represents a fractional Gaussian noise process with drift μ, volatility σ, and Hurst index h.


represents a fractional Gaussian noise process with drift 0, volatility 1, and Hurst index h.


  • FractionalGaussianNoiseProcess is a continuous-time and continuous-state random process.
  • FractionalGaussianNoiseProcess is a Gaussian process with mean function and covariance function  sigma^2 (TemplateBox[{{t, -, s, -, 1}}, Abs]^(2 h)-2 TemplateBox[{{t, -, s}}, Abs]^(2 h)+TemplateBox[{{t, -, s, +, 1}}, Abs]^(2 h))/2.
  • FractionalGaussianNoiseProcess[μ,σ,h] is equivalent to TransformedProcess[x[t+1]-x[t],xFractionalBrownianMotionProcess[μ,σ,h],t].
  • FractionalGaussianNoiseProcess allows μ to be any real number, σ to be any positive real number, and h to be a real number between 0 and 1.
  • FractionalGaussianNoiseProcess can be used with such functions as RandomFunction and CovarianceFunction.


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Basic Examples  (3)

Simulate a fractional Gaussian noise process:

Plot the path:

Mean and variance functions are constant:

Covariance function:

Scope  (11)

Basic Uses  (6)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for various Hurst indices:

Process parameter estimation:

Correlation function as a function of lags:

Absolute correlation function:

Process Slice Properties  (5)

Univariate SliceDistribution:

First-order probability density function does not depend on time:

Compare with the density function of a normal distribution:

Multivariate slice distribution:

Slice distribution of higher order will not autoevaluate:

Second-order PDF:

Compute the expectation of an expression:

Calculate the probability of an event:

Skewness and kurtosis are constant:


Generating functions:

CentralMoment and its generating function:


Cumulant and its generating function:

Generalizations & Extensions  (1)

Useful shortcut evaluates to its full-form counterpart:

Applications  (1)

Consider the time series of yearly minimal water levels of the Nile River for the years 6221281:

Fit a fractional Gaussian noise process:

Compare means of the data and the model:

Compare covariance functions:

Simulate the values for the following 100 years:

Properties & Relations  (7)

FractionalGaussianNoiseProcess is weakly stationary:

Fractional Gaussian noise process is mean ergodic:

The process is weakly stationary:

Calculate absolute correlation function:

Find value of the strip integral:

Check if the limit of the integral is 0 to conclude mean ergodicity:

Fractional Gaussian noise does not have independent increments for :

Compare to the product of expectations:

Fractional Gaussian noise has a long memory for a Hurst parameter greater than 1/2:

Covariance function for a long memory process is not summable:

For a short memory, process is summable:

Conditional cumulative probability distribution:

Fractional Gaussian noise is self-similar:

Calculate scaled sums of each path:

Fit a normal distribution:

Compare to the slice distribution of the process:

Probability density histogram and functions:

Compare to non-weakly stationary FractionalBrownianMotionProcess:

Neat Examples  (3)

Simulate a fractional Gaussian noise process in two dimensions:

Study the behavior of fractional Gaussian noise depending on the Hurst parameter:

Simulate paths from a fractional Gaussian noise process:

Take a slice at 1 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 1:

Wolfram Research (2014), FractionalGaussianNoiseProcess, Wolfram Language function,


Wolfram Research (2014), FractionalGaussianNoiseProcess, Wolfram Language function,


Wolfram Language. 2014. "FractionalGaussianNoiseProcess." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2014). FractionalGaussianNoiseProcess. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_fractionalgaussiannoiseprocess, author="Wolfram Research", title="{FractionalGaussianNoiseProcess}", year="2014", howpublished="\url{}", note=[Accessed: 29-May-2024 ]}


@online{reference.wolfram_2024_fractionalgaussiannoiseprocess, organization={Wolfram Research}, title={FractionalGaussianNoiseProcess}, year={2014}, url={}, note=[Accessed: 29-May-2024 ]}