*1.1. Monofractal vs. Multifractal Processes*

The multifractal system was introduced as a generalisation of the fractal system to describe more complicated dynamics in time series. Multifractals are common in nature and have enjoyed grea<sup>t</sup> application in finance and science, e.g., modelling the turbulence in fluid dynamics (Sreenivasan 1991). In the finance field, multifractal processes are able to capture many stylised characteristics of high-volatility financial assets.

Multifractal processes are defined based on the scaling property of their moments, when they are finite, in Mandelbrot et al. (1997).

**Definition 1.** *(Multifractal Stochastic Process) A stochastic process* {*X*(*t*), *t* ≥ 0} *is multifractal if it has stationary increments and there exist functions c*(*q*) > 0 *and τ*(*q*) > 0 *and positive constants* Q *and* T *suchthat*

$$\mathbb{E}|X(t)|^q = \mathfrak{c}(q)t^{\tau(q)}, \text{for all } q \in [0, \mathbb{Q}] \text{ and } t \in [0, \mathcal{T}].\tag{1}$$

*The function τ*(*q*) *is called the scaling function.*

The multifractal processes are said to be *multiscaling* when the scaling function is nonlinear. For a multifractal process, the scaling function *τ*(*q*) is always concave when *q* > 0 on a bounded interval as is shown in Sly (2006).

In contrast, the multifractal process is called *uniscaling* or a monofractal process when the scaling function is linear, *τ*(*q*) = *Hq*. Monofractal processes enjoy a self-similarity.<sup>2</sup>

**Theorem 1** (Lamperti (1962))**.** *If* {*X*(*t*), *t* ≥ 0} *is self-similar and stochastically continuous at t = 0, then there exists a unique H* > 0 *such that for all a* > 0*,*

$$\{X(at)\} \stackrel{d}{=} \{a^H X(t)\}\tag{2}$$

*where equality is of finite dimensional distributions.*

<sup>1</sup> "False positive" corresponds to the scenario that we mistakenly detect multifractality when the underlying process does not possess the multifractal property.

<sup>2</sup> For convenience, in the rest of this paper, we call uniscaling multifractal processes *monofractal processes* while referring to multiscaling multifractal processes as *multifractal processes*.

Here, *H* ≥ 0 is a constant known as the *Hurst parameter*. *H* measures in some way the persistence of a process. Given all moments of the process are finite, the process is long-range dependent if *H* ∈ ( 12 , <sup>1</sup>), while anti-persistence occurs if *H* ∈ (0, 12 ) (Embrechts and Maejima 2000). Typical examples of monofractal processes are the fractional Brownian motions and the stable Lévy processes. The fractional Brownian motion *BH*(*t*) introduced in Mandelbrot et al. (1997) is a unique class of self-similar Gaussian process with stationary increments possessing a dependence structure described by the following covariance equation. When *H* = 12 , the fractional Brownian motion becomes Brownian motion with independent increments.
