtaining the wrapped stable family of distributions. The performance of the estimator obtained is better than the existing estimators in the literature in terms of both consistency and efficiency. The estimator is applied to model some real life financial datasets. A mixture of normal and Cauchy distributions is compared with the stable family of distributions when the estimate of the parameter α lies between 1 and 2. A similar approach can be adopted when α (or its estimate) belongs to (0.5,1). In this case, one may compare with a mixture of Laplace and Cauchy distributions. A new measure of goodness of fit is proposed for the above family of distributions.

On the maximum of sums of random variables and the supremum functional for stable processes

1969 ◽  
Vol 6 (2) ◽  
pp. 419-429 ◽  
Author(s):  
C.C. Heyde
Keyword(s):  
Limit Theorem ◽  
Stable Distribution ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Law ◽  
Sums Of Random Variables ◽  
Symmetric Stable Distribution ◽  
General Limit ◽  
General Limit Theorem

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

STATIC STATISTICAL APPROACH TO THE BTW SANDPILE MODEL

Fractals ◽  
2006 ◽  
Vol 14 (01) ◽  
pp. 55-61
Author(s):  
DAHUI WANG ◽  
WEITING CHEN ◽  
QIANG YUAN ◽  
ZENGRU DI
Keyword(s):  
Power Law ◽  
Stable Distribution ◽  
Particle Number ◽  
Statistical Approach ◽  
Negative Temperature ◽  
Two Dimensional ◽  
Power Law Distribution ◽  
Sandpile Model ◽  
Total Particle ◽  
The One

A static statistical approach to the Bak, Tang and Wiesenfeld (BTW) sandpile model is proposed. With this approach, the exact avalanche distribution of the one-dimensional BTW sandpile is given concisely. Furthermore, we investigate the two-dimensional BTW sandpile and obtain some interesting results. First, the total particle number of the two-dimensional BTW sandpile obeys some kind of stable distribution. With the increase of the sandpile scale, the stable distribution transits from Gamma to Normal distribution. Second, when the total number of particles is fixed, the avalanche distribution is not power law. The system, however, shows a kind of "negative temperature" phenomenon when the particle number increases. Third, power law distribution of the avalanche could be viewed as the result of the superposition of a series of weighted distributions which do not yield power law.

Maturity Guarantees Revisited: Allowing for Extreme Stochastic Fluctuations using Stable Distributions

1997 ◽  
Vol 3 (2) ◽  
pp. 411-482 ◽  
Author(s):  
G.S. Finkelstein
Keyword(s):  
Stable Distribution ◽  
Working Party ◽  
Stable Distributions ◽  
Investment Model ◽  
Stochastic Fluctuations ◽  
Stable Family ◽  
Family Of Distributions

ABSTRACTThe paper examines the suitability of the stable family of distributions with the Maturity Guarantees Working Party's stochastic investment model (Ford et al, 1980). It then examines the effect of replacing the Gaussian assumption made by the working party with a more general stable distribution. It also explains how the appropriate stable distribution can be fitted.

Names and Pseudonyms

Philosophy ◽  
1995 ◽  
Vol 70 (274) ◽  
pp. 487-512 ◽  
Author(s):  
Lloyd Humberstone
Keyword(s):  
Negative Answer ◽  
Affirmative Answer ◽  
Lewis Carroll ◽  
Charles Dodgson ◽  
Formed Part ◽  
Literary Output

Was there such a person as Lewis Carroll? An affirmative answer is suggested by the thought that Lewis Carroll was Charles Dodgson, and since there was certainly such a person as Charles Dodgson, there was such a person as Lewis Carroll. A negative answer is suggested by the thought that in arguing thus, the two names ‘Lewis Carroll’ and ‘Charles Dodgson’ are being inappropriately treated as though they were completely on a par: a pseudonym is, after all, a false or fictitious name. Perhaps we should say instead that there was really no such person as Lewis Carroll, but that when Charles Dodgson published under that name, he was pretending that there was, and further, pretending that the works in question formed part of the literary output of this pretendedly real individual. Whether or not this is correct for the case of ‘Lewis Carroll’, I will be suggesting that an account of this second style–a fictionalist account, for short–is appropriate for at least a good many pseudonyms. We shall get to reasons why it might nonetheless not be especially appropriate in the present case in due course: one advantage of the ‘Lewis Carroll’/‘Charles Dodgson’ example, such qualms notwithstanding, is that everyone (likely to be reading this) is familiar not only with both names but with which of them is the pseudonym. Another is that, as we shall have occasion to observe below, Dodgson himself had some interesting views on this particular case of pseudonym(it)y.

scholarly journals A direct approach to the stable distributions

2016 ◽  
Vol 48 (A) ◽  
pp. 261-282 ◽  
Author(s):  
E. J. G. Pitman ◽  
Jim Pitman
Keyword(s):  
Functional Equation ◽  
Characteristic Function ◽  
Integral Representation ◽  
Explicit Form ◽  
Regular Variation ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Direct Approach ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible

AbstractThe explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

scholarly journals REAL ZEROS OF ALGEBRAIC POLYNOMIALS WITH STABLE RANDOM COEFFICIENTS

2008 ◽  
Vol 85 (1) ◽  
pp. 81-86 ◽  
Author(s):  
K. FARAHMAND
Keyword(s):  
Stable Distribution ◽  
Random Coefficients ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Symmetric Stable Distribution ◽  
Number Of Real Zeros ◽  
Significant Interest ◽  
Special Case

AbstractWe consider a random algebraic polynomial of the form Pn,θ,α(t)=θ0ξ0+θ1ξ1t+⋯+θnξntn, where ξk, k=0,1,2,…,n have identical symmetric stable distribution with index α, 0<α≤2. First, for a general form of θk,α≡θk we derive the expected number of real zeros of Pn,θ,α(t). We then show that our results can be used for special choices of θk. In particular, we obtain the above expected number of zeros when $\theta _k={n\choose k}^{1/2}$. The latter generate a polynomial with binomial elements which has recently been of significant interest and has previously been studied only for Gaussian distributed coefficients. We see the effect of α on increasing the expected number of zeros compared with the special case of Gaussian coefficients.

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