  We develop a quasi-likelihood analysis procedure for a general class of
multivariate marked point processes. As a by-product of the general method, we
establish under stability and ergodicity conditions the local asymptotic
normality of the quasi-log likelihood, along with the convergence of moments of
quasi-likelihood and quasi-Bayesian estimators. To illustrate the general
approach, we then turn our attention to a class of multivariate marked Hawkes
processes with generalized exponential kernels, comprising among others the
so-called Erlang kernels. We provide explicit conditions on the kernel
functions and the mark dynamics under which a certain transformation of the
original process is Markovian and $V$-geometrically ergodic. We finally prove
that the latter result, which is of interest in its own right, constitutes the
key ingredient to show that the generalized exponential Hawkes process falls
under the scope of application of the quasi-likelihood analysis.
