Simulate a fractional Gaussian noise process:

Plot the path:

Mean and variance functions are constant:

Covariance function:

Useful shortcut evaluates to its full-form counterpart:

Consider the time series of yearly minimal water levels of the Nile River for the years 622–1281:

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:

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:

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: