The Domain of Normal Attraction of a Stable Probability Measure on a Nilpotent Group

1991 ◽  
pp. 239-247 ◽  
Author(s):  
Yu. S. Khokhlov
Keyword(s):  
Probability Measure ◽  
Nilpotent Group ◽  
Normal Attraction ◽  
Domain Of Normal Attraction

The Bounds Based on the Functions of Observations for Maximum of Stable Law

1972 ◽  
Vol 15 (2) ◽  
pp. 285-287
Author(s):  
A. K. Basu ◽  
M. T. Wasan
Keyword(s):  
Stable Law ◽  
Partial Sum ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

Gnedenko and Kolmogorov [3, pp. 181-182] have shown that if Xn with law F(x) belong to the domain of normal attraction of a stable law of index 0<α<2, i.e. if partial sum Sn/an1/α converges in distribution to some stable law Vα, a>0 then there exist c1 and c2 such thatand

A local limit theorem for attractions under a stable law

1980 ◽  
Vol 87 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Sujit K. Basu ◽  
Makoto Maejima
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Stable Law ◽  
Absolutely Continuous ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.

scholarly journals Global fixed points for nilpotent actions on the torus

2019 ◽  
Vol 40 (12) ◽  
pp. 3339-3367
Author(s):  
S. FIRMO ◽  
J. RIBÓN
Keyword(s):  
Fixed Point ◽  
Probability Measure ◽  
Nilpotent Group ◽  
Fixed Points ◽  
Nilpotent Groups ◽  
Rotation Vector

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

scholarly journals The Domain of Normal Attraction of an Operator-Stable Law

1983 ◽  
Vol 11 (1) ◽  
pp. 178-184 ◽  
Author(s):  
William N. Hudson ◽  
J. David Mason ◽  
Jerry Alan Veeh
Keyword(s):  
Stable Law ◽  
Normal Attraction ◽  
Operator Stable ◽  
Domain Of Normal Attraction

scholarly journals Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

2020 ◽  
Vol 52 (1) ◽  
pp. 237-265 ◽  
Author(s):  
Vytautė Pilipauskaitė ◽  
Viktor Skorniakov ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Stable Distribution ◽  
Stability Index ◽  
Random Coefficient ◽  
Infinite Variance ◽  
Stable Limit ◽  
Normal Attraction ◽  
Tail Distribution ◽  
Increase Rate ◽  
Domain Of Normal Attraction

AbstractWe discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an $\alpha$ -stable distribution, $0< \alpha \le 2$ , as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$ , we show that, for $\beta < \max (\alpha, 1)$ , the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$ , $\beta$ and the mutual increase rate of N and n. The paper extends the results of Pilipauskaitė and Surgailis (2014) from $\alpha =2$ to $0 < \alpha < 2$ .

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