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

Uchiyama, Kôhei

J. Math. Soc. Japan 63, No. 2, 675-713 (2011).

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 \(|x|\to \infty\).

Reviewer: Ion Cuculescu (Bucureşti)

MSC:

60G50	Sums of independent random variables; random walks
60J45	Probabilistic potential theory
60G51	Processes with independent increments; Lévy processes

Keywords:

random walk on \(\mathbb Z\); identically distributed random variables, absorption; harmonic function; asymptotics; period; moments; characteristic function

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