Geodesic
Convergence to the local time of Brownian meander
Diskretnaya Matematika, Tome 29 (2017) no. 4, pp. 28-40

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\left\{ S_{n},\;n\geq 0\right\}$ be integer-valued random walk with zero drift and variance $\sigma^2$. Let $\xi(k,n)$ be number of $t\in\{1,\ldots,n\}$ such that $S(t)=k$. For the sequence of random processes $\xi(\lfloor u\sigma \sqrt{n}\rfloor,n)$ considered under conditions $S_{1}>0,\ldots ,S_{n}>0$ a functional limit theorem on the convergence to the local time of Brownian meander is proved.
Keywords: Brownian meander, local time of Brownian meander, sojourn time of random walk, functional limit theorems. 
V. I. Afanasyev. Convergence to the local time of Brownian meander. Diskretnaya Matematika, Tome 29 (2017) no. 4, pp. 28-40. http://geodesic.mathdoc.fr/item/DM_2017_29_4_a1/
@article{DM_2017_29_4_a1,
     author = {V. I. Afanasyev},
     title = {Convergence to the local time of {Brownian} meander},
     journal = {Diskretnaya Matematika},
     pages = {28--40},
     year = {2017},
     volume = {29},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2017_29_4_a1/}
}
TY  - JOUR
AU  - V. I. Afanasyev
TI  - Convergence to the local time of Brownian meander
JO  - Diskretnaya Matematika
PY  - 2017
SP  - 28
EP  - 40
VL  - 29
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/DM_2017_29_4_a1/
LA  - ru
ID  - DM_2017_29_4_a1
ER  - 
%0 Journal Article
%A V. I. Afanasyev
%T Convergence to the local time of Brownian meander
%J Diskretnaya Matematika
%D 2017
%P 28-40
%V 29
%N 4
%U http://geodesic.mathdoc.fr/item/DM_2017_29_4_a1/
%G ru
%F DM_2017_29_4_a1

[1] Billingsley P., Convergence of Probability Measures, New York: John Wiley Sons, 1968 | MR | MR | Zbl

[2] Borodin A. N., “Brownian local time”, Russian Math. Surveys, 44:2 (1989), 1–51 | DOI | MR | Zbl

[3] Borodin A. N., “On the asymptotic behavior of local times of recurrent random walks with finite variance”, Theory Probab. Appl., 26:4 (1982), 758–772 | DOI | MR | Zbl | Zbl

[4] Bulinskiy A.V., Shashkin A.P., “Limit theorems for associated random fields and related systems”, Adv. Ser. Statist. Sci. Appl. Probab., 10 (2007), World Scientific Publ., Singapore, 448 pp. | DOI | MR

[5] Iglehart D.L., “On a functional central limit theorems for random walks conditioned to stay positive”, Ann. Probab., 2:4 (1974), 608–619 | DOI | MR | Zbl

[6] Bolthausen E., “Functional central limit theorems for random walks conditioned to stay positive”, Ann. Probab., 4:3 (1976), 480–485 | DOI | MR | Zbl

[7] Takacs L., “Limit distributions for the Bernoulli meander”, J. Appl. Probab., 32:2 (1995), 375–395 | DOI | MR | Zbl

[8] Takacs L., “Brownian local times”, J. Appl. Math. Stoch. Anal., 8:3 (1995), 209–232 | DOI | MR | Zbl

[9] Csaki E., Mohanty S.G., “Some joint distributions for conditional random walks”, Canad. J. Statist., 14:1 (1986), 19–28 | DOI | MR | Zbl