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This paper discusses the estimation of autocorrelation function (ACF) of fractional Gaussian noise (fGn) with long-range dependence (LRD). A variance bound of ACF estimation of one block of fGn with LRD for a given value of the Hurst parameter (

ACF analysis, or equivalently spectral analysis according to the Wiener-Khintchine theorem, plays a role in many areas of sciences and technologies (see, e.g., [

The literature of error analysis (mainly, bias, and variance) of ACF/PSD estimation of an ordinary random process is quite rich; see, for example, [

Note that processes with LRD or long-memory substantially differ from ordinary processes [

In the field, [

An ACF is usually estimated on a block-by-block basis [

Intuitively, if the size of one block is large enough, the ACF estimation will be independent of the start point for sectioning the block. Let

The remaining article is organized as follows. Section

Let

Following Mandelbrot and van Ness [

FGn is the increment process of fBm. It is stationary and self-affine with parameter

Recall that a stationary Gaussian process with ACF

Note that the expression 0.5[(^{2H } − 2^{2H } + (^{2H }] described in (^{2H }. Approximating it with the second-order differential of 0.5

FGn contains three subclasses of time series. In the case of 0.5 <

Note that if

Now, in the case of

Figures

Plots of ACF of fractional Gaussian noise. (a) ACF of fGn with LRD. Solid line is for

In practical terms, the number of measured data points within a sample of fGn is finite. Let a positive integer

Let

Let

Mathematically,

Let

Let

As Var(

Since ACF is an even function, the above expression is written by

Now, replacing

The above formula represents an upper bound of Var(

Error bound

From Figure

Recall that processes with LRD substantiality differ from those with SRD [

Suppose that we have a block of fGn with

Case study. Solid line: theoretic ACF of fGn. Dot line: ACF estimate. (a) FGn with

Assume the block size

Now we increase the block size such that

Solid line: theoretic ACF of FGN. Dot line: ACF estimate. (a) ACF estimate of fGn with

From the above, one sees that the accuracy of ACF estimate of fGn with LRD can be increased if the block size increases. Therefore, in addition to the direct way to increase the record length, increasing the sampling rate in measurement of fGn to be processed may yet be a way to increase the accuracy of the ACF estimation in the case that the block size is given.

The previous discussions regarding ACF estimation of fGn with LRD do not relate to averaging. In fact, once the block size

In the field of fractional order signal processing (see, e.g., [

Finally, we note that the ACF estimate expressed by (

We have established an error bound of ACF estimation of one block of fGn with LRD. It has been shown that the error does not depend on the absolute length of the sample but only relies on the number of data points, that is, the block size

This work was supported in part by the National Natural Science Foundation of China under the project Grant nos. 60573125, 60873264, and 60873102.