In the above definition two separate requirements are mixed together as if they would be one. Here are the two independent requirements: (i) stationarity of the increments, x(t+T)-x(t)=x(T)-x(0) in distribution. This is the condition that yields longtime autocorrelations. (ii) Self-similarity of the stochastic process then yields variance scaling, but is not needed for longtime memory. E.g., both Markov processes (i.e., memory-free processes) and fractional Brownian motion scale at the level of 1-point densities (simple averages), but neither scales at the level of pair correlations or, correspondingly, the 2-point probability density.