  We consider discrete-time Markov bridges, chains whose initial and final
states coincide. We derive exact finite-time formulae for the joint probability
distributions of additive functionals of trajectories. We apply our theory to
time-integrated currents and frenesy of enzymatic reactions, which may include
absolutely irreversible transitions. We discuss the information that frenesy
carries about the currents and show that bridges may violate known uncertainty
relations in certain cases. Numerical simulations are in perfect agreement with
our theory.
