Anomalous Scaling of Stochastic Processes and the Moses Effect

TL;DR

This paper investigates the causes of anomalous scaling in stochastic processes, introducing the Moses effect as a new mechanism, and applies the analysis to financial data to identify its role in market fluctuations.

Contribution

The paper defines and relates the Moses effect to other known effects, providing methods to measure each and demonstrating their relevance in financial time series analysis.

Findings

Financial data's anomalous scaling is due to the Moses effect.

The Joseph exponent, not the Hurst exponent, tests market efficiency.

Methods enable independent measurement of scaling effects.

Abstract

The state of a stochastic process evolving over a time $t$ is typically assumed to lie on a normal distribution whose width scales like $t^{1/2}$. However, processes where the probability distribution is not normal and the scaling exponent differs from $\frac{1}{2}$ are known. The search for possible origins of such "anomalous" scaling and approaches to quantify them are the motivations for the work reported here. In processes with stationary increments, where the stochastic process is time-independent, auto-correlations between increments and infinite variance of increments can cause anomalous scaling. These sources have been referred to as the $\it{Joseph}$ $\it{effect}$ the $\it{Noah}$ $\it{effect}$, respectively. If the increments are non-stationary, then scaling of increments with $t$ can also lead to anomalous scaling, a mechanism we refer to as the $\it{Moses}$ $\it{effect}$. Scaling exponents quantifying the three effects are defined and related to the Hurst exponent that characterizes the overall scaling of the stochastic process. Methods of time series analysis that enable accurate independent measurement of each exponent are presented. Simple stochastic processes are used to illustrate each effect. Intraday Financial time series data is analyzed, revealing that its anomalous scaling is due only to the Moses effect. In the context of financial market data, we reiterate that the Joseph exponent, not the Hurst exponent, is the appropriate measure to test the efficient market hypothesis.

Tables

Table 1. Table 1: A list of abbreviations used in the text and their descriptions

Abbreviations	Process or Index	Description
SIP	stationary increment process	increments: identical
NIP	nonstationary increment process	increments: different
CLT	central limit theorem
R/S	re-scaled range statistics	Eqn. (7)
BM	Brownian motion	increments: independent, Gaussian distributed
LM	Lévy motion	increments: independent, Lévy distributed
FBM	fractional Brownian motion	increments: identical, correlated, Gaussian distributed
FLM	fractional Lévy motion	increments: identical, correlated, Lévy distributed
SBM	scaled Brownian motion	increments: scaled, independent, Gaussian distributed
SLM	scaled Lévy motion	increments: scaled, independent, Lévy distributed
SFBM	scaled fractional Brownian motion	increments: scaled, correlated, Gaussian distributed
SFLM	scaled fractional Lévy motion	increments: scaled, correlated, Lévy distributed
VDP	variable diffusion processes	Eqns. (18) and  (19)
DIA	Dow Jones Industrial Average
SPY	S&P500 Index
QQQ	PowerShare NASDAQ-100 Index

Table 2. Table 2: Estimates of the exponents for several classes of self similar processes using the form y ​ ( t ) = a ​ t Ω ′ + b ​ t Ω ′ − c 𝑦 𝑡 𝑎 superscript 𝑡 superscript Ω ′ 𝑏 superscript 𝑡 superscript Ω ′ 𝑐 y(t)=at^{\Omega^{\prime}}+bt^{\Omega^{\prime}-c} for finite-time corrections. Note that when J < 1 2 𝐽 1 2 J<{\frac{1}{2}} , it cannot be evaluated using the R/S method  Embrechts and Maejima ( 2002 ) .

Processes	$J$	$L$	$M$	$J+L+M-1$	$H$
BM	0.4997(5)	0.5000(1)	0.5000(1)	0.4997(5)	0.501(1)
SBM($M=0.3$)	0.4996(5)	0.5000(1)	0.3000(1)	0.2996(5)	0.300(3)
SBM($M=0.4$)	0.4999(5)	0.5000(1)	0.4000(1)	0.3999(5)	0.400(1)
SBM($M=0.6$)	0.4995(4)	0.5000(1)	0.6000(1)	0.5995(5)	0.601(1)
SBM($M=0.7$)	0.4993(4)	0.5000(1)	0.7000(1)	0.6993(5)	0.700(2)
LM($L=0.71$)	0.4998(2)	0.7139(6)	0.5001(2)	0.7139(6)	0.714(2)
LM($L=0.58$)	0.4996(3)	0.5883(2)	0.5000(1)	0.588(3)	0.588(3)
LM($L=0.53$)	0.5004(4)	0.5266(2)	0.5000(1)	0.5270(5)	0.5256(9)
SLM($L=0.53$, $M=0.3$)	0.4998(3)	0.5268(3)	0.3000(1)	0.3266(6)	0.3254(4)
SLM($L=0.53$, $M=0.4$)	0.4999(7)	0.5263(2)	0.4000(1)	0.4263(9)	0.426(2)
SLM($L=0.53$, $M=0.6$)	0.4989(6)	0.5265(2)	0.6000(1)	0.6253(8)	0.626(2)
SLM($L=0.53$, $M=0.7$)	0.5000(4)	0.5265(2)	0.7000(1)	0.7265(5)	0.726(1)
SLM($L=0.77$, $M=0.3$)	0.5000(2)	0.769(5)	0.3004(4)	0.569(5)	0.569(1)
SLM($L=0.77$, $M=0.4$)	0.4998(2)	0.768(2)	0.4009(5)	0.669(2)	0.669(2)
SLM($L=0.77$, $M=0.6$)	0.5002(2)	0.769(5)	0.6006(5)	0.870(4)	0.869(2)
SLM($L=0.77$, $M=0.7$)	0.4997(2)	0.768(4)	0.7006(5)	0.969(3)	0.970(1)
FBM($J=0.3$)	0.2994(5)	0.5000(1)	0.5000(1)	0.2994(5)	0.300(1)
FBM($J=0.4$)	0.4000(5)	0.5000(1)	0.5000(1)	0.3999(5)	0.400(1)
FBM($J=0.6$)	0.6001(3)	0.5000(1)	0.5000(1)	0.6001(3)	0.598(2)
FBM($J=0.7$)	0.6998(2)	0.5000(1)	0.5000(1)	0.6998(2)	0.700(1)
SFBM($J=0.3$, $M=0.3$)	0.2988(7)	0.5000(1)	0.3000(1)	0.0988(8)	0.097(3)
SFBM($J=0.3$, $M=0.4$)	0.3004(5)	0.5000(1)	0.4000(1)	0.2004(5)	0.201(2)
SFBM($J=0.3$, $M=0.6$)	0.2997(5)	0.5000(1)	0.6000(1)	0.3998(6)	0.400(2)
SFBM($J=0.3$, $M=0.7$)	0.2989(6)	0.5000(1)	0.7000(1)	0.4989(6)	0.500(1)
SFBM($J=0.4$, $M=0.3$)	0.3990(6)	0.5000(1)	0.3000(1)	0.1990(6)	0.2007(5)
SFBM($J=0.4$, $M=0.4$)	0.3991(6)	0.5000(1)	0.4000(1)	0.2991(6)	0.3006(4)
SFBM($J=0.4$, $M=0.6$)	0.3993(4)	0.5000(1)	0.6000(1)	0.4993(4)	0.500(1)
SFBM($J=0.4$, $M=0.7$)	0.3994(4)	0.5000(1)	0.7000(1)	0.59994(4)	0.6001(5)
SFBM($J=0.6$, $M=0.3$)	0.5998(5)	0.5000(1)	0.3000(1)	0.3998(5)	0.4004(6)
SFBM($J=0.6$, $M=0.4$)	0.5998(3)	0.5000(1)	0.4000(1)	0.4998(3)	0.5004(5)
SFBM($J=0.6$, $M=0.6$)	0.5998(2)	0.5000(1)	0.6000(1)	0.6998(2)	0.700(1)
SFBM($J=0.6$, $M=0.7$)	0.6000(3)	0.5000(1)	0.7000(1)	0.8000(3)	0.800(2)
SFBM($J=0.7$, $M=0.3$)	0.6989(8)	0.5000(1)	0.3000(1)	0.4989(8)	0.498(2)
SFBM($J=0.7$, $M=0.4$)	0.6999(3)	0.5000(1)	0.4000(1)	0.5999(3)	0.5987(7)
SFBM($J=0.7$, $M=0.6$)	0.7000(2)	0.5000(1)	0.6000(1)	0.7999(2)	0.7997(4)
SFBM($J=0.7$, $M=0.7$)	0.6999(1)	0.5000(1)	0.7000(1)	0.8999(2)	0.900(2)
FLM($J=0.4$, $L=0.60$)	-	0.600(4)	0.4999(1)	-	0.4991(5)
FLM($J=0.6$, $L=0.60$)	0.5996(3)	0.600(3)	0.4999(1)	0.699(3)	0.699(1)
FLM($J=0.4$, $L=0.56$)	-	0.5560(3)	0.4999(1)	-	0.4564(7)
FLM($J=0.6$, $L=0.56$)	0.5996(3)	0.5558(4)	0.5000(1)	0.6553(7)	0.6547(4)
SFLM($J=0.4$, $L=0.60$, $M=0.3$)	-	0.601(2)	0.2998(1)	-	0.300(1)
SFLM($J=0.4$, $L=0.60$, $M=0.4$)	-	0.600(3)	0.4000(1)	-	0.401(3)
SFLM($J=0.4$, $L=0.60$, $M=0.6$)	-	0.6003(4)	0.5999(1)	-	0.5995(4)
SFLM($J=0.4$, $L=0.60$, $M=0.7$)	-	0.6002(4)	0.6999(1)	-	.6992(9)
SFLM($J=0.6$, $L=0.60$, $M=0.3$)	0.5990(6)	0.602(3)	0.2998(2)	0.500(4)	0.499(1)
SFLM($J=0.6$, $L=0.60$, $M=0.4$)	0.6000(2)	0.600(2)	0.4001(1)	0.600(2)	0.600(1)
SFLM($J=0.6$, $L=0.60$, $M=0.6$)	0.5996(4)	0.6003(4)	0.5998(1)	0.7998(7)	0.798(1)
SFLM($J=0.6$, $L=0.60$, $M=0.7$)	0.6000(3)	0.5993(8)	0.7001(1)	0.900(4)	0.898(1)
VDP($H=0.3$)	0.4999(5)	0.498(4)	0.302(4)	0.300(5)	0.2996(7)
VDP($H=0.4$)	0.4999(5)	0.500(3)	0.400(3)	0.400(4)	0.4003(5)
VDP($H=0.6$)	0.4994(4)	0.500(2)	0.600(1)	0.599(2)	0.6008(4)
VDP($H=0.7$)	0.4993(4)	0.499(5)	0.701(4)	0.699(5)	0.6999(7)

Table 3. Table 3: Estimates for the exponents for exchange traded funds using the form y ​ ( t ) = a ​ t Ω ′ + b ​ t Ω ′ − c 𝑦 𝑡 𝑎 superscript 𝑡 superscript Ω ′ 𝑏 superscript 𝑡 superscript Ω ′ 𝑐 y(t)=at^{\Omega^{\prime}}+bt^{\Omega^{\prime}-c} for finite-time corrections. SPY, DIA, and QQQ are abbreviations for the Standard and Poor 500 Index, the Dow Jones Industrial Average, and the PowerShare NASDAQ-100 Index respectively.

Processes	$J$	$L$	$M$	$J+L+M-1$	$H$
SPY(30:190)	0.500(2)	0.503(4)	0.290(2)	0.294(4)	0.298(7)
DIA(30:190)	0.502(2)	0.497(6)	0.287(4)	0.286(4)	0.29(1)
QQQ(30:190)	0.500(2)	0.501(6)	0.295(2)	0.296(5)	0.29(1)
SPY(260:380)	0.500(2)	0.511(9)	0.552(4)	0.564(7)	0.55(1)
DIA(260:380)	0.499(1)	0.50(1)	0.552(2)	0.56(1)	0.55(1)
QQQ(260:380)	0.501(1)	0.50(1)	0.550(4)	0.56(1)	0.55(1)

Equations

$${\delta }_t(\tau )=X_{t+\tau }-X_t$$

$$X_t=\sum _{s=0}^{t-1}{\delta }_s,$$

$$Y_t=\sum _{s=0}^{t-1}|{\delta }_s|,$$

$$Z_t=\sum _{s=0}^{t-1}{\delta }_s^2.$$

$$X_{at}\stackrel{d}{=}{a}^H{X}_t$$

$$w[X_t]\sim t^H.$$

$$R_t$$

$$S_t^2$$

$$E\left[R_t/S_t\right]\sim t^J,$$

$$\underset{x\to \infty }{\lim }Pr(|{\delta }_t|>x)\sim x^{-\gamma },$$

$$m\left[Z_t\right]\sim t^{2L+2M-1}$$

$$E\left[\left|{\delta }_t-E({\delta }_t)\right|\right]\sim t^{M-\frac{1}{2}}$$

$$m\left[Y_t\right]\sim t^{M+\frac{1}{2}}.$$

$$H=J+L+M-1.$$

$${\mathcal{B}}_t^{(J,\frac{1}{2})}\propto$$

$${\mathcal{B}}_t^{(J,M)}={\int }_0^t{s}^{M-\frac{1}{2}}d{\mathcal{B}}_s^{(J,\frac{1}{2})},$$

$$E[\exp (i\theta \delta )]=\exp \left[-|\theta {|}^{(1/L)}\right].$$

$${\mathcal{L}}_t^{(J,L,\frac{1}{2})}\propto$$

$${\mathcal{L}}_{at}^{(J,L,\frac{1}{2})}\stackrel{d}{\propto }$$

$${\mathcal{L}}_t^{(J,L,M)}={\int }_0^t{s}^{M-\frac{1}{2}}d{\mathcal{L}}_s^{(J,L,\frac{1}{2})},$$

$$X=\frac{\sin (ϵ/L)}{(\cos ϵ{)}^L}{\left(\frac{\cos ((L-1)ϵ/L)}{\Phi }\right)}^{(L-1)/L}$$

$$W(X_t)=t^{-H}\mathcal{F}(u),$$

$$D(X_t,t)=t^{2H-1}\mathcal{D}(u).$$

$$\frac{\partial }{\partial (t)}W(X_t)=\frac{1}{2}\frac{{\partial }^2}{\partial Y^2}\left[D(X_t,t)W(X_t)\right].$$

$$\mathcal{F}(u)=\frac{C}{\mathcal{D}(u)}\exp \left(-2H\int \frac{udu}{\mathcal{D}(u)}\right)$$

$$X_t={\int }_0^t{s}^{H-\frac{1}{2}}\sqrt{\frac{2H}{{ϵ}^2}\left(1+ϵ\left|\frac{X_s}{s^H}\right|\right)}d{\mathcal{B}}_s.$$

$$y(t)/t^{\Omega }=a+b{t}^{-c},$$

$$y(t)=a{t}^{{\Omega }^{\prime }}+b{t}^{{\Omega }^{\prime }-c}$$

$$t(i)=round\left[t_{min}{\left(\frac{t_{max}}{t_{min}}\right)}^{\frac{i}{t_{\#}}}\right],$$

$$X_t=\ln \frac{P_t}{P_0}$$

$${\delta }_t=X_t-X_{t-1}=\ln \frac{P_t}{P_{t-1}}.$$

Full text

PACS numbers

02.50.Fz, 02.50.Ey, 89.65.Gh, 89.75.Da

††preprint: APS/123-QED

I Introduction

A stochastic process is a sequence of random variables indexed either continuously or discretely, through a parameter often interpreted as time. Stochastic processes have been used to model a range of phenomena from stock prices Black and Scholes (1973); Mandelbrot (1963); Mantegna and Stanley (1995) to precipitation levels Le Cam (1961); Richardson (1981); Wheater et al. (2005) and animal locomotion Viswanathan et al. (1999); Sims et al. (2008); Soibam et al. (2014). A standard example is Brownian motion, which is described by the Wiener process ${\cal B}_{t}$. It has Gaussian distributed increments ${\cal B}_{t+\tau}-{\cal B}_{t}$ that are uncorrelated and independent of $t$. The probability distribution of ${\cal B}_{t}$ is also Gaussian, but with a width that grows as $t^{1/2}$. Thus, ${\cal B}_{t}$, also referred to as “normal” diffusive motion Mandelbrot (2002), is said to scale as $t^{{\frac{1}{2}}}$. More generally, processes are found that scale as $t^{H}$, where $H$ is referred to as the self-affine exponent or Hurst exponent Embrechts and Maejima (2002).

Following experimental observations including in biological systems Iannaccone and Khokha (1995), financial markets Mantegna and Stanley (1995), and turbulence Castaing et al. (1989), it is of considerable interest to understand the nature of stochastic processes that scale anomalously. For example, $H\neq 1/2$ has been associated with the failure of the efficient market hypothesis (EMH) Malkiel and Fama (1970); Malkiel (2003), namely that asset prices do not fully reflect all pertinent information on the market Alvarez-Ramirez et al. (2008); Eom et al. (2008); Wang and Liu (2010).

For processes whose increments lie on stationary, time-independent probability distributions, Mandelbrot identified two causes of anomalous scaling, which he referred to as the Noah Effect and the Joseph Effect, and furthermore defined scaling exponents to characterize them Mandelbrot and Wallis (1968); Mandelbrot (2002); Eliazar and Shlesinger (2013). The Noah effect represents the occurance of large increments with anomalously large frequency resulting in a probability distribution with infinite variance. It is quantified by the latent exponent $L$; increment distributions with $L>1/2$ have “fat tails” and exhibit anomalous scaling. The Joseph effect occurs when increments are correlated and is quantified by the Joseph exponent $J$. When $J\neq 1/2$ increment correlations can result in anomalous scaling. Mandelbrot’s nomenclature are biblical references: Noah built an ark to save mankind and other creatures from the great flood bib (1982a), an occurrence of an anomalously large event, and Joseph, interpreting a dream of Pharaoh’s, counceled him concerning what he predicted would be a correlated sequence of years of abundance, followed by years of famine bib (1982b).

In this paper, we extend the characterization of scaling of stochastic processes to include those with non-stationary increments. Such processes can model the intraday prices of financial markets Bassler et al. (2007); Müller et al. (1990); Dacorogna et al. (1993), daily precipitation levels Kantelhardt et al. (2006, 2003); Brunsell (2010); Rehman (2009); Richardson (1981), the abundance of solar flares Lepreti et al. (2000); Oliver and Ballester (1998); Rehman (2009); Richardson (1981), and temperature fluctuations in turbulence Castaing et al. (1989); Rehman (2009); Richardson (1981). An additional mechanism that can produce anomalous scaling is identified. It is referred to as the Moses effect and characterized with the Moses exponent $M$, which quantifies the growth of the increment distribution. The nomenclature continues Mandlebrot’s tradition: Moses led the Israelites after their Exodus from Egypt as they wandered through the wilderness having no stationary settlements bib (1982c).

The many abbreviations used in the text are summarized in Table 1.

We will generalize the definitions of $H$, $L$, and $J$ and show that the sum of increments scale as $t^{H}$ with $H=L+J+M-1$. For normal diffusive processes with stationary increments $L=J=M=1/2$ and thus $H=1/2$. Anomalous scaling can occur when any of the exponents $L$, $J$, or $M$ differ from $1/2$. We argue the even for processes with a Moses effect, as we have previously discussed McCauley et al. (2008a), the EMH should be validated by measuring the Joseph exponent $J$, not the Hurst exponent $H$, since the EMH relates to the absence of correlations in market returns. A measurement of $J\neq 1/2$, indicating the presence of the Joseph effect, would violate the EMH, but anomalous scaling resulting from a combination of Noah or Moses effects can still be consistent with the EMH.

The principal result of the work is the decomposition of the overall scaling $H$ of the probability distributions into Joseph, Noah, and Moses effects via $H=J+L+M-1$. Accurate numerical methods for establishing the four scaling exponents independently will be presented. They account for finite-time corrections to the scaling behavior, and will be applied to standard and scaled versions of example stochastic processes to highlight the roles played by $H$, $L$, $J$, and $M$. Finally, exponents for a model of intraday trading in financial markets, variable diffusion processes Gunaratne et al. (2005); Bassler et al. (2006); McCauley et al. (2007, 2008b); Alejandro-Quiñones et al. (2006); Bassler et al. (2007); McCauley (2009); Seemann et al. (2012); Hua et al. (2015), will be computed and compared to the results with an empirical analysis of financial market data.

The paper is organized as follows. Definitions of the scaling exponents and methods to quantify them are given in Section 2. A collection of example stochastic processes that will be studied and the numerical methods to simulate them are given in Section 3. Section 4 presents the analysis of scaling in the various processes, demonstrating how the different effects can combine to yield an overall anomalous scaling. The methods used to accurately measure scaling indices are also presented in this section. In section 5, empirical financial market data is analyzed and compared to that of variable diffusion processes. The results are discussed in Section 6.

II Scaling in Stochastic Processes

Consider a one-dimensional stochastic process $\{X_{t};t\in\mathcal{T}\}$ where the set $\mathcal{T}$ can either be a subset of the real numbers or a subset of the integers. If the increments of the process

[TABLE]

have a probability distribution that is independent of $t$, then the stochastic process is said to be a stationary increment process (SIP). If instead, the probability distribution of $\delta_{t}(\tau)$ depends on $t$, the process is referred to as a non-stationary increment process (NIP). Statistical analyses of SIPs can be preformed on a single time series of data, as the time independence of the increments permits a statistical ensemble to be constructed by time translations of the starting point. Statistical analyses of NIPs, on the other hand, generally can not be performed on a single time series, requiring instead an ensemble of time series. If, however, a NIP begins anew at certain times, such as after a triggering event, then corresponding time translations may be used to construct a statistical ensemble from a single time-series McCauley (2008); Bassler et al. (2007); McCauley et al. (2008c). Although, this perhaps can only be done if the time between renewals has a finite average Sibani (2013), since, more generally, weak ergodicity breaking Bouchaud (1992); Bel and Barkai (2005) may prevent time averages being equated with ensemble averages in diffusive processes that scale anomalously.

The generalization of time series analyses to include NIPs necessitates generalizing the definitions of indices used to characterize scaling in SIPs. Consider an ensemble of realizations of a stochastic process $\mathcal{X}=\{X^{(p)}_{t},p=1,2,....\}$, each of which starts at the origin, i.e., $X^{(p)}_{0}=0$, and has increments $\delta_{t}^{(p)}(\tau)$. The increments are random variables with probability distributions that can depend on $t$, but are the same for all realizations $p$. These realizations can either have continuous or discrete time, but for the purposes of analyzing the scaling of the process assume that they are sampled at regular intervals of time $\tau$, which can be taken to be unity. The sampled times are then $t=1,2,3,\ldots$. Then

[TABLE]

where here and in what follows the $p$ superscript is suppressed for simplicity and $\delta_{t}=\delta_{t}(1)$. Define also the following random variables: the the sum of the absolute values of increments

[TABLE]

and the sum of increment squares

[TABLE]

Probability distributions of these variables and of $X_{t}$ over the ensemble $\mathcal{X}$ will be used to characterize and quantify the scaling of NIPs. The definitions of the scaling exponents that follow, which involve ensemble averages, i.e., over $p$, become equivalent to standard definitions for processes with stationary increments.

A stochastic processes $X$ is self-similar if, for any $a>0$, there exists an exponent $H\geq 0$ such that

[TABLE]

where “$\stackrel{{\scriptstyle d}}{{=}}$” represents equality “in distribution;” $H$ quantifies the scaling of the overall process. Note that, in general, only the one-point probability distribution $W(X_{t})$ scales in self-similar processes; higher-order, multi-point distributions do not necessarily scale. Operationally, scaling of a suitable “width” of the probability distribution of $X_{t}$ can be used to estimate $H$

[TABLE]

In this paper, we use the difference of the 75th quantile and the 25th quantile of the ensemble probability distribution as the measure of this width. For SIPs this definition of $H$ becomes equivalent to Mandelbrot’s Embrechts and Maejima (2002); Mandelbrot (2002); Eliazar and Shlesinger (2013) and can be estimated by a variety of means, including de-trended fluctuation analysis Peng et al. (1994). Using the quantiles circumvents difficulties in cases where moments of the probability distribution diverge.

Since $X_{t}$ is the sum of increments [Eq. (2)], it follows from the central limit theorem (CLT) that if the increment probability distributions are

• (a)

uncorrelated,

• (b)

have finite variance, and

• (c)

identical, independent of time,

then the process will scale “normally,” with $H=\frac{1}{2}$. Anomalous scaling in stochastic processes ($H\neq\frac{1}{2}$) may originate from the failure of one or more of these conditions. Joseph, Noah and Moses effects are associated with the failures of conditions (a), (b), and (c) respectively, which are analyzed in following subsections.

II.1 Joseph effect

The Joseph effect is associated with the failure of condition (a) and can be quantified through a variant of rescaled range statistics (R/S) Mandelbrot and Wallis (1969, 1968); Avram and Taqqu (2000); Mandelbrot (2002); Hurst (1951). Estimate the range $R_{t}$ and standard deviation $S_{t}$ of a stochastic process as

[TABLE]

Then the ensemble averaged ratio of $R_{t}$ and $S_{t}$ scales as

[TABLE]

where $J$ is the Joseph exponent. Negatively dependent processes have $J\in(0,1/2)$, positively dependent processes have $J\in(1/2,1)$, and independent processes have $J=1/2$.

II.2 Noah effect

The Noah effect refers to the failure of condition (b), and is quantified by the latent exponent $L$ Mandelbrot and Wallis (1968); Mandelbrot (2002); Eliazar and Shlesinger (2013). Suppose the tails of the increment distributions decay as

[TABLE]

where $\gamma\in(0,\infty)$. Then $L(t):=max(\frac{1}{2},\frac{1}{\gamma})$. In this paper, we only consider processes for which $L$ is independent of time. Note that CLT condition (b) fails when $L>\frac{1}{2}$, because the variance of $\delta_{t}$ is then infinite. If the increment distribution is Gaussian, log-normal or is any distribution with $\gamma\geq 2$, then $L=\frac{1}{2}$. If instead the increment distribution has fat tails with $\gamma<2$, then $L=\frac{1}{\gamma}$. We further limit our considerations to processes with $L<1$, as otherwise the increment distributions have infinite mean Samoradnitsky and Taqqu (1994).

For time series analyses, a more convenient and stable way to estimate the latent exponent is from the scaling of ensemble probability distribution of the sum of increment squares, which can be estimated by the scaling of the median of the probability distribution of $Z_{t}$

[TABLE]

where $M$ is Moses exponent introduced below. A proof for Eqn. (10) is given in the Supplemental Material at [URL will be inserted by publisher].

II.3 Moses effect

We define the failure of condition (c) as the Moses effect. It occurs when the increment distribution is time dependent. We consider processes with increment distributions whose mean absolute deviation scale as

[TABLE]

where, as before, $E$ is the ensemble average. For SIPs, $M=\frac{1}{2}$, whereas for NIPs, $M\neq\frac{1}{2}$.

In time series analyses, a convenient and more robust way to estimate the Moses exponent is from the scaling of the ensemble probability distribution of the sum of the absolute value of increments, which can be estimated by the scaling of the median of the probability distribution of $Y_{t}$

[TABLE]

A proof for Eq. (12) and a discussion of the effects on changes in the measurement frequency are given in the Supplemental Material at [URL will be inserted by publisher]. Typically, we find that varying the increment interval, $\tau$ in Eqn. 1, does not affect the leading scaling behavior of $Y_{t}$.

The anomalous scaling of NIPs can arise due to combination of all three effects listed above. Equations (6), (8), (10), and (12) provide independent estimates of the four exponents. However, they are related through

[TABLE]

This scaling relation provides a useful independent check of the estimates of the four exponents.

III Examples of Self similar processes

In this section several model stochastic processes are introduced that will be used to illustrate the relationship (13). However, we emphasize that the relationship is expected to be valid for other stochastic processes as well.

III.1 Processes with Gaussian Increments

The classic example is Brownian motion (BM) which consist of a sequence of identical, independent Gaussian increments Feller (1957). BM can be generalized by including (a) correlations between increments, and (b) time-dependent increments. One way to include long-term correlations is through fractional Brownian motion (FBM), denoted ${\cal B}^{(J,\frac{1}{2})}_{t}$, and defined below. These “correlations” can be characterized by an index $J$ which, as shown below, is the Joseph exponent. As for time-dependent increments, we limit consideration to processes whose increments scale in time. As shown below, the growth can be characterized by the Moses exponent $M$. A stochastic process consisting of correlated and scaled Gaussian increments is denoted ${\cal B}^{(J,M)}_{t}$. In this notation, Brownian motion ${\cal B}_{t}$ is denoted ${\cal B}^{(\frac{1}{2},\frac{1}{2})}_{t}$.

Fractional Brownian motions ${\cal B}^{(J,\frac{1}{2})}_{t}$ that scale in time have the form

[TABLE]

where the the proportionality constant is $1/{\Gamma(J+{\frac{1}{2}})}$ and $d{\cal B}_{s}\equiv{\cal B}_{s+ds}-{\cal B}_{s}={\cal B}_{ds}$. The Gaussian increments of FBM are correlated Mandelbrot (1975); Avram (1986), and the scaling exponents are derived as follows:

• (i)

Using Donsker’s theorem Donsker (1951) and continuous mapping therom Billingsley (2013); Whitt (2002), Avram and et al. proved that the index $J$ is the Joseph exponent Mandelbrot (1975); Avram (1986); Avram and Taqqu (2000).

• (ii)

Since individual increments are Gaussian-distributed, $L=1/2$.

• (iii)

Since FBM is a SIP, $M=\frac{1}{2}$.

• (iv)

To derive $H$, let $v=s/t$, then $ds=tdv$, $d{\cal B}_{s}\stackrel{{\scriptstyle d}}{{=}}{\cal B}_{ds}\stackrel{{\scriptstyle d}}{{=}}{\cal B}_{tdv}\stackrel{{\scriptstyle d}}{{=}}t^{\frac{1}{2}}d{\cal B}_{v}$. Thus, ${\cal B}^{(J,\frac{1}{2})}_{t}=\int_{-\infty}^{t}(t-s)^{J-\frac{1}{2}}d{\cal B}_{s}\stackrel{{\scriptstyle d}}{{=}}\int_{-\infty}^{1}(1-v)^{J-\frac{1}{2}}t^{J-\frac{1}{2}}t^{\frac{1}{2}}d{\cal B}_{v}\stackrel{{\scriptstyle d}}{{=}}t^{J}\int_{-\infty}^{1}(1-v)^{J-\frac{1}{2}}d{\cal B}_{v}\stackrel{{\scriptstyle d}}{{=}}t^{J}{\cal B}^{(J,\frac{1}{2})}_{1}$. From definition  (5) $H=J$.

Observe that $H=J+L+M-1$.

A scaled FBM is an NIP, except when $M=\frac{1}{2}$, and is defined as

[TABLE]

where $d{\cal B}^{(J,\frac{1}{2})}_{s}=d{\cal B}^{(J,\frac{1}{2})}_{s+ds}-d{\cal B}^{(J,\frac{1}{2})}_{s}$. Since increments $d{\cal B}^{(J,M)}_{t}=t^{M-\frac{1}{2}}d{\cal B}^{(J,\frac{1}{2})}_{t}$ and FBM is a SIP, $M$ is the Moses exponent of ${\cal B}^{(J,M)}_{t}$. Furthermore, since ${\cal B}^{(J,M)}_{t}=t^{M-\frac{1}{2}}{\cal B}^{(J,\frac{1}{2})}_{t}=t^{J+M-\frac{1}{2}}{\cal B}^{(J,\frac{1}{2})}_{1}$ [see (iv) above], the self-similarity exponent is $H=J+M-\frac{1}{2}$. Consequently, $H=J+L+M-1$.

Generation of FBM and scaled FBM: There are several methods proposed to generate FBM, three of them, the Hosking method Brockwell and Davis (2013), the Cholesky method Dieker (2004); Bardet et al. (2003), and the Davies-Harte method Davies and Harte (1987); Dietrich and Newsam (1997); Wood and Chan (1994), are exact. We used the Davies-Harte method, which requires $O(N\log\ N)$ operations, due to its computational efficiency. The algorithm is predicated on computing the square root of the covariance matrix using the circulant matrix and a fast Fourier transform instead of much slower lower-upper triangular decomposition Dieker (2004); Bardet et al. (2003).

III.2 Processes with Lévy Increments

Independent and identically distributed increments in the classic Lévy motions can be characterized by a single index $L$ Samoradnitsky and Taqqu (1994), which, as shown below, turns out to be the latent index. As before we can generalize the underlying process by including correlations and time-scaling of increments. The resulting process will be denoted ${\cal L}^{(J,L,M)}_{t}$.

Increments of the Lévy motion ${\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{t}$ are stochastic variates from a probability distribution whose characteristic function is Gnedenko et al. (1954); Uchaikin and Zolotarev (1999); Nolan (2015)

[TABLE]

The scaling exponents for Lévy motions are evaluated as follows:

• (i)

It has been shown that ${{\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{t}}/{t^{L}}\stackrel{{\scriptstyle d}}{{=}}{\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{1}$, where $t\geq 0$ Embrechts and Maejima (2002); Samoradnitsky and Taqqu (1994). Consequently, $R(t)\sim t^{L}$. For $L>{\frac{1}{2}}$, increments of LM have the property that $Pr(|\delta_{t}|>x)\sim x^{-1/L}$, which implies that $Pr(\delta_{t}^{2}>x^{2}=y)\sim x^{-1/L}=y^{-1/2L}$. Applying CLT for processes with infinite variance Uchaikin and Zolotarev (1999); Gnedenko et al. (1954), we obtain that $Z(t)\sim t^{2L}$, $S(t)=\sqrt{Z(t)/t}\sim t^{L-1/2}$. Thus, $E_{e}\left[R(t)/S(t)\right]\sim t^{1/2}$, and $J=\frac{1}{2}$ Mandelbrot (2002).

• (ii)

Increments satisfy $Pr(|\delta_{t}|>x)\sim x^{-1/L}$ implying that $L$ is the latent exponent.

• (iii)

For $\delta_{t}$ with $L<1$, the first order moment is bounded; since LM is a SIP, we have $E[|\delta_{t}-E(\delta_{t})|]\sim t^{0}$, therefore $M=\frac{1}{2}$.

• (iv)

$H=L$  Embrechts and Maejima (2002); Samoradnitsky and Taqqu (1994).

Note that $H=L+J+M-1$.

As with the Brownian case, correlations between the variates can be induced using the Fractional Lévy motion (FLM) defined as

[TABLE]

where $d{\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{s}={\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{s+ds}-{\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{s}={\cal L}^{({\frac{1}{2}},L,{\frac{1}{2}})}_{ds}$ and the proportionality constant is $1/{\Gamma(J+{\frac{1}{2}})}$. Only the exponents $J$ and $H$ of a FLM differ from those of the corresponding Lévy motion. The Joseph exponent cannot be defined using R/S statistics when $J<\frac{1}{2}$, since the process is nowhere bounded in that case Embrechts and Maejima (2002); Samoradnitsky and Taqqu (1994); Eliazar and Shlesinger (2013). Thus, we restrict consideration to $J\geq\frac{1}{2}$. Avram et al. proved that the index $J$ here is Joseph exponent Avram (1986); Avram and Taqqu (2000).

In order to estimate H, setting $s=av$, Eqn. (17) can be expressed as

[TABLE]

where the proportionality constant is the same as before. It follows that ${\cal L}^{(J,L,{\frac{1}{2}})}_{at}\stackrel{{\scriptstyle d}}{{\propto}}a^{J+L-{\frac{1}{2}}}{\cal L}^{(J,L,{\frac{1}{2}})}_{t}$, and hence that $H=J+L-{\frac{1}{2}}$.

Finally, time-dependent increments can be generated through scaled FLM is defined via

[TABLE]

where $d{\cal L}^{(J,L,{\frac{1}{2}})}_{s}={\cal L}^{(J,L,{\frac{1}{2}})}_{s+ds}-{\cal L}^{(J,L,{\frac{1}{2}})}_{s}$. Only the values of $M$ and $H$ of a scaled FBM differ from the corresponding exponents of the associated FBM. Specifically, using a calculation similar to that of $H$ for FBM, ${\cal L}^{(J,L,M)}_{t}\stackrel{{\scriptstyle d}}{{=}}t^{J+L+M-1}{\cal L}^{(J,L,M)}_{1}$, thus $H=J+L+M-1$.

Generation of Lévy-stable random variables. Let $\epsilon$ be a uniform random variate on $(-\frac{\pi}{2},\frac{\pi}{2})$ and let a random variable $\Phi$ be exponential with mean 1. Assume $\epsilon$ and $\Phi$ to be independent. Then the random variable

[TABLE]

is known to be distributed ${\cal L}({\frac{1}{2}},L,{\frac{1}{2}})$  Samoradnitsky and Taqqu (1994); DuMouchel (1971); Chambers et al. (1976).

The Davies-Harte method cannot be used to generate FLM variates since the associated correlation function does not exist. We used an approach introduced by Stoev and Wu Stoev and Taqqu (2004); Wu et al. (2004) that takes advantage of the circulant matrix and the fast Fourier transform to generate FLM. It requires $O(N\log N)$ operations. However, the algorithm can only simulate FLM approximately.

III.3 Variable Diffusion Process

Processes with Gaussian or Lévy increments, discussed above, have one common characteristic, that the increments are from a stable process and independent of the stochastic variable $X_{t}$. In this subsection we introduce a set of diffusive processes whose increments depend on $X_{t}$ as well. Variable diffusion processes (VDPs) was introduced as a model for intraday variations in financial markets Bassler et al. (2007); Seemann et al. (2012); Hua et al. (2015). Here $\{X_{t},t\geq 0\}$ satisfies the stochastic differential equation: $dX_{t}=\sqrt{D[X_{t},t]}d{\cal B}_{t}$, where $D(X_{t},t)$ is the diffusion coefficient. If the probability distribution function $W(X_{t})$ at time $t$ is self-similar (as given by Eqn. (5)),

[TABLE]

where the scaling variable is $u={X_{t}}/{t^{H}}$. Variable diffusion processes exhibit many stylized facts (i.e., common statistical features) reported in financial markets Gunaratne et al. (2005).

For variable diffusion processes with finite variance, Eqn. (18) shows that $E[X_{t}]\sim t^{H}$ and $E[X_{t}^{2}]\sim t^{2H}$, and hence that $Var[X_{t}]=t^{2H}Var[X_{1}]$. The self-similarity of the probability distribution implies further that the diffusion coefficient scales as Gunaratne et al. (2005)

[TABLE]

The probability distribution $W(X_{t})$ satisfies the Fokker-Planck equation:

[TABLE]

Using Eqns. (18) and  (19), we obtain $2H(uf(u))^{\prime}+(D(u)f(u))^{\prime\prime}=0$, whose solution is

[TABLE]

As an example, if ${\cal D}(u)$ is constant $D_{0}$, ${\cal F}(u)=C_{0}\exp\left(-\frac{1}{2D_{0}}u^{2}\right)$. If ${\cal D}(u)=D_{0}(1+\epsilon|u|)$ and $D_{0}=\frac{2H}{\epsilon^{2}}$, where $\epsilon$ is a constant, then ${\cal F}(u)=\frac{\epsilon}{2}\exp(-\epsilon|u|)$, the bi-exponential distribution. The corresponding variable diffusion process $\{X_{H}(t),t\geq 0\}$ is given by

[TABLE]

The associated exponents are:

• (i)

Since VDP is a Markov process $J=\frac{1}{2}$.

• (ii)

For VDP with finite variance $L=\frac{1}{2}$.

• (iii)

$dX_{t}=\sqrt{D(X_{t},t)}d{\cal B}_{t}=t^{H-\frac{1}{2}}\sqrt{{\cal D}(u)}d{\cal B}_{t}$, thus $M=H$.

• (iv)

Since $W(X_{t})=W(t^{H}X_{1})$, the Hurst exponent is $H$.

Once again, $H=J+L+M-1$.

IV Results From Simulations

IV.1 Finite-Size Corrections

Rescaled range statistics (R/S) analysis has been used extensively in studying persistence and long-term dependence in natural time series. The classical approach using the best linear relationship between $\log[E(R/S(t))]$ and $\log(t)$ yields a biased estimate unless $t$ is large Caccia et al. (1997); Bisaglia and Guégan (1998); Taqqu et al. (1995); Hamed (2007). Corresponding approaches to measure $L$, $M$, and $H$ also suffer from analogous finite-time corrections. Ref. Hamed (2007) showed that the first order finite-time corrections take the form

[TABLE]

where $a,b,c$ are constants. As an example, Figure 1 shows the nonlinear fit given by Eqn. (22) for a SFLM with parameters $J=0.6$, $\alpha=1.6666$, and $M=0.6$ for the (known) indices $J$, $2(L+M-{\frac{1}{2}})$, $M+{\frac{1}{2}}$, and $H$. Here, the reciprocal of the variance has been used as the weight of a point.

When exponents for a stochastic process are unknown (e.g., financial markets) we use the form

[TABLE]

for finite-time corrections, and estimate $\Omega^{\prime}$ as well. For the remainder of the paper, we use this approach to estimate the exponents $H$, $J$, $L$, and $M$.

IV.2 Exponents for different processes

For model stochastic processes, we use an ensemble of 100,000 stochastic realizations, each of length $t_{max}=1$ million. The lower cutoff is chosen to be $t_{min}=50$. We select $t_{\#}=500$ points between $t_{min}$ and $t_{max}$; the i-th point $t(i)$ ($i\in[1,2,...t_{\#}]$) is

[TABLE]

where $round[x]$ represents the integer nearest to $x$; these points are (approximately) uniformly distributed in log scale. $J$, $L+M$, $M$, $H$, estimated independently, are given in Table 2. The reported standard errors are obtained through a bootstrap method. The relation $H=L+M+J-1$ is found to hold for each process.

V Application to financial markets

V.1 Financial Markets Data

The data used for the analyses were one-minute valuations for the most actively traded exchange-traded funds (ETFs) in the US market extracted from PiTrading.com. We restrict consideration to the most recent 2500 trading days ($\sim$ 10 years). Intra-day trading is assumed to be a realization of the same stochastic process. The data provide open, close, high, and low prices within every minute; we use close price for our analysis. Missing data, perhaps due to technical problems or errors, are replaced by the last recorded price. We studied the three most traded ETFs, the Dow Jones Industrial Average (DIA), S$\&$P 500 (SPY), and PowerShare NASDAQ-100 (QQQ).

Stochastic processes underlying financial time series are represented using the return

[TABLE]

where $P_{t}$ is the price of a financial asset at time $t$, and $P_{0}$ is a reference price, typically the price at the start of a session. The one-minute increments are

[TABLE]

Note that $X_{t=0}=0$, which is the necessary condition for $X_{t}$ to scale, i.e. $X_{t}\stackrel{{\scriptstyle d}}{{=}}X_{1}$.

V.2 Scaling regions

Intraday seasonality. The analysis is predicated on the assumption that intra-day variations of the return follow a same stochastic process each day. Consequently, return data from the 2500 trading days constitute the ensemble. In order to eliminate any drift within the trading day, the data is “de-trended” by subtracting ensemble average $E_{e}(\delta_{t})$ at each $t$. The intraday pattern of $E_{e}(|\delta_{t}|)$ shows that the stochastic behavior is non-stationary within the day. $E_{e}(|\delta_{t}|)$ of SPY (as well as for the other ETFs) appears to scale as a power law within two intervals during the day, the first following the opening of the market and and the second in the afternoon, see Figure 2. The horizontal bars indicate the start and end of the two scaling intervals. The first ranges from 30 to 190 minutes and the second from 260 to 380 minutes from the start of the trading day.

V.3 Estimation of the exponents

The duration of the two scaling intervals are 180 and 120 minutes respectively. The lower cutoff $t_{min}$ for the finite-time analysis [Section 4(b)] was set to 10 minutes. The total number of points used for the analysis, with intervals given by Eqn. (24), is $t_{\#}=60$. Methods outlined in the previous section were used for the analysis and Figure 3 shows the nonlinear fit for SPY(30:190). The indices extracted from the analyses are given in Table 3, and conclusions include (1) $L\approx\frac{1}{2}$, implying that increments of the prices of ETFs are not from fat-tailed distributions, and (2) $J\approx\frac{1}{2}$ implying the absence of long-term memory. The latter is validated using the auto-correlation function which vanishes for time delays larger than 1 minute. Furthermore, the relation $H=J+L+M-1$ is validated.

The analyses outlined in Section IV were carried out for 100,000 ensembles of length 1 million. In contrast, financial market analysis was conducted for an ensemble of size 2500, and the intervals were 180 and 120 minutes respectively. One may inquire if the information extracted from these short processes through finite-size corrections are reliable. In order to address this issue, we re-computed the indices for variable diffusion process over a time interval $t_{max}$. As described below, $t_{max}$ is evaluated using the relaxation of the finite-time indices. Specifically, we write Eqn. (22) as

[TABLE]

where $\tau=(-\frac{b}{a})^{\frac{1}{c}}=0.18(1)$ is a time-scale for convergence of the index. In order to estimate $t_{max}$ for the VDP, we note that the nonlinear fit for $J$ for SPY(30:190) gives $\tau=0.42(5)$. The corresponding analysis of the VDP ($H=0.3$) should be of length $t_{max}=\frac{0.42}{0.18}*160\approx 370$, $t_{min}=\frac{0.42}{0.18}*10\approx 20$. Figure 4 shows the nonlinear fit for this VDP with $t_{max}=370$. The estimated exponents are $J=0.499(3)$, $L=0.497(6)$, $M=0.300(4)$, $J+L+M-1=0.297(5)$, $H=301(2)$. We thus infer that exponents computed for financial markets are reliable.

VI Conclusions

There are increasing numbers of examples, ranging from variations in biological systems Iannaccone and Khokha (1995) and thermal fluctuations in turbulence Castaing et al. (1989) to price variations in financial markets Mantegna and Stanley (1995), where probability distributions associated with a stochastic process exhibit anomalous scaling and are non-Gaussian. The anomalies can have different origins, and the goal of the work reported here is to disentangle them. Previous studies on stationary processes Mandelbrot (2002); Eliazar and Shlesinger (2013) had established that infinite variance of increments (Noah effect) and long-time correlations between increments (Joseph effect) are two sources of the anomaly. However, these studies failed to recognize that time-dependence of the increments themselves can also be a source of anomalous scaling. In this paper, we showed how scaling of the increments with time, referred to as the Moses effect, can also contribute to anomalous scaling. Noah, Joseph, and Moses effects, characterized by $L$, $J$ and $M$ respectively, are independent and the overall scaling of probability distributions, quantified by the Hurst exponent $H$, is given by $H=L+J+M-1$. As was emphasized, definitions of the exponents require the use of an ensemble of (nominally identical) stochastic trajectories when the underlying processes are time dependent.

Numerical approaches of time series analysis to accurately estimate each of the four scaling exponents independently were introduced. They are based on the use of medians and quantiles, which is especially appropriate for probability distributions lacking finite variance. These methods account for finite-time power-law corrections to scaling. The fact that the four indices can be measured independently allows the scaling relation that connects them to be verified, providing a stringent numerical check on the accuracy of the time-series analysis. The new numerical techniques were applied to a variety of different stochastic processes, including ones with both stationary and non-stationary increments, with and without long-time auto-correlations, and with both finite and infinite increment variance, to demonstrate the role of each effect toward anomalous scaling.

Financial time series of exchange-traded funds (ETFs) were analyzed as an application of the methods introduced here. As has been found to be the case for other financial markets Bassler et al. (2007); Seemann et al. (2012); Hua et al. (2015), the intraday prices of ETFs can be considered to be governed by non-stationary stochastic processes that repeat each trading day. We find two intervals where the underlying stochastic process scales anomalously with $H\neq 1/2$, which is often associated with a violation of the efficient market hypothesis (EMH). However, we find that $L=1/2$ and $J=1/2$; i.e., neither Noah nor Joseph effects are observed in financial markets. The deviation from $H=1/2$ results solely from the Moses effect ($M\neq 1/2$). Previously, it was proposed that the true test of the EMH should be the lack of correlations, i.e., $J=1/2$, and not $H=1/2$ McCauley et al. (2008a). Therefore, our analysis reiterates that ETF markets satisfy the EMH, despite the fact that they exhibit anomalous scaling. Finally, consistent with other recent studies of intraday trades in financial markets Bassler et al. (2007); McCauley et al. (2008b, c), we found that a variable diffusion process accurately models the scaling behavior in the two scaling intervals.

Although the scaling exponents defined here characterize the sources of anomalous scaling of the distribution functions, they do not uniquely identify higher order statistic or the underlying stochastic processes themselves. As an example, a scaled Brownian process and a variable diffusion process can have the same exponents as the first stage of stock markets (i.e., with $J={\frac{1}{2}}$, $H={\frac{1}{2}}$, and $M=0.3$). As discussed in the Supplemental Material at [URL will be inserted by publisher], VDP exhibits volatility clustering (i.e., the absolute values and the squares of increments exhibit long-time correlations) while the corresponding scaled BM does fails to do so. Volatility clustering is one of the well-known stylized facts on financial market dynamics Cont et al. (1997).

It would be interesting to apply the methods of time series analysis developed here to other, more physical, recurring stochastic processes with non-stationary increments. For example, the amount of daily precipitation recorded at a fixed location Kantelhardt et al. (2006, 2003); Brunsell (2010); Rehman (2009); Richardson (1981) may be amenable to such analysis. If underlying (stochastic) process is assumed to repeat itself each year, an ensemble can be constructed using the data for each year. Similarly, the daily, or monthly abundance of solar flares Lepreti et al. (2000); Oliver and Ballester (1998); Rehman (2009); Richardson (1981) may also be amenable to our methods of analysis. Solar activity is known to have an eleven year cycle, so that the days or months at the same phase each cycle may form an ensemble. The approach may also be useful for analyzing hard turbulence Castaing et al. (1989); here the temperature variation at a given location may be taken to be a stochastic process with non-stationary increments Rehman (2009); Richardson (1981). The process may then be considered to repeat after a non-periodic triggering event, such as a boundary layer separation Castaing et al. (1989). In this case, the temperatures at a given time following the triggering event form an ensemble. In each of these systems, it would be interesting to learn if the scaling is anomalous and, if so, which of the Noah, Joseph, and Moses effects, or what combination thereof, leads to the anomaly.

This work was supported by NSF-DMR-1507371 (KB) and NSF-IOS-1546858 (KB).