
This is a two-part article. In part 2 (This part), I will go over random processes ( stochastic processes), their properties, and their response to the linear time-invariant (LTI) channel. In part 1, I discussed probabilities, random variables, and their properties. If you have not read the first part yet, please read it here first:
Comprehensive Overview of Random Variables, Random Processes, and Their Properties (Part 1)
The first part is fundamental to understand this part since random processes are the general extension of random variables and random vectors. Random variables and random processes play important roles in the real-world. They are used extensively in various fields such as Machine Learning, signal processing, digital communication, statistics, etc.
The following is the outline of this article:
Outline
- Review of Probability. (Part 1)
- Random Variable. (Part 1)
- Properties of Random Variable. (Part 1)
- Important Theorems in Probability. (Part 1)
- Random Vector. (Part 1)
- Random Process. (Part 2)
- Properties of Random Process. (Part 2)
- Different Classes of Random Process. (Part 2)
- Random Process Through LTI System. (Part 2)
- Important Random Processes in Machine Learning, AI, and Signal Processing. (Part 3)

Random Process
Random process X(w;t) is a time-varying function of t and w where the t is usually time and w is the element of the sample space (Ω). Therefore, There are two ways to look at the random process. Case 1: t (Time) is fixed, and w is changing:

If t is fixed to t = tᵢ, then we obtain a random variable for every value of i while w is changing. Therefore if w is changing, a random process is a collection of random variables. Case 2: w is fixed, and t is changing:

If w is fixed to wᵢ, then we get a time-domain signal or a realization. Therefore a random process is a deterministic time-domain function for a fixed w.
Observation 1: The randomness in a random process is due to w, not t. Hence, w is usually omitted in the definition of a random process (It is assumed that w is changing); thus, we sometimes referred to the random process X(w,t) as X(t).
Intuition: Consider the random process as a bag of time-domain signals (functions, realizations) every time you put your hand in the bag and grab one of the signals, but you do not know which one you are taking (That is where the randomness is coming from).
Observation 2: You can think of a random process as a general version of an n-dimensional random vector where n approaches infinity.
You can find more about random processes on my Youtube channel:
Properties of Random Process
A random process is described by some properties such as the mean, autocorrelation, cross-correlation, autocovariance, power spectral density, and average power. Mean, or the average of a random process X(t) (recall, w is missing in the notation since the assumption is that it is changing), is defined similarly to a random variable and a random vector (Please refer to Part 1).
Observation 3: Recall when w is changing, a random process is a random variable for each time t. Therefore at each point in time, the expectation of the random variable is the mean. The mean, in general, is a function of time.
Autocorrelation of a random process X(t) is defined as the correlation of the random process X(t) with itself (hence the word auto) at different points in time.

Autocovariance of a random process X(t) is defined as the covariance of the random process X(t) with itself (hence the word auto) at different points in time.
For the definition of the covariance of two random variables, refer to part 1 article (link is given above in the introduction).
The cross-correlation between random processes X(t) and Y(t) is a measure of the correlation between them at two different time points.
The power spectral density (PSD) is defined as the amount of power in each frequency band and is the Fourier transform of the autocorrelation function. I will discuss more about the PSD and the total power in the next section when discussing wide sense stationary (WSS) processes. If t takes any value in a real-line then X(t) is called continuous-time random process (a Stochastic Process) and if t is a member of a countable set, then X(t) is a random sequence. Observation 4: { X(t), t ∈ ℝ} is called continuous-time random process and {X(t), t ∈ ℕ} is called a discrete-time random process or a random sequence.
Different Classes of Random Process
Stationary Process
Process X(t) is said to be a stationary process if its statistical properties (CDF, PDF, etc) do not change over time. For example, X(t) is stationary if and only if:
Observation 5: What the above equation says is that the CDF of the stationary process remains the same over time for any t and any Δ.
Also for the stationary process, the joint distribution of X(t₁) and X(t₂) remains the same over time. Mathematically this is defined as follows:
The stationary process is also known as a strict sense stationary process (SSS) if for all t₁, t₂, . . . , tₙ and any Δ, the following condition is satisfied:
In general, it is very hard not impossible to prove if a process is SSS therefore we focus on weaker notation.
Weak/Wide Sense Stationary (WSS)
X(t) is a WSS process if for all t₁, t₂ ∈ ℝ and all Δ ∈ ℝ the following two conditions hold:
Observation 5: A process is WSS if its mean is constant and its autocorrelation is a function of the time lag (t₁- t₂). The time lag is usually denoted by τ.
From now on I focus on the WSS process since most real-world processes are modeled as WSS processes.
Please visit my YouTube channel for more contents about random processes:
Properties of the Autocorrelation of the WSS Process
1- It is the only function of the time lag.
2- It is an even function.
3- The autocorrelation at lag (τ) of zero gives the average power.
4- Maximum value of the autocorrelation occurs when the time lag is zero.
5- If X(t) is a real-valued random process then autocorrelation is a real-valued function.
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Jointly WSS Process
Two random processes X(t) and Y(t) are said to be jointly WSS if the following two conditions hold:
Observation 6: What the above two conditions say is that for processes X(t) and Y(t) to be jointly WSS then each one should be WSS (constant mean and the autocorrelation should be the function of the time lag only) and the cross-correlation (second condition) should be the function of time lag only.
Power Spectral Density (PSD)
PSD is defined as the Fourier transform of the autocorrelation of the WSS process. It is denoted by the symbol S.
Properties of PSD of the WSS Process
1- It is a real-valued function if X(t) is a real-valued random process.
2- It is an even function.
3- Area under PSD curve is the average power.
Property 3 is the most important one since it provides three different methods to calculate the average power of the random process.
Observation 7: The average power of a signal is computed by integrating the area under the PSD curve, calculating the autocorrelation function at the lag of zero, and calculating the expected square of the random process.
Random Process Through LTI (Linear Time-Invariant) System
A linear system is uniquely represented by the impulse response function h(t). If X(t)and Y(t) are input and output of linear system then Y(t) can be calculated as the convolution of X(t) and impulse response h(t).

If X(t) is a random process and is input to an LTI system then output Y(t) is also a random process.
Theorem 1: If X(t) is a WSS random process and is input to the LTI system with impulse response h(t) then X(t) and Y(t) are jointly WSS processes.
Since Y(t) is a WSS random process then its mean is constant and its autocorrelation is only the function of the time lag. The followings are the expressions for the mean, autocorrelation of Y(t), and the cross-correlation between X(t) and Y(t).
Observation 7: Since X(t) is a WSS process then it has a constant mean therefore the expected value of Y(t) is also constant (Equation 1). Equation 2 says that the cross-correlation between X(t) and Y(t) is only the function of the time lag. Equation 3 says that the autocorrelation of the output is only the function of the time lag. (Equations 1 and 3 claim that Y(t) is a WSS process and Equations 1,2 and 3 state that X(t) and Y(t) are jointly WSS processes.)
Frequency Domain Analysis
The above equations can be analyzed in the frequency domain given that H(f) is the frequency response (Fourier Transform of impulse response)of the LTI system, Sx(f), and Sy(f) (Fourier transform of the autocorrelation of the input and the output) are the power spectral densities of the input and the output respectively.
Observation 8: In finding the power spectral density of the joint X(t) and Y(t) (Equation 5), we use the property that H(-f) is the conjugate of H(f) (Fourier Transform properties.)
Conclusion
In this article, I discuss random processes, their properties, different classes of random processes, and random processes through the LTI system. In the next part (Part 3), I will go over Important Random Processes in Machine Learning, AI, and Signal Processing such as Markov chain, Gaussian Random process, hidden Markov model (HMM), etc.