The geometricity of the limiting distributions in queues and dams

1993 ◽  
Vol 30 (02) ◽  
pp. 438-445
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walks ◽  
Limiting Distribution ◽  
Duality Relation ◽  
Limiting Distributions ◽  
Initial Term

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

The geometricity of the limiting distributions in queues and dams

1993 ◽  
Vol 30 (2) ◽  
pp. 438-445
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walks ◽  
Limiting Distribution ◽  
Duality Relation ◽  
Limiting Distributions ◽  
Initial Term

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

scholarly journals Genealogy for supercritical branching processes

2006 ◽  
Vol 43 (04) ◽  
pp. 1066-1076 ◽  
Author(s):  
Andreas Nordvall Lagerås ◽  
Anders Martin-Löf
Keyword(s):  
Marginal Distribution ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Random Trees ◽  
Limiting Distributions ◽  
Marginal Distributions ◽  
Special Cases ◽  
Supercritical Branching Processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

RANDOM WALKS WITH HEAVY TAILS AND LIMIT THEOREMS FOR BRANCHING PROCESSES WITH MIGRATION AND IMMIGRATION

2012 ◽  
Vol 12 (01) ◽  
pp. 1150007 ◽  
Author(s):  
YAQIN FENG ◽  
STANISLAV MOLCHANOV ◽  
JOSEPH WHITMEYER
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Heavy Tails ◽  
Stochastic Dynamics ◽  
Branching Processes ◽  
Spatial Dynamics ◽  
Limiting Distributions ◽  
Central Result

The central result of this paper is the existence of limiting distributions for two classes of critical homogeneous-in-space branching processes with heavy tails spatial dynamics in dimension d = 2. In dimension d ≥ 3, the same results are true without any special assumptions on the underlying (non-degenerated) stochastic dynamics.

scholarly journals Law of large numbers for the drift of the two-dimensional wreath product

2021 ◽  
Author(s):  
Anna Erschler ◽  
Tianyi Zheng
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Wreath Product ◽  
Law Of Large Numbers ◽  
Abelian Groups ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Lamplighter Group ◽  
Large Numbers ◽  
Asymptotic Geometry

AbstractWe prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $$(2+\epsilon )$$ ( 2 + ϵ ) -moment. This result is in contrast with classical examples of abelian groups, where the displacement after n steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised n-step displacement admits a density whose support is $$[0,\infty )$$ [ 0 , ∞ ) . We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

scholarly journals Genealogy for supercritical branching processes

2006 ◽  
Vol 43 (4) ◽  
pp. 1066-1076 ◽  
Author(s):  
Andreas Nordvall Lagerås ◽  
Anders Martin-Löf
Keyword(s):  
Marginal Distribution ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Random Trees ◽  
Limiting Distributions ◽  
Marginal Distributions ◽  
Special Cases ◽  
Supercritical Branching Processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Random collision processes and their limiting distribution using the discrimination information

1974 ◽  
Vol 11 (2) ◽  
pp. 266-280 ◽  
Author(s):  
Nishimura Shōichi
Keyword(s):  
Statistical Mechanics ◽  
Exponential Family ◽  
Limiting Distribution ◽  
The Other ◽  
Limiting Distributions ◽  
The Family ◽  
Discrimination Information ◽  
Collision Processes

By analogy with statistical mechanics random collision processes are considered. In two cases we derive their limiting distributions using discrimination information. In one case the limiting distribution is unique and in the other one the family of limiting distributions is an exponential family.

APPROXIMATION TO THE LIMITING DISTRIBUTION OF t- AND F-STATISTICS IN TESTING FOR SEASONAL UNIT ROOTS

2001 ◽  
Vol 17 (4) ◽  
pp. 711-737 ◽  
Author(s):  
Seiji Nabeya
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Unit Roots ◽  
Limiting Distribution ◽  
Limiting Distributions ◽  
Discussion Paper ◽  
Seasonal Unit Roots ◽  
Series Analysis ◽  
The Moment ◽  
F Statistics

Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), Beaulieu and Miron (1993, Journal of Econometrics 55, 305–328), Ghysels, Lee, and Noh (1994, Journal of Econometrics 62, 415–442), Smith and Taylor (1998, Journal of Econometrics 85, 269–288; 1999, Journal of Time Series Analysis 20, 453–476; 1999, Discussion paper 99-15 in economics, University of Birmingham), and Taylor (1998, Journal of Time Series Analysis 19, 349–368) have developed a method of testing for seasonal unit roots of zero and nonzero frequencies. They propose to use t- and F-statistics as criteria that are obtained from an auxiliary regression and find their limiting distributions as the number of observations becomes large. Their limiting distributions are expressed by means of Brownian motions. In this paper the moment generating functions associated with the limiting distributions are derived, and it is shown, as in Nabeya (2000, Econometric Theory 16, 200–230), that the limiting distribution of t is well approximated by a distribution given in Gram–Charlier series. The limiting distribution of F is also well approximated by another type of distribution.

scholarly journals Non-crossing trees revisited: cutting down and spanning subtrees

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Alois Panholzer
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Limiting Distributions ◽  
Brownian Excursion ◽  
International Audience ◽  
Two Parameters

International audience Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.

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