On the local time of a recurrent random walk on ℤ²

2021 ◽  
Vol 105 (0) ◽  
pp. 69-78
Author(s):  
V. Bohun ◽  
A. Marynych
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Process ◽  
Local Time ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Content Type ◽  
Finite Set ◽  
Number Of Visits ◽  
Recurrent Random Walk

We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 \mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 P\subset \mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.

Functional limit theorem for the local time of stopped random walk

2020 ◽  
Vol 30 (3) ◽  
pp. 147-157
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Local Time ◽  
Functional Limit Theorem ◽  
Finite Variance ◽  
Functional Limit ◽  
Number Of Visits ◽  
The Moment ◽  
Half Axis ◽  
Brownian Meander

AbstractInteger random walk {Sn, n ≥ 0} with zero drift and finite variance σ2 stopped at the moment T of the first visit to the half axis (-∞, 0] is considered. For the random process which associates the variable u ≥ 0 with the number of visits the state ⌊uσ$\begin{array}{} \displaystyle \sqrt{n} \end{array}$⌋ by this walk conditioned on T > n, the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.

Convergence to the local time of Brownian meander

2019 ◽  
Vol 29 (3) ◽  
pp. 149-158 ◽  
Author(s):  
Valeriy. I. Afanasyev
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Local Time ◽  
Random Processes ◽  
Functional Limit Theorem ◽  
Zero Drift ◽  
Functional Limit ◽  
Brownian Meander

Abstract Let {Sn, n ≥ 0} be integer-valued random walk with zero drift and variance σ2. Let ξ(k, n) be number of t ∈ {1, …, n} such that S(t) = k. For the sequence of random processes $\begin{array}{} \xi(\lfloor u\sigma \sqrt{n}\rfloor,n) \end{array}$ considered under conditions S1 > 0, …, Sn > 0 a functional limit theorem on the convergence to the local time of Brownian meander is proved.

scholarly journals Path to survival for the critical branching processes in a random environment

2017 ◽  
Vol 54 (2) ◽  
pp. 588-602 ◽  
Author(s):  
Vladimir Vatutin ◽  
Elena Dyakonova
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Limit Theorem ◽  
Levy Process ◽  
Initial Part ◽  
Functional Limit ◽  
Critical Branching Process

Abstract A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

Functional limit theorem for a stopped random walk attaining a high level

2017 ◽  
Vol 27 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Functional Limit Theorem ◽  
Zero Drift ◽  
Convergence In Distribution ◽  
Functional Limit ◽  
High Level

AbstractFor a stopped random walk with zero drift conditioned to attain a high level the theorem on the convergence in distribution to the Brownian high jump in the space

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