  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.
