scholarly journals Asymptotic behaviour of non-isotropic random walks with heavy tails

2017 ◽  
Vol 4 (1) ◽  
pp. 79-89
Author(s):  
Mark Kelbert ◽  
Enzo Orsingher
Keyword(s):  
Asymptotic Behaviour ◽  
Random Walks ◽  
Heavy Tails ◽  
Isotropic Random

scholarly journals Limit theorems for isotropic random walks on triangle buildings

2002 ◽  
Vol 73 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Marc Lindlbauer ◽  
Michael Voit
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional ◽  
Littlewood Polynomials ◽  
Large Numbers ◽  
Moment Functions ◽  
Vertex Sets ◽  
Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

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 Transition phenomena for ladder epochs of random walks with small negative drift

2009 ◽  
Vol 41 (4) ◽  
pp. 1189-1214
Author(s):  
Vitali Wachtel
Keyword(s):  
Asymptotic Behaviour ◽  
Random Walks ◽  
Growth Rates ◽  
Negative Drift

For a family of random walks {S(a)} satisfying E S1(a)=-a<0, we consider ladder epochs τ(a)=min {k≥1: Sk(a)<0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)>n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

scholarly journals Empirical processes for recurrent and transient random walks in random scenery

2020 ◽  
Vol 24 ◽  
pp. 127-137
Author(s):  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Martin Wendler
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Stable Distribution ◽  
Random Variables ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Random Scenery ◽  
Recurrent Random Walk ◽  
Transient Random Walk

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

scholarly journals Transition phenomena for ladder epochs of random walks with small negative drift

2009 ◽  
Vol 41 (04) ◽  
pp. 1189-1214
Author(s):  
Vitali Wachtel
Keyword(s):  
Asymptotic Behaviour ◽  
Random Walks ◽  
Growth Rates ◽  
Negative Drift

For a family of random walks {S (a)} satisfying E S 1 (a)=-a&lt;0, we consider ladder epochs τ(a)=min {k≥1: S k (a)&lt;0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)&gt;n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

Isotropic random walks in a building of type

2004 ◽  
Vol 247 (1) ◽  
pp. 101-135 ◽  
Author(s):  
Donald I. Cartwright ◽  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Isotropic Random
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