Bounds of memory strength for power-law series

Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents $alpha$. By permuting the independently drawn samples from a power-law distribution, we present non-trivial bounds on the memory strength (1st-order autocorrelation) as a function of $alpha$, which are markedly different from the ordinary $pm 1$ bounds for Gaussian or uniform distributions. When $1 < alpha leq 3$, as $alpha$ grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when $alpha > 3$, the upper bound remains +1 while the lower bound descends below 0. Theoretical bounds agree well with numerical simulations. Based on the posts on Twitter, ratings of MovieLens, calling records of the mobile operator Orange, and browsing behavior of Taobao, we find that empirical power-law distributed data produced by human activities obey such constraints. The present findings explain some observed constraints in bursty time series and scale-free networks, and challenge the validity of measures like autocorrelation and assortativity coefficient in heterogeneous systems.