  We study the long-time behavior of an additive functional that takes into
account the jumps of a symmetric Markov process. This process is assumed to be
observed through a biased observation scheme that includes the survival to
events of extinction and the Feynman-Kac weight by another similar additive
functional. Under conditioning for the convergence to a quasi-stationary
distribution and for two-sided estimates of the Feynmac-Kac semigroup to be
obtained, we shall discuss general assumptions on the symmetric Markov process.
For the law of additive functionals, we will prove a quasi-ergodic theorem,
namely a conditional version of the ergodic theorem and a conditional
functional weak law of large numbers. As an application, we also establish a
large deviation principle for the mean ratio of additive functionals.
