Functional limit theorem for the local time of stopped random walk

Integer 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

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.

Limit distributions for the Bernoulli 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.

Limit distributions for the Bernoulli 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.

Brownian local times

In this paper explicit formulas are given for the distribution functions and the moments of the local times of the Brownian motion, the reflecting Brownian motion, the Brownian meander, the Brownian bridge, the reflecting Brownian bridge and the Brownian excursion.