On the asymptotic behavior of mixed Poisson processes
A mixed Poisson process \(N(t)\) \((t>0)\) is considered which is a Poisson process with a random intensity \(\lambda\). The following formula is proved \[ \limsup_{t\to\infty}{N(t)-A(t)\over\sqrt{2t\log\log t}}=\sqrt\lambda, \] where \(A(t)\) is a compensator of the process \(N\) such that \(M(t)=N(t)-A(t)\) is a martingale. A transformation of an \(N\)-predictable process \(\varphi\) of the form \[ X_t(\varphi)=\int^t_0\varphi(s)dM(s)\quad(t\geq 0) \] is investigated. A law of iterated logarithms is proved for this process as \(t\to\infty\) and also an asymptotics of a variance of \(X_t(\varphi)\) is derived under assumption \(\varphi\) increases as \(t^\gamma\) \((\gamma\in[0,\infty))\).
scientific article; zbMATH DE number 1135556
scientific article