Geodesic
Functional limit theorem for the local time of stopped random walk
Diskretnaya Matematika, Tome 31 (2019) no. 1, pp. 7-20

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Integer random walk $\left\{ S_{n},\,n\geq 0\right\} $ with zero drift and finite variance $\sigma ^{2}$ stopped at the moment $T$ of the first visit to the half axis $\left( -\infty ,0\right] $ is considered. For the random process which associates the variable $u\geq 0$ with the number of visits the state $\left\lfloor u\sigma \sqrt{n}\right\rfloor $ by this walk conditioned on $T>n$, the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.
Keywords: conditioned Brownian motions, local time of conditioned Brownian motions, functional limit theorems. 
V. I. Afanasyev. Functional limit theorem for the local time of stopped random walk. Diskretnaya Matematika, Tome 31 (2019) no. 1, pp. 7-20. http://geodesic.mathdoc.fr/item/DM_2019_31_1_a1/
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[1] Afanasyev V.I., “Functional limit theorem for a stopped random walk attaining a high level”, Discrete Math. Appl., 27:5 (2017), 269–276 | DOI | DOI | MR | Zbl

[2] Hooghiemstra G., “Conditioned limit theorems for waiting-time processes of the M/G/1 queue”, J. Appl. Probab., 20:3 (1983), 675-688 | DOI | MR | Zbl

[3] 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

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

[5] Afanasev V.I., “Skhodimost k lokalnomu vremeni brounovskoi izviliny”, Diskretnaya matematika, 29:4 (2017), 28–40 | DOI | MR

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

[7] Bulinski A.V., Shashkin A.P., Limit theorems for associated random fields and related systems, World Scientific, Singapore, 2007, 436 pp. | MR

[8] Borodin A.N., Stochastic Processes, Birkhäuser Basel, 2017, 626 pp. | MR | Zbl

[9] 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

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