Asymptotic expansions in multivariate renewal theory
Let P be the probability distribution on \({\mathbb{R}}^{m+1}\) of a random vector \((X,Y)\in {\mathbb{R}}^{m+1}\) with \(X\in {\mathbb{R}}\), \(Y\in {\mathbb{R}}^ m\) and let \(R=\sum^{\infty}_{n=0}P^{n*}\) be the renewal measure associated with P. Let \(x\in {\mathbb{R}}\), \(y\in {\mathbb{R}}^ m\). When x is large, \(R(A+(x,y))\) is approximated by \(\hat R(A+(x,y))\), where the measure \(\hat R\) is the product of \(\nu^{-1}\) times Lebesgue measure on the x-axis and an m-dimensional Gaussian measure for the y-coordinates. Here \(\nu =E X>0\). Let \(F_ a\) be the set of positive Borel functions \(f: {\mathbb{R}}^{m+1}\to [0,1]\) with \(f(x,y)=0\) for \(x\not\in [a,a+1].\) The paper proves two estimates for \(\int fdR-\int fd\hat R\) as \(a\to \infty\), involving integrals of the oscillation of f over exponentially decreasing distance with coefficients bounded by order terms, uniformly with respect to \(f\in F_ a\). It is allowed that \(Y=(Y_ 1,Y_ 2)\) where \(Y_ 2\) is arithmetic and \(E \exp(is X+ip\cdot Y_ 1+iq\cdot Y_ 2)\) satisfies a strengthened version of Cramér's condition for \(| s| +\| p\| \to \infty\). There are some moment conditions and \(Y-\nu^{-1}X E Y\) is required to have a finite nonsingular covariance matrix. A corollary gives a global limit theorem for \(R([a,a+1]\times B)\). Proofs apply elaborate Fourier techniques. The paper extends previous results of the author for \(m=1\) in Probab. Theory Relat. Fields 80, No.1, 1-20 (1988; Zbl 0638.60053).
scientific article; zbMATH DE number 4138906
scientific article