*1.4. Multifractality and Heavy Tails*

While a concave scaling function is necessary for multifractality, it is not sufficient. Several papers have explored other potential drivers of concavity in the scaling function. One is heavy-tailedness (Sly 2006; Heyde 2009 and Grahovac and Leonenko 2014). Sly (2006) has shown that, after removing the extreme values of the S&P500 price process, the scaling function approaches linearity. He proceeded further to study the asymptotic behaviour of the estimated scaling functions and proved the following theorem.

**Theorem 3** (Sly 2006)**.** *Let X be a random variable with* E(*X*) = 0 *such that the distribution function of* |*X*| *has a regularly varying tail of order* −*α where α* > 2*; that is,*

$$P(|X| > \mathbf{x}) = \mathbf{x}^{-a} L(\mathbf{x}),$$

*where <sup>L</sup>*(*x*) *is slowly varying. Then, for an i.i.d. sequence with distribution X and q* > *α, for each s* ∈ (0, <sup>1</sup>)*, with Sq as defined in Equation (4) and the time increment* Δ*t* = *ns,*

$$\frac{\ln \mathcal{S}\_{\emptyset}(n, n^s)}{\ln n} \stackrel{p}{\rightarrow} \max \left( s + \frac{q}{n} - 1, \frac{sq}{2} \right) \tag{8}$$

*as n* → <sup>∞</sup>*, where p*→ *stands for convergence in probability.*

<sup>3</sup> We compared the scaling functions of the Bitcoin and other financial assets with the ones of BMMTs simulated using multiplicative cascades with Poisson distribution, Gamma distribution and Normal distribution. The scaling function of the BMMT simulated through log-Normal multiplicative cascade displays the most similar behaviour.

Grahovac and Leonenko (2014) generalised this result to a type E set of stochastic processes and extended Sly's (2006) work by investigating the relationship between the asymptotic behaviour of scaling functions and tail indices.

**Definition 3** (*Type* E *stochastic processes*)**.** *A stochastic process* {*X*(*t*), *t* ≥ 0} *is said to be of type* E *if Y*(*t*) = *X*(*t*) − *X*(*t* − <sup>1</sup>)*, t* ∈ N*, is a strictly stationary sequence having heavy-tailed marginal distribution with index α, satisfying the strong mixing property with an exponentially decaying rate and such that* <sup>E</sup>*Y*(*t*) = 0 *when α* > 1*.*

**Theorem 4** (Grahovac and Leonenko 2014)**.** *Suppose* {*X*(*t*), 0 ≤ *t* ≤ *T*} *is of type* E *and suppose* Δ*ti is of the form T iN for i* = 1, ..., *N. Then, for every q* > 0 *and s* ∈ (0, 1)

$$\lim\_{N \to \infty} \mathop{\rm plim}\_{T \to \infty} \mathop{\rm plim}\_{T \to \infty} (q) = \tau\_{\infty}(q), \tag{9}$$

*where* plim *stands for limit in probability and*

$$
\pi\_{\infty}(q) = \begin{cases}
\frac{q}{a} & \text{if } 0 < q \le a \text{ and } a \le 2 \\
1 & \text{if } q > a \text{ and } a \le 2 \\
\frac{q}{2} & \text{if } 0 < q \le a \text{ and } a > 2 \\
\frac{q}{2} + \frac{2(a-q)^2(2a+4q-3aq)}{a^3(q-2)^2} & \text{if } q > a \text{ and } a > 2.
\end{cases} \tag{10}
$$

Besides the heavy-tailed effects, other factors are discussed in a number of papers. Matia et al. (2003) concludes that price fluctuations can contribute to concavity of scaling function. Bouchaud et al. (2000) states that another factor is the long range nature of the volatility correlations. Nevertheless, neither of these scenarios excludes the effect of heavy-tailedness. Therefore, we regard the heavy-tailedness as a main contributor to the possible misdetection of multifractality in this paper.

Figure 1 displays the situation considered in Theorem 4. To illustrate the heavy-tailed effect, the estimated scaling functions for both S&P500 Index and 20 Student *t*4-distributed processes before and after truncating heavy tails are shown in Figure 2.<sup>4</sup>

**Figure 1.** Asymptotic scaling functions for various values of *α*. The grey line is the *q*/2 reference line.

<sup>4</sup> The choice of Student's *t*-distribution with 4 degrees of freedom is suggested by the findings of Platen and Rendek (2008).

**Figure 2. Left**—Scaling Functions of the original S&P 500 open price data (black solid line), the S&P 500 open price data after removing 19 October 1987 (dotted line), the S&P 500 open price data after truncating any return larger than 4 standard deviations from the mean (grey solid line). **Right**—Scaling Functions of the Student *t*-distributed processes before (black lines) and after truncation (grey lines). For comparison, the reference lines *q*/2 (black dashed line) are included.
