Some relations between harmonic renewal measures and certain first passage times
Let \(X_ 1,X_ 2,..\). be i.i.d. nondegenerate at 0 with E \(X_ 1=\mu \geq 0\). Let \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\), \(n\geq 1\) and \(\tau (t)=\inf \{n\geq 1:\;S_ n>t\},\) \(t\geq 0\). The author connects moments of \(\tau\) (t) to the harmonic renewal measure \(U(A)=\sum^{\infty}_{n=1}n^{-1}P(S_ n\in A).\) Combining ordinary renewal theory and Spitzer's theory of ladder variables he obtains a number of relations and estimates for U. Examples: \(U(t)=U((-\infty,t])=E \Psi (\tau (t))+\gamma\) when \(X_ 1\geq 0\), where \(\Psi (z)=d \log \Gamma (z)/dz\) and \(\gamma\) is Euler's constant. If \(\mu >0\), then \(0\leq U(t)- \log E \tau (t)<\gamma,\) \(t\geq 0\), giving \(U(t)-\log \mu^{-1}t-\gamma \to 0\) as \(t\to \infty\). If also \(\Delta =E (X^+_ 1)^ 2<\infty\), then \(0\leq U(t)-\log \mu^{-1}t\leq t^{-1}\Delta +\gamma\) for \(t\geq 0\). Inequalities and estimates for \(U([-s,t])\) for \(s\geq 0\), \(t\geq 0\) when \(\mu =0\) also are given. Asymptotics for \(U(\phi,t)=\sum^{\infty}_{n=1}\phi (n)P(S_ n\leq t)\) when \(X_ 1\geq 0\) are derived from the relation \(U(\phi,t)=E \Phi (t)\) where \(\Phi (t)=\int^{t}_{0}\phi (x)dx\) with \(\phi (x)=0\), \(x<1\), \(\phi (x)=\phi ([x])\), \(x\geq 1\).
scientific article; zbMATH DE number 4213169
scientific article