Ensemble fluctuations matter for variances of macroscopic variables

Extending recent work on stress fluctuations in complex fluids and amorphous solids we describe general terms the ensemble average $$v(\Delta t)$$ standard deviation $$\delta v(\Delta of variance $$v[{\mathbf {x}}]$$ time series $${\mathbf {x}}$$ a stochastic process x(t) measured over finite sampling $$\Delta t$$ . Assuming stationary, Gaussian ergodic process, v$$ is given by functional v_{\mathrm {G}}[h]$$ autocorrelation function h(t). shown to become large similar if corresponds fast relaxation process. Albeit v = \delta does not hold for non-ergodic systems, deviations common systems with many microstates are merely finite-size corrections. Various issues illustrated shear-stress simple coarse-grained model systems.