2000 character limit reached
Expansions for random walks conditioned to stay positive (2401.09929v1)
Published 18 Jan 2024 in math.PR
Abstract: We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(\tau_x>n)$ of the first passage times $\tau_x:=\inf{n\ge1:x+S_n\le0}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,\tau_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.
- On conditioning a random walk to stay nonnegative. Ann. Probab., 22:2152–2167, 1994.
- Regular variation. Cambridge University Press, 1987.
- Borovkov, A.A. New limit theorems in boundary-value problems for sums of independent terms. Siberian Math. J. 3:645–694, 1962.
- Borovkov, A.A. Probability Theory. Springer-Verlag, London, 2013.
- Polyharmonic functions and random processes in cones. International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, 2020.
- Ordered exponential random walks. ALEA, Lat. Am. J. Probab. Math. Stat. 20:1211–1246, 2023.
- Doney, R.A. Local behaviour of first passage probabilities. Probab. Theory Relat. Fields, 152:559–588, 2012.
- Logarithmic terms in discrete heat kernel expansions in the quadrant. ArXiv: 2309.15209.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. Wiley, New York, 1971.
- Analytic Combinatorics. Cambridge University Press, 2009.
- Conditioned local limit theorems for random walks on the real line ArXiv preprint: 2110.05123.
- Gelfond, A. O. An estimate for the remainder term in the limit theorem for recurrent events.(Russian. English summary) Teor. Verojatnost. i Primenen., 9:327–331, 1964.
- Koroljuk, V. S. Asymptotic analysis of distributions of maximum deviation on a lattice random walk. Teor. Verojatnost. i Primenen. 7: 393–409, 1962.
- Nagaev, S. V. Asymptotic expansions for the distribution function of the maximum of the sums of independent identically distributed random variables. Siberian Math. J. 11:288–309, 1970.
- Nessmann, A. Polyharmonic functions in the quarter plane. ArXiv: 2212.07258.
- Nessmann, A. Full asymptotic expansions for orbit-summable quadrant walks and discrete polyharmonic functions. ArXiv: 2307.11539.
- Petrov, V. V. Sums of independent random variables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975.
- Rogozin B.A. On the distribution of the first ladder moment and height and fluctuations of a random walk. Theory Probab. Appl., 16:575-595, 1971.
- Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields, 143:177–217, 2009.