We like to walk on the comb
The paper presents simpler cases of results obtained by \textit{E. Csáki}, \textit{M. Csörgő}, \textit{A. Földes} and \textit{P. Révész} in the three papers [Electron.\ J.\ Probab.\ 14, 2371--2390, electronic only (2009; Zbl 1190.60020)], [Publ.\ Math.\ 76, No. ~3, 255--270 (2010; Zbl 1224.60191)] and [Stochastic Processes Appl.\ 121, No. ~6, 1290--1314 (2011; Zbl 1223.60025)]. A random walk is considered, \(C(n)=(C_{1}(n),C_{2}(n))\) on \({\mathbb Z}^{2}\), whose nonnull transition probabilities are \(p((x,y),(x,y\pm 1))=1/2\) for \(y\neq 0\), \(p((x,0),(x,\pm 1))=1/4\), \(p((x,0),(x\pm 1,0))=1/4\). The proof of the following main fact is given: for some independent Wiener processes \(W_{1}\), \(W_{2}\), if \(\eta_{2}\) is the local time of \(W_{2}\) then, for every \(\varepsilon >0\), when \(n\rightarrow \infty\), \(n^{-1/4}| C_{1}(n)-W_{1}(\eta_{2}(0,n))| + n^{-1/2}| C_{2}(n)-W_{2}(n) | =O(n^{\varepsilon -1/8})\) holds a.s. The presented results concerning \(W_{1}\), \(W_{2}\) are the following. As \(t\rightarrow \infty\), \(((W_{1}(xt)/b(t),W_{2}(xt)/b(t),\eta_{2}(0,xt)/b(t));0\leq x\leq 1)\), \(b(t)=(2t\log\log t)^{1/2}\), is almost surely relatively compact in the space \(C([0,1],{\mathbb R}^{3})\) and the set of its limit points \((f,g,h)\) is described by: absolutely continuous, null in \(0\), \(\int_{_{0}}^{^{1}}(\dot{f}^{2}(x)+\dot{g}^{2}(x)+\dot{h}^{2}(x))dx\leq 1\), \(g(x)\dot{h}(x)=0\) a.e., nondecreasing \(h\). A similar result with two components, the first \(W_{1}(\eta_{2}(0,xt))/b_{1}(t)\), the second \(W_{2}(xt)/b(t)\) and a set of limit points denoted by \(S^{(2)}\), is shown. For a nonincreasing \(\beta >0\), \(\liminf_{t\rightarrow \infty }sup_{0\leq s\leq t}| u_{j}(\eta_{2}(0,s))| /t^{1/4}\beta (t)=0\) or \(\infty\) when \(\int_{_{1}}^{^{\infty }}(\beta^{j}(t)/t)dt=\infty\), \(<\infty\) respectively, where \(j=1, 2\), \(u_{1}=\eta_{1}(0,\cdot)\), \(u_{2}=W_{1}\). As \(t\rightarrow \infty\), \(P(\sup_{x\in {\mathbb R}}\eta_{1}(x,\eta_{2}(0,t)) >t^{1/4}f(t)\) i.o.\()\) is \(0\) or \(1\) if \(\int_{_{1}}^{^{\infty}}(f^{2} (t)/t) h(t)dt\) is \(<\infty\), \(=\infty\) respectively, where \(h(t)=e^{-(3/2^{5/3})f^{4/3}(t)}\) and the same with \(<\) under \(P\), but with \(h(t)=1\). We list results involving \(C(n)\) and \(\Xi(x,n)=\text{card}\{0<k\leq n;C(k)=x\}\) (the local time of \(C\) in \(x\in {\mathbb Z}^{2}\)). \(((C_{1}(xn)/2^{3/4}n^{1/4}(\log\log n)^{3/4},C_{2}(xn)/b(n));0\leq x\leq 1)\) is, for \(n\rightarrow \infty\), almost surely relatively compact in the space \(C([0,1],R^{2})\) and its set of limit points is \(S^{(2)}\). For \(x=1\), the set of limit points is \(\cup_{K\in(0,1)} \{(u,v):F(| u| ,| v| ,K)\leq 1,0<| u| ,| v| <1\}\), \(F(B,A,K)=3B^{4/3}/2^{2/3} K^{1/3}+A^{2}/(1-K)\). For a non-increasing \(\beta >0\), \(\liminf_{n\rightarrow \infty }(\max_{0\leq k\leq n}C_{1}(k)) /n^{1/4}\beta (n)=0\) or \(\infty\) and \(\liminf_{n\rightarrow \infty } (\max_{0\leq k\leq n}C_{2}(k))/n^{1/2}\beta (n)=0\) or \(\infty\), these according as the series \(\sum_{n\geq 1}\beta (n)/n\) diverges or converges. \(\liminf_{n\rightarrow \infty} (\max_{0\leq k\leq n}| C_{1} (k)| ) /n^{1/4}\beta (n)=0\) or \(\infty\) a.s.\ according as the series \(\sum_{n\geq 1}\beta^{2}(n)/n\) diverges or converges. For any \(0<\varepsilon < 1/4\), a.s., \(\max_{| x| \leq n^{1/4 -\varepsilon }}| \Xi ((x,0),n)-2\eta_{1}(x,\eta_{2}(0,n))| = O(n^{1/8+\varepsilon })\) and \(\max_{0<| y| \leq n^{1/4-\varepsilon},| x| \leq n^{1/4-\varepsilon }}| \Xi ((x,y),n)-\eta_{1}(0,\eta_{2}(0,n))| =O(n^{1/8+\varepsilon })\). As \(n\rightarrow \infty\), for fixed \(x\) and \(y\) in \({\mathbb Z}^2\), \((\Xi ((x,0),zn)/2^{9/4}3^{-3/4}n^{1/4}(\log\log n)^{3/4};0\leq z\leq 1)\) and, for \(y\neq 0\), \((\Xi ((x,y),zn)/2^{5/4}3^{-3/4}n^{1/4} (\log\log n)^{3/4};0\leq z \leq 1)\), are a.s.\ relatively compact in the space \(C([0,1],R)\), and the set of their limit points is defined by \(f(0)=0\), \(f\) nondecreasing, \(\int_{_{0}}^{^{1}}| \dot{f}(x)| ^{4/3}dx\leq 1\). If \(a(n)\) is a non-decreasing sequence, then, as \(n\rightarrow \infty\), \(P(\sup_{x\in {\mathbb Z}}\Xi ((x,0),n)>n^{1/4}a(n)\) i.o.\()=0\) or \(1\) according as \(\sum_{n\geq 1}(a^{2}(n)/n)e^{-3a^{4/3}(n)/2^{5/3}}<\infty\) or \(=\infty\); the same for \(<\) under \(P\) and \(\sum_{n\geq 1} (a^{2}(n)/n)\). A more general setup is a \(C\) with \(p((x,y),(x,y+k)) =p_{2}(k)\) for \(y\neq 0\), \(p((x,0),(x,+ k))=p_{2}(k)/2\), \(p((x,0),(x+k,0)) =p_{1}(k)/4\), where \(p_{1}\), \(p_{2}\) are probability measures on \({\mathbb Z}\) with null means, finite third moments; let \(\sigma_{j}^{2}= \int k^{2}dp_{j}(k)\). If \(X_{n}^{j}\), \(j=1,2\), \(n\geq 1\), are independent, \(p_{j}\)-distributed, \(S_{n}^{j}=\sum_{k\leq n} X_{k}^{j}\) are aperiodic and \(\xi_{j}\) is the local time of \(S^{j}\) and if there exist two independent Wiener processes \(W_{1}\), \(W_{2}\) with local times \(\eta_{1}\), \(\eta_{2}\) such that, for some \(\alpha , \beta \in (0,1/2)\) and for every \(\varepsilon >0\) a.s., \(\lim_{n}n^{-\alpha -\varepsilon }| S_{j}(n)-\sigma_{j}W_{j}(n)| =0\), \(\lim_{n}n^{-\beta -\varepsilon}\sup_{x\in {\mathbb Z}}| \xi_{j}(x,n)-\sigma_{j}^{-2} \eta_{j} (x,n\sigma_{j}^{2})| =0\), then \(| C_{1}(n)-\sigma_{1}W_{1}(\sigma_{2}^{-2} \eta_{2} (0,n\sigma_{2}^{2}))| =O(n^{\varepsilon +\theta /2})\), \(| C_{2}(n)-\sigma_{2} W_{2}(n)| =O(n^{\varepsilon +\alpha^{+}})\), \(\alpha^{+}=\max(\alpha ,1/4)\), \(\theta =\max(\alpha^{+},\beta )\), \(\sup_{x\in {\mathbb Z}}| \Xi ((x,0),n)-2\sigma_{1}^{-} \eta_{1} (x,\sigma_{1}^{2}\sigma_{2}^{-2} \eta_{2}(0,\sigma_{2}^{2} n))| = O(n^{\varepsilon +\max(\beta ,1/4)/2})\). The paper contains also consequences and the statements of many results (with corresponding quotations) related to or implying some of the above.
scientific article; zbMATH DE number 5990589
scientific article