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. 
@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},
     publisher = {mathdoc},
     volume = {29},
     number = {4},
     year = {2017},
     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
PB  - mathdoc
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
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2017_29_4_a1/
%G ru
%F DM_2017_29_4_a1
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/

[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