Local limit theorem for the first crossing time of a~fixed level by a~random walk

A. A. Mogul'skii

Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 126-138.

Résumé

Let $X,X(1),X(2),\dots$ be independent identically distributed random variables with mean zero and a finite variance. Put $S(n)=X(1)+\dots+X(n)$, $n=1,2,\dots$, and define the Markov stopping time $\eta_y=\inf\{n\ge1\colon S(n)\ge y\}$ of the first crossing a level $y\ge0$ by the random walk $S(n)$, $n=1,2,\dots$. In the case $\mathbb E|X|^3\infty$ the following relation was obtained in [5]: $\mathbb P(\eta_0=n)=\frac1{n\sqrt n}(R+\nu_n+o(1))$, $n\to\infty$, where the constant $R$ and the bounded sequence $\nu_n$ were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $H(y):=\lim_{n\to\infty}n^{3/2}\mathbb P(\eta_y=n)$ for every fixed $y\ge0$, and there was found a representation for $H(y)$. The present paper was motivated by the following reason. In [5], the authors unfortunately did not cite papers [8,9] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [8] the existence of the limitа $\lim_{n\to\infty}n^{3/2}\mathbb P(\eta_y=n)$ for every fixed $y\ge0$ under the condition $\mathbb EX^2\infty$ only. In [9], an explicit form of the limit $\lim_{n\to\infty}n^{3/2}\mathbb E(\eta_0=n)$ was found under тthe same condition $\mathbb EX^2\infty$ in the case when the summand $X$ has an arithmetic distribution. In the present paper, we prove that the main assertion in [8] fails and we correct the original proof. It worth noting that this corrected version was formulated in [5] as a conjecture.

Bibliographie

[1] Borovkov A. A., “O faktorizatsionnykh tozhdestvakh i svoistvakh raspredeleniya supremuma posledovatelnykh summ”, Teoriya veroyatnostei i ee primeneniya, 15:3 (1970), 377–418 | MR | Zbl

[2] Borovkov A. A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Nauka, M., 1972 | MR

[3] Borovkov A. A., “Ob asimptotike raspredelenii vremen pervogo prokhozhdeniya. I”, Mat. zametki, 75:1 (2004), 24–39 | MR | Zbl

[4] Gnedenko B. V., “O lokalnoi predelnoi teoreme teorii veroyatnostei”, Uspekhi mat. nauk, 3:3(25) (1948), 187–194 | MR

[5] Mogulskii A. A., Rogozin B. A., “Lokalnaya teorema dlya momenta dostizheniya fiksirovannogo urovnya sluchainym bluzhdaniem”, Mat. trudy, 8:1 (2005), 43–70 | MR | Zbl

[6] Rogozin B. A., “Ob odnoi otsenke funktsii kontsentratsii”, Teoriya veroyatnostei i ee primeneniya, 6:3 (1961), 103–105 | MR | Zbl

[7] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, T. II, Mir, M., 1967 | Zbl

[8] Eppel M. S., “Lokalnaya predelnaya teorema dlya momenta pervogo pereskoka”, Sib. mat. zhurn., 20:1 (1979), 181–191 | MR

[9] Alili L., Doney R. A., “Wiener–Hopf factorization revisited and some applications”, Stochastics Stochastics Rep., 66:1–2 (1999), 87–102 | MR | Zbl

[10] Shepp L. A., “A local limit theorem”, Ann. Math. Statist., 35 (1964), 419–423 | DOI | MR | Zbl

[11] Stone C., “On local and ratio limit theorems”, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, V. II: Contributions to Probability Theory, Part 2, Univ. California Press, Berkeley, Calif., 1965/1966, 217–224 | MR

[12] Vatutin V. A., Wachtel V., “Local limit theorems for ladder epochs”, Probab. Theory Related Fields, 143 (2009), 177–217 | DOI | MR | Zbl