The fractional Gaussian noise/fractional Brownian motion framework (fGn/fBm) has been widely used for modeling and interpreting physiological and behavioral data. The concept of 1/

During the last decades, there has been a considerable interest in the use of stochastic fractal models for interpreting physiological or behavioral data. These models have been applied to various processes, including in the physiological domain heart-beat variability [

The most popular formalization of this approach refers to the concepts of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn), initially introduced by Mandelbrot and Van Ness [

The success of the fGn/fBm concept is mainly related to the presence, in some parts the model, of series possessing long-range correlation properties. Long-range correlations are characterized by a very slow decay of the autocorrelation function and suggest that the system possesses a long-term, multiscale memory of its previous states that affects its current behavior. Long-range correlations have been popularized through the concept of

Most analysis methods of long-range correlated processes are based on the statistical properties of fGn and fBm. However, we think that a number of issues about these processes remain unclear, often leading to erroneous interpretations. In this paper, we propose a formal analysis of the correlational properties of fBm and fGn, focusing on the limit behavior of these processes when approaching their definition boundaries. Considering the state of the art about this model, our main aim is to derive an analytical expression of the autocorrelation function of the discrete version of fBm.

Fractional Brownian motion (fBm), denoted by

In experimental and engineering applications, researchers often deal with sampled data, and such sampling leads to a discrete-time version of fBm,

By definition, a dfGn is the difference of a dfBm, and conversely the cumulative sum of dfGn gives a dfBm. Each dfBm series is then related to a specific dfGn, and both are characterized by the same

For

dfBm and dfGn are characterized by some essential basic properties [

Another self-similarity property characterizes the variance of a fixed lag difference between dfBm values:

The autocovariance function of a dfGn is given by [

The main aim of this paper is to derive an expression of the autocorrelation function of fBm in the discrete-time case. A well-known expression of the expected covariance of a continuous-time fBm series between two times

This autocovariance function depends explicitly on

The autocorrelation of lag

For simplifying (

In the simplest case

Developing the preceding equation, we get

And replacing

All terms containing

Using similar calculations for

Replacing

One can easily show that if (

Combining (

Equations (

One can easily show that

When

We present in Figures

Theoretical lag-one autocorrelation for dfGn (a), based on (

dfGn and dfBm were originally defined as two distinct families, which can be considered superimposed, with their relationships of summing/differencing. A number of authors, however, have proposed to consider these two families as a continuum, surrounding the mythical border of “ideal”

This conception has been favored by the existence of analysis methods which can be applied indifferently on both families and provide continuous metrics for characterizing the series. One of these methods is the

dfGn series are characterized by

In this methodological framework,

In contrast, our present results suggest a clear discontinuity between dfGn and dfBm, around this supposed

Theoretical autocorrelation functions, up to lag 30, for dfGn for

Stochastic fractal processes can also be defined in the frequency domain, on the basis of a scaling law that relates power (i.e., squared amplitude) to frequency according to an inverse power function, with an exponent

This scaling law defines a family of processes called

For

The present paper essentially questions the supposed equivalence between the dfGn/dfBm and the

Beyond its theoretical interest, this result has interesting implications in more applied perspectives. Fractal models currently find useful applications in biomedical engineering, especially for conceiving real-time monitoring devices allowing analyzing physiological fluctuations, for example, in brain activity, heart rate, or gait [

This paper highlighted some formal aspects of discrete-time fractional processes, which were rarely considered in the literature, which generally focuses on continuous processes. The derivation of expressions for dfBm variance (see (

Our main claim is that the dfGn/dfBm model cannot be considered a continuum. This point is clearly evidenced by the formal analysis of the correlation properties of the two classes of processes. dfGn converges towards a straight line process as

The author declares that there is no conflict of interests regarding the publication of this paper.