  Conventional wisdom indicates that initial memory should decay away
exponentially in time for general (noncritial) equilibration processes. In
particular, time-integrated quantities such as heat are presumed to lose
initial memory in a sufficiently long-time limit. However, we show that the
large deviation function of time-integrated quantities may exhibit initial
memory effect even in the infinite-time limit, if the system is initially
prepared sufficiently far away from equilibrium. For a Brownian particle
dynamics, as an example, we found a sharp finite threshold rigorously, beyond
which the corresponding large deviation function contains everlasting initial
memory. The physical origin for this phenomenon is explored with an intuitive
argument and also from a toy model analysis.
