×

One dimensional lattice random walks with absorption at a point/on a half line. (English) Zbl 1234.60051

Let \(Y_{n}\), \(n\geq 1\), be independent, identically distributed, \({\mathbb Z}\)-valued random variables with \(\operatorname{E}Y= 0\), \(\sigma^{2}=\operatorname{E}(Y^{2})<\infty\) (\(Y\) having the same law as \(Y_1\)), \(S_{n}=x+Y_{1}+\dots +Y_{n}\), \[ q^{n}(x,y)=\operatorname{P}(S_{n}=y,S_{i}\neq 0,i=1,\dots ,n-1), \] \(a_{0}>1\), \(p^{n}(x)=\operatorname{P}(Y_{1}+\dots +Y_{n}=x)\), \(d_{0}\) be the period, \(a(x)=\sum_{n\geq 0}(p^{n}(x)-p^{n}(-x))\), \(a^{\ast }(x)= 1_{\{0\}}(x)+a(x)\), \(g_{n}(u)=(2\pi n\sigma^{2})^{-1/2}\operatorname{E}^{- u^{2}/2\sigma^{2}n},\) \(T=\inf\{n \geq 1,S_{n}\leq 0\}\), \[ q_{(-\infty ,0]}^{n}(x,y)= \operatorname{P}(S_{n}=y,n<T), \] \(f_{+}\), \(f_{-}\) be positive harmonic on \(x>0\), for \(S_{n}\), \(-S_{n}\) respectively, absorbed on \((-\infty ,0]\), and satisfying \(\lim_{x\to \infty }x^{-1}f_{\pm }(x)=1\), \(h(n,y)=\operatorname{P}(S_{T}=y,T=n)\), \(H(y)=2\operatorname{E}(f_{-}(y-Y);Y<y)/\sigma^{2}\).
The first result (all are uniformly in \(x\),\(y\)) is, for \(o(n^{1/2})=\min(|x|,|y|)\), \(\max(|x|,|y|)<a_{0}n^{1/2}\) we have \[ q^{n}(x,y)=(\sigma^{2}n)^{-1}(\sigma^{4}a^{\ast }(x)a(-y)+xy)p^{n}(y-x)+ o(n^{-3/2}|y|\max(|x|,1)), \] for \(a_{0}^{-1}n^{1/2}<|x|,|y|<a_{0}n^{1/2}\), \[ q^{n}(x,y)=o(n^{-1/2}) \] plus, if \(xy>0\), \(d_{0}1_{{\mathbb C}\{0\}}(p^{n}(y-x))(g_{n}(y-x)-g_{n}(y+x))\) and, for \(0<\min(|x|,|y|)<n^{-1/2}<\max(|x|,|y|)\) and \(\operatorname{E}(|Y|^{2+\delta })<\infty\) for some \(\delta \geq 0\), \[ q^{n}(x,y)=O\left(\frac{\min(|x|,|y|)}{\max(|x|,|y|)}g_{4n}(\max(|x|,|y|))\right)+ o\left(\frac{\min(|x|,|y|)}{\max(|x|,|y|)^{2+\delta}}\right). \] In the case \(\operatorname{E}(|Y|^{3};Y<0)<\infty\), for \(y<0<x\), \(\max(|x|,|y|)\leq a_{0}n^{1/2}\), \(\min(|x|,|y|)\to \infty\), we have \[ q^{n}(x,y)=C^{+}\frac{|x|+|y|}{\sigma^{2}n}p^{n}(y-x)+o(n^{-3/2}\max(|x|,|y|)) \] where \(C^{+}=\lim_{x\to \infty }(\sigma^{2}a(x)-x)\).
The third is, for \(0<x,y\leq a_{0}n^{1/2}\), \(xy/n\to 0\), \[ q_{(-\infty ,0]}^{n}(x,y)= \frac{2f_{+}(x)f_{-}(y)}{\sigma^{2}n}p^{n}(y-x)(1+o(1)). \] The fourth result, appearing as an important step, is, if \(\operatorname{E}(|Y|^{2+\delta };Y<0)<\infty\), \(d_{0}=1\), when \(y\leq 0<x<a_{0}n^{1/2}\), \[ h(n,y)=n^{-1}f_{+}(x)g_{n}(x)H(y)(1+o(1))+x\alpha_{n}(x,y)n^{-3/2} \] with \(\alpha_{n}(x,y)=o(\max(|y|,n^{1/2})^{-1-\delta })\), \(\sum_{y\leq 0}|\alpha_{n}(x,y)|=o(n^{-\delta /2})\), \(\sum_{y\leq 0}|\alpha_{n}(x,y)|y|^{\delta }=o(1)\), and, when \(x\geq n^{1/2}\), \(y\leq 0\), \[ h(n,y)\leq C(n^{-1/2}g_{4n}(x)+o(x^{-(2 +\delta )}))H(y)+Cn^{-1/2}\operatorname{P}(Y<y-(x/2)). \] The fifth is \[ \lim_{n}\sum_{x\geq 1}\sum_{y\leq -1}q^{n}(x,y)=C^{+}/2. \] Different corollaries are also stated. The case when the absorption in \(0\) takes place only with a certain constant probability is considered.
In the last paragraph the author proves that, if \(\operatorname{E}(|Y|^{2+\delta })<\infty\), then \[ a(x)=(2\pi )^{-1}\int_{-\pi}^{\pi}Re(\frac{1-e^{ixr}}{1-\operatorname{E}(e^{irY})})\,dr \] and obtains estimates for \(\sigma^{2}a(x)-|x|\) when \(|x|\to \infty\).

MSC:

60G50 Sums of independent random variables; random walks
60J45 Probabilistic potential theory
60G51 Processes with independent increments; Lévy processes
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] K. L. Chung, A course in probability theory, Second edition, Probability and Mathematical Statistics, 21 , Academic Press, 1974. · Zbl 0345.60003
[2] W. Feller, An introduction to probability theory and its applications, 2 , John Wiley and Sons, Inc., 1966. · Zbl 0138.10207
[3] W. Hoeffding, On sequences of sums of independent random vectors, Fourth Berkeley Sympos. Math. Statist. and Prob., II , 1961, pp.,213-226. · Zbl 0211.20605
[4] H. Kesten, Sums of independent random variables without moment conditions, Ann. Math. Statist., 43 (1972), 701-732. · Zbl 0267.60053
[5] F. Spitzer, Principles of random walk, The University Series in Higher Mathematics, D Van Nostrand Co., Inc., Princeton, 1964. · Zbl 0119.34304
[6] E. C. Titchmarsh, The theory of functions, Second Edition, Oxford, 1939. · JFM 65.0302.01
[7] K. Uchiyama, Asymptotic estimates of Green functions and transition probabilities for Markov additive processes, Electron. J. Probab., 12 (2007), 138-180. · Zbl 1134.60055
[8] K. Uchiyama, The first hitting time of a single point for random walks, 2010, http://www.math.titech.ac.jp/ tosho/Preprints/index-j.html · Zbl 1245.60050
[9] K. Uchiyama, The Green functions of two dimensional random walks absorbed on a line, · Zbl 1226.60069
[10] K. Uchiyama, The random walks on the upper half plane, in preparation. · Zbl 1296.60119
[11] L. Alili and R. Doney, Wiener-Hoph factorization revisited and some applications, Stochastics Stochastics Rep., 66 (1999), 87-102. · Zbl 0928.60067
[12] A. Bryn-Jones and R. Doney, A functional limit theorem for random walk conditioned to stay positive, J. London Math. Soc. (2), 74 (2006), 244-258. · Zbl 1120.60047
[13] F. Caravenna, A local limit theorem for random walks conditioned to stay positive, Probab. Theory Rel. Fields, 133 (2005), 508-530. · Zbl 1080.60045
[14] R. A. Doney, Local behaviour of first passage probabilities, Probab. Theory Rel. Fields, DOI: 10.1007/s00440-010-0330-7
[15] V. A. Vatutin and V. Wachtel, Local probabilities for random walks conditioned to stay positive, Probab. Theory Rel. Fields, 143 (2009), 177-217. · Zbl 1158.60014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.