A random walk in a quadratic random scenery (Q2758436)

Let \(X_1,X_2,\dots,\) be a sequence of i.i.d. \(Z^d\)-valued random vectors defined on a probability space \((\Omega,{\mathcal F},P)\) and let \(S_n=X_1+\cdots+X_n\) \((n\geq 0)\). A random scenery is a sequence \(\xi_\alpha\) \((\alpha\in Z^d)\) of independent symmetric \(R^m\)-valued random vectors defined on a probability space \((E,{\mathcal A},Q)\). Put NEWLINE\[NEWLINEG(t)=\sup_{x\neq y}Q(\|\xi_x\|\|\xi_y\|\geq t)\quad (t\geq 0)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEQ_n=\sum^n_{i=0}\sum^n_{j=0} a^{(n)}_{i,j} \xi_{S_i}\cdot \xi_{S_j} \quad (n\geq 0)NEWLINE\]NEWLINE for fixed real numbers \(a^{(n)}_{i,j}=a^{(n)}_{j,i}\), \(n,i,j=0,1,2,\dots\) (\(x\cdot y\) denoting the scalar product of \(x,y\in R^m\)). A random scenery is called a \(\delta\)-random scenery \((0<\delta<1)\) if, for some constant \(L>0\), \(t^{1+\delta}G(t)<L\), \(t>0\). The random walk \((S_n)\) is called \(\delta\)-universally representative \((0<\delta<1)\) if there exists \(\Omega_0\subset \Omega\) with \(P(\Omega_0)=1\) such that for every \(\omega\in \Omega_0\) the following holds: For every \(\delta\)-random scenery \((\xi_\alpha)\), \(Q_n(\omega)\) converges \(Q\)-a.s. to 0. It is assumed that the generating function \(U(\lambda)=u_0+\lambda^1u_1+\lambda^2u_2+\cdots\) \((0\leq \lambda\leq 1)\) of the sequence \(u_n=P(S_n=0)\) satisfies the condition NEWLINE\[NEWLINEU(\lambda)\sim (1-\lambda)^{-\rho}H((1-\lambda)^{-1})\quad \text{as }\lambda\uparrow 1\tag{*}NEWLINE\]NEWLINE for some constant \(0\leq \rho < 1\) and some function \(H\geq 0\) defined on \([0,\infty[\), varying slowly at infinity. Furthermore it is assumed that, for some constant \(M>0\), NEWLINE\[NEWLINE\sum^\infty_{i=0}\sum^\infty_{j=0} |a^{(n)}_{i,j}|^{1/2} \leq M.NEWLINE\]NEWLINE Finally, put \(a_n=\sup_{i,j}|a_{i,j}^{(n)}|.\) The author derives the following two results:NEWLINENEWLINENEWLINETheorem 1.1. Let \(0< \delta<1\). The random walk \((S_n)\) satisfying (*) is \(\delta\)-universally representative if, for some \(\rho<\beta<1\), \(\sum^\infty_{n=0}(a_nn^{2\beta})^{\delta/2}<\infty\).NEWLINENEWLINENEWLINETheorem 1.2. Let \(0<\delta<1\). The simple symmetric random walk on \(Z^d\) is \(\delta\)-universally representative if NEWLINE\[NEWLINE\begin{gathered}\sum_n(a_n\log\log n)^{\delta/2}<\infty\text{ if }d=1,\\ \sum_n(a_n(\log n)^4)^{\delta/2}<\infty\text{ if }d=2,\\ \sum_n(a_n(\log n)^2)^{\delta/2}<\infty\text{ if }d\geq 3.\end{gathered}NEWLINE\]NEWLINE{}.