Geodesic
On the Local Time of a Stopped Random Walk Attaining a High Level
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 11-31.

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An integer-valued random walk $\{S_i,\, i\geq 0\}$ with zero drift and finite variance $\sigma ^2$ stopped at the time $T$ of the first hit of the semiaxis $(-\infty ,0]$ is considered. For the random process defined for a variable $u>0$ as the number of visits of this walk to the state $\lfloor un\rfloor $ and conditioned on the event $\max _{1\leq i\leq T}S_i>n$, a functional limit theorem on its convergence to the local time of the Brownian high jump is proved.
Keywords: conditional Brownian motion, local time, functional limit theorem. 
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V. I. Afanasyev. On the Local Time of a Stopped Random Walk Attaining a High Level. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 11-31. http://geodesic.mathdoc.fr/item/TM_2022_316_a2/

[1] V. I. Afanas'ev, “Local time of a random walk up to the first passage to the semiaxis”, Math. Notes, 48:6 (1990), 1173–1177 | DOI | MR | MR

[2] V. I. Afanasyev, “On a conditional invariance principle for a critical Galton–Watson branching process”, Discrete Math. Appl., 15:1 (2005), 17–32 | DOI | MR | Zbl

[3] V. I. Afanasyev, “Galton–Watson process attaining a high level”, Theory Probab. Appl., 52:3 (2008), 509–515 | DOI | MR | MR | Zbl

[4] V. I. Afanasyev, “Brownian high jump”, Theory Probab. Appl., 55:2 (2011), 183–197 | DOI | MR | Zbl

[5] V. I. Afanasyev, “Functional limit theorem for a stopped random walk attaining a high level”, Discrete Math. Appl., 27:5 (2017), 269–276 | DOI | MR | Zbl

[6] V. I. Afanasyev, “Functional limit theorem for the local time of stopped random walk”, Discrete Math. Appl., 30:3 (2020), 147–157 | DOI | MR | Zbl

[7] P. Billingsley, Convergence of Probability Measures, J. Wiley and Sons, New York, 1968 | MR | Zbl

[8] Cohen J.W., Hooghiemstra G., “Brownian excursion, the $M/M/1$ queue and their occupation times”, Math. Oper. Res., 6:4 (1981), 608–629 | DOI | MR | Zbl

[9] W. Feller, An Introduction to Probability Theory and Its Applications, v. 2, J. Wiley and Sons, New York, 1971 | MR | MR | Zbl

[10] 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

[11] Lindvall T., “On the maximum of a branching process”, Scand. J. Stat., 3:4 (1976), 209–214 | MR | Zbl

[12] Pakes A.G., “A limit theorem for the maxima of the para-critical simple branching process”, Adv. Appl. Probab., 30:3 (1998), 740–756 | DOI | MR | Zbl

[13] Shimura M., “A class of conditional limit theorems related to ruin problem”, Ann. Probab., 11:1 (1983), 40–45 | DOI | MR | Zbl

[14] V. A. Vatutin, V. Wachtel, and K. Fleischmann, “Critical Galton–Watson processes: The maximum of total progenies within a large window”, Theory Probab. Appl., 52:3 (2008), 470–492 | DOI | MR | Zbl