Interrelation between precisions on integrated currents and on recurrence times in Markov jump processes

For Markov jump processes on irreducible networks with a finite number of sites, we derive a general and explicit expression of the squared coefficient of variation for the net number of transitions from one site to a connected site in a given time window of observation (i.e., an 'integrated current' as dynamical output). Such an expression, which in itself is particularly useful for numerical calculations, is then elaborated to obtain the interrelation with the precision on the intrinsic timing of the recurrences of the forward and backward transitions. In biochemical ambits, such as enzyme catalysis and molecular motors, the precision on the timing is quantified by the so-called randomness parameter, and the above connection is established in the long time limit of monitoring and for an irreversible site-site transition; the present extension to finite time and reversibility adds a new dimension. Some kinetic and thermodynamic inequalities are also derived.