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.

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.

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 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

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.

Limit distributions for the Bernoulli meander

1995 ◽  
Vol 32 (2) ◽  
pp. 375-395 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Limit Behavior ◽  
Explicit Formulas ◽  
Limit Distributions ◽  
Brownian Meander

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.

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