The Domains of Attraction of Semistable Laws

“…Grinevich and Khokhlov [8] also showed that, in the sufficiency part of the theorem above, k n can be chosen as follows. First, choose a sequence {Ã n } such that…”

Section: )mentioning

confidence: 99%

Non-Gaussian semi-stable laws arising in sampling of finite point processes

Chaudhuri¹,

Pipiras²

2016

Bernoulli

View full text Add to dashboard Cite

A finite point process is characterized by the distribution of the number of points (the size) of the process. In some applications, for example, in the context of packet flows in modern communication networks, it is of interest to infer this size distribution from the observed sizes of sampled point processes, that is, processes obtained by sampling independently the points of i.i.d. realizations of the original point process. A standard nonparametric estimator of the size distribution has already been suggested in the literature, and has been shown to be asymptotically normal under suitable but restrictive assumptions. When these assumptions are not satisfied, it is shown here that the estimator can be attracted to a semi-stable law. The assumptions are discussed in the case of several concrete examples. A major theoretical contribution of this work are new and quite general sufficient conditions for a sequence of i.i.d. random variables to be attracted to a semi-stable law.

show abstract

“…Note that such i.i.d. sequences belong to the domain of geometric attraction of semistable laws, see Grinevich and Khokhlov [7] for a precise characterization of this class in terms of characteristic functions. Also, as shown by Csörgö and Megyesi [8], for partial sums of i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Strong approximation of the St. Petersburg game

Berkes

2017

Statistics

View full text Add to dashboard Cite

Let X, X 1 , X 2 , . . . be i.i.d. random variables with P(X = 2 k ) = 2 −k (k ∈ N) and let S n = n k=1 X k . The properties of the sequence S n have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let {Z(t), t ≥ 0} be a semistable Lévy process with underlying Lévy measure k∈Z 2 −k δ 2 k . For a suitable version of (X k ) and Z(t), we prove the strong approximation S n = Z(n) + O(n 5/6+ε ) a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws. ARTICLE HISTORY

show abstract

The Domains of Attraction of Semistable Laws

Cited by 19 publications

References 5 publications

The domain of normal attraction of a semistable distribution on a semidirect product compact group and R d

The domain of normal attraction of a semistable distribution on a semidirect product compact group and R d

Non-Gaussian semi-stable laws arising in sampling of finite point processes

Strong approximation of the St. Petersburg game

Contact Info

Product

Resources

About