On the multi-dimensional renewal theorem
Let \(\nu\) be the renewal measure of the random walk in \({\mathbb{R}}^ n\) with steps \(X_ i\) having moment vector (0,...,0,\(\mu)\), \(\mu>0\), and characteristic function f with \(\lim \inf \quad | 1-f(t)|>0\), \(\| t\| \to \infty\). It is known that under suitable moment conditions for sufficiently regular \(Q\subset {\mathbb{R}}^ n\) we have (*) \(\nu(x+Q)\sim \mu^{-1}| Q| f(x)\) as \(x_ n\to \infty\), uniformly in \(\{\| x'\|<cx_ n^{1/2}\}\) where \(x'=(x_ 1,...,x_{n-1})\). Here f(x) for fixed \(x_ n\) is the (n-1)-dimensional normal density with zero expectation vector and with covariance matrix having elements \(\mu^{-1}x_ nEX_{1i}X_{1j}\), \(i,j=1,...,n-1\). Under the assumption that \(E \exp(c\| X_ 1\|)<\infty\) for some \(c>0\) the authors prove by Fourier methods in \({\mathbb{C}}^ n\) that (*) still holds uniformly in any region \(\{\| x'\| \leq h(x_ n)\}\) with \(h(y)=o(y^{2/3})\), \(y\to \infty\), and that the asymptotic behaviour of \(\nu(x+Q)\) is different if \(x\to \infty\) such that \(\| x'\|<\alpha x_ n\) but \(\| x'\|\) is not \(o(x_ n^{2/3})\).
scientific article; zbMATH DE number 3864250