Simulate the telegraph process:

Mean and variance functions:

Covariance function:

The collision times for a particle moving between two barriers are distributed exponentially, with a mean of 3 microseconds. If the particle starts at the right-hand barrier () and moves towards the left-hand barrier (), then simulate the collision process for 100 microseconds:

Variance for the slice distribution:

Compare with the variance of a random sample:

TelegraphProcess is a jump process:

The telegraph process is not weakly stationary:

The absolute correlation depends only on time differences:

But the mean function is not constant:

It is, however, asymptotically weakly stationary:

The number of jumps at time t follows a PoissonDistribution:

Generate a random sample of paths from a telegraph process and record their lengths:

The times between jumps follow an ExponentialDistribution:

Generate a random sample of paths from a telegraph process:

Transition probability:

TelegraphProcess is a transformation of a PoissonProcess:

Simulate the process:

Probability density function for a time slice of the process:

Compare with the PDF for TelegraphProcess:

Compare CovarianceFunction:

Simulate paths from a telegraph process:

Take a slice at 20 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 20: