  Progressive quenching (PQ) is the stochastic process in which the system's
degrees of freedom are sequentially fixed. While such process does not satisfy
the local detailed balance, it has been found that the some physical observable
of a complete spin network exhibits the martingale property. We studied
system's response to the perturbation given at intermediate stages of the PQ.
The response at the final stage reveals the persistent memory, and we show that
this persistence is a direct consequence of the martingale process behind. Not
only the mean response, the shape of the probability distribution at the stage
of perturbation is also memorized. Using the hidden martingale process we can
predict the final bimodal distribution from the early-stage unimodal
distribution in the regime where the unfrozen spins are paramagnetic. We
propose a viewpoint that the martingale property is a stochastic conservation
law which is supported behind by some stochastic invariance.
