The exponential moment tail of inhomogeneous renewal process

Let θ1,θ2,… be a sequence of nonnegative not necessarily identically distributed and independent random variables having finite means and satisfying some additional conditions. We consider the asymptotic behavior of the quantity E(bΘ(t)1Θ(t)>at), where a and b are suitable positive constants, and Θ(t) is an inhomogeneous renewal process generated by the sequence θ1,θ2,…. We also present a few corollaries concerning elementary renewal theorems for the above process.