Non-uniform Estimates in Asymptotic Expansions with a Stable Limit Distribution

1991 ◽  
Vol 153 (1) ◽  
pp. 257-272 ◽  
Author(s):  
G. Christoph
Keyword(s):  
Limit Distribution ◽  
Asymptotic Expansions ◽  
Stable Limit ◽  
Uniform Estimates ◽  
Stable Limit Distribution

scholarly journals The local limit theorem for sums of random variables defined on a homogeneous Markov chain in the case of the stable limit distribution law

1961 ◽  
Vol 1 (1-2) ◽  
pp. 7-16
Author(s):  
A. Aleškevičienė
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Distribution ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Homogeneous Markov Chain ◽  
Stable Limit ◽  
Distribution Law ◽  
Sums Of Random Variables ◽  
Stable Limit Distribution

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: A. Алешкявичене. Локальная предельная теорема для сумм случайных величин, связанных в однородную цепь Маркова в случае устойчивого предельного распределения A. Aleškevičienė. Lokalinė ribinė teorema atsitiktinių dydžių, surištų homogenine Markovo grandine, sumoms stabilaus ribinio dėsnio atveju  

scholarly journals The multi-dimensional local limit theorem for the Markov chain in the case of the stable limit distribution law

1962 ◽  
Vol 2 (1) ◽  
pp. 6-8
Author(s):  
A. Aleškevičienė
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Limit Distribution ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Stable Limit ◽  
Distribution Law ◽  
Stable Limit Distribution

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. Алешкявичене, Многомерная локальная предельная теорема для однородной цепи Маркова в случае устойчивого предельного закона A. Aleškevičienė, Daugiamatė lokalinė ribinė teorema homogeninei Markovo grandinei stabilaus ribinio dėsnio atveju

scholarly journals On Asymptotics of the Beta Coalescents

2014 ◽  
Vol 46 (2) ◽  
pp. 496-515 ◽  
Author(s):  
Alexander Gnedin ◽  
Alexander Iksanov ◽  
Alexander Marynych ◽  
Martin Möhle
Keyword(s):  
Asymptotic Expansions ◽  
Branch Length ◽  
Wasserstein Distance ◽  
Stable Limit ◽  
Initial Number ◽  
Stable Law ◽  
Total Branch Length ◽  
Coalescent Tree ◽  
Number Of Particles

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

scholarly journals On Asymptotics of the Beta Coalescents

2014 ◽  
Vol 46 (02) ◽  
pp. 496-515 ◽  
Author(s):  
Alexander Gnedin ◽  
Alexander Iksanov ◽  
Alexander Marynych ◽  
Martin Möhle
Keyword(s):  
Asymptotic Expansions ◽  
Branch Length ◽  
Wasserstein Distance ◽  
Stable Limit ◽  
Initial Number ◽  
Stable Law ◽  
Total Branch Length ◽  
Coalescent Tree ◽  
Number Of Particles

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1,b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instanceb= 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a,b)-coalescents with 0 &lt;a&lt; 1 leads to a simplified derivation of the known (2 -a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1,b)-coalescent by exploiting the method of sequential approximations.

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