Summary

Random processes are very useful tools for analyzing random variables over time. In this chapter, we have introduced some of the most basic mechanisms:

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• Statistical versus temporal analysis: The statistical analysis of a random process looks at the random process vertically. It treats \(X(t)\) as a random variable and studies the randomness across different realizations. The temporal analysis is the horizontal perspective. It treats \(X(t)\) as a function in time with a fixed random index. In general, statistical average \(\not=\) temporal average.
• Mean function \(\mu_X(t)\): The mean function is the expectation of the random process. At every time \(t\), we take the expectation to obtain the expected value \(\E[X(t)]\).
• Autocorrelation function \(R_X(t_1,t_2)\): This is the joint expectation of the random process at two different time instants \(t_1\) and \(t_2\). The corresponding values \(X(t_1)\) and \(X(t_2)\) are two random variables, and so the joint expectation measures how correlated these two variables are.
• Wide-sense stationary (WSS): This is a special class of random processes in which \(\mu_X(t)\) is a constant and \(R_X(t_1,t_2)\) is a function of \(t_1-t_2\). When this happens, the autocorrelation function (which is originally a 2D function) will have a Toeplitz structure. We write \(R_X(t_1,t_2)\) as \(R_X(\tau)\), where \(\tau = t_1-t_2\).
• Power spectral density (PSD): This is the Fourier transform of the autocorrelation function \(R_X(\tau)\), according to the Einstein-Wiener-Khinchin theorem. It is called the power spectral density because we can integrate it in the Fourier space to retrieve the power. This provides us with some convenient computational tools for analyzing data.
• Random process through a linear time-invariant (LTI) system: This tells us how a random process behaves after going through an LTI system. The analysis can be done at the realization level, where we look at each random process, or at the statistical level, where we look at the autocorrelation function and the PSD.
• Optimal linear filter: A set of techniques that can be used to retrieve signals by using the statistical information of the data and the system. We introduced two specific approaches: the Yule-Walker equation for a finite-length filter and the Wiener filter for an infinite-length filter. We demonstrated how these techniques could be applied to forecast a time series and recover a time series from corrupted measurements.

While we have covered some of the most basic ideas in random processes, there are also several topics we have not discussed. These include, but are not limited to: strictly stationary process, a more restrictive class of random process than WSS; Poisson process, a useful model for arrival analysis; Markov chain, a discrete-time random process where the current state only depends on the previous state. Readers interested in these materials should consult the references listed at the end of this chapter.