ss of stock $H^-\approx 0.51-0.52$. This means that intraday fluctuations of traded value are nearly uncorrelated.

2.  For long time fluctuations the data are correlated, but the strength of correlations depends strongly on the liquidity of the stock. As one moves to groups of larger $\ev{f}$, the strength of correlations ($H^+$) grows, up to $H^+\approx 0.8$.

3.  If one shuffles the time series, correlations are destroyed, and $H_{\rm shuff} = 0.5$.

The same phenomenon can be characterized by directly plotting the dependence of $H^\pm$ on $\ev{f}$, as done in Fig. \[fig:hurst\]. Such a dependence is well described by a logarithmic law: $$\begin{aligned}
H_i^{\pm} = H_0^\pm + \gamma^\pm \log \ev{f_i},
\label{eq:hurst_scaling}\end{aligned}$$ where $\gamma^-=0.00\pm0.01$, and $\gamma^+=0.053\pm0.01$. For the shuffled time series $\gamma_{\rm shuff}=0$.

These results indicate, at least in the case of traded value, the absence of universal behavior. Liquidity (or, analogously, company size) is a relevant quantity, which acts as a *continuous* parameter of empirical observables, in particular the strength of correlations and the distribution of $f$. Related results can be found in Refs. [@eisler.liquidity; @bonanno.dynsec; @ivanov.itt; @eisler.sizematters].

![The dependence of the scaling exponent $\alpha$ on the window size $\Delta t$ for the years $1994-1995$. The lighter shaded intervals have well-defined Hurst exponents and values of $\gamma$, the crossover is indicated with a darker background. Without shuffling ($\blacksquare$) there are two linear regimes: For shorter windows $\alpha = 0.74 \pm 0.02$, the slope is $\gamma^-=\gamma(\Delta t<60$ min$)=0.00\pm 0.01$ (solid line), while for longer windows $\alpha$ grows logaritmically, with a slope $\gamma^+=\gamma(\Delta t>390$ min$)=0.052\pm 0.01$ (dashed line). For shuffled data (O) the exponent is independent of window size, $\alpha (\Delta t)=0.74\pm0.02$.[]{data-label="fig:alpha"}](EislerFig4){width="255pt"}

Fluctuation scaling {#sec:alpha}
============