*2.2. Simulation Results*

We explore the possible range of behaviours for scaling functions for simulated multifractal processes. The findings are compared with some simulated monofractal processes (e.g., Brownian motions and fractional Brownian motions) and Student *t*-distributed processes. Both Brownian motion and fractional Brownian motion yield convex scaling functions only, which puts them out of consideration in the case of multifractality detection. The distributions of the concavity measures of the simulated multifractal processes and Student *t*-distributed processes are found to differ significantly in terms of their ranges and skewness. This provides us with a way to design a hypothesis test. As a result of these simulations, we construct a look-up table for a multifractality hypothesis test.

The simulated distributions of both the global and localised<sup>5</sup> simplex statistics are quite similar. Here, we only present the simulation results based on the global simplex statistics. The simulation results are shown in Figures 3 and 4. The distributions for the Student *t*-distributed process and a simulated multifractal process both span [−1, 1] but display different characteristics. A detailed comparison for the specific tail index = 3.06 is shown in Figure 4. For the simulated BMMTs, more than 20% of the scaling functions have global simplex statistics of exactly −1 indicating strict concavity. For the Student *t*-distributed process, the concavity statistic spans [−1, 1] with its median at around −0.36 in comparison to the one for the simulated BMMTs, −0.45.

**Figure 3.** Distribution of global simplex statistics for Brownian motion (**left**) and fractional Brownian motion (**right**).

**Figure 4.** Distributions of global simplex statistics for Student *t*-distributed process (black) and the simulated BMMT with log-normal multiplicative cascade (grey) around tail index 3.06. The two vertical lines represent the 95th percentile for the simulated BMMTs (**left**) and 5th percentiles for the Student *t*-distributed processes, respectively (**right**).

If the concavity of the observed scaling function for a given process is more severe than for the Student *t*-distributed processes, we are led to conclude that the concavity is driven by multifractality instead of heavy tails. Therefore, hypothesis tests can be constructed using the concavity measures as the test statistic.
