Random (Stochastic) Processes — Fundamentals

A stochastic (random) process is a family of random variables

$$\{X(t),\; t \in T\}$$

defined on a common probability space $(\Omega, \mathcal{F}, P)$ and indexed by a parameter $t$ belonging to an index set $T$.

Interpretations and terminology:

1. Fixed $t$, varying $\omega$: For a fixed time $t_0 \in T$, $X(t_0) = X(t_0, \omega)$ is an ordinary random variable on $(\Omega, \mathcal{F}, P)$.

2. Fixed $\omega$, varying $t$: For a fixed outcome $\omega_0 \in \Omega$, the function $t \mapsto X(t, \omega_0)$ is a deterministic function of time called a sample path (or realisation) of the process.

3. Both varying: The full object $X(t, \omega)$ is a function of two variables --- time and randomness.

Classification by index set:

Classification by state space:

• Continuous-valued: $X(t) \in \mathbb{R}$ or $\mathbb{C}$ (e.g., thermal noise voltage).
• Discrete-valued: $X(t) \in \{s_1, s_2, \ldots\}$ (e.g., the state of a Markov chain modelling a fading channel).

In this text, unless stated otherwise, $X(t)$ denotes a continuous-time, complex-valued stochastic process.

Let $\{X(t),\; t \in T\}$ be a stochastic process.

Mean function: $$\mu_X(t) = E[X(t)].$$

Autocorrelation function: $$R_X(t_1, t_2) = E[X(t_1)\,X^*(t_2)],$$ where $^*$ denotes complex conjugation.

Autocovariance function: $$C_X(t_1, t_2) = E\bigl[(X(t_1) - \mu_X(t_1))(X(t_2) - \mu_X(t_2))^*\bigr] = R_X(t_1, t_2) - \mu_X(t_1)\,\mu_X^*(t_2).$$

Cross-correlation between two processes $X(t)$ and $Y(t)$: $$R_{XY}(t_1, t_2) = E[X(t_1)\,Y^*(t_2)].$$

The autocorrelation function is the fundamental second-order descriptor of a stochastic process. It captures how the process at time $t_1$ is statistically related to the process at time $t_2$.

A stochastic process $\{X(t),\; t \in T\}$ is strict-sense stationary (SSS) if its complete statistical description is invariant under time shifts. Formally, for every positive integer $n$, every set of time instants $t_1, t_2, \ldots, t_n \in T$, and every time shift $\tau$ such that $t_1+\tau, \ldots, t_n+\tau \in T$, the joint distribution of $(X(t_1+\tau), X(t_2+\tau), \ldots, X(t_n+\tau))$ is identical to that of $(X(t_1), X(t_2), \ldots, X(t_n))$:

$$F_{X(t_1+\tau),\ldots,X(t_n+\tau)}(x_1,\ldots,x_n) = F_{X(t_1),\ldots,X(t_n)}(x_1,\ldots,x_n)$$

for all $(x_1, \ldots, x_n) \in \mathbb{R}^n$ (or $\mathbb{C}^n$) and all valid $\tau$.

Consequences of SSS:

• Setting $n = 1$: the first-order distribution $F_{X(t)}$ does not depend on $t$. In particular, $\mu_X(t) = \mu$ (constant mean) and $\mathrm{Var}(X(t))$ is constant.

• Setting $n = 2$: the joint distribution of $(X(t_1), X(t_2))$ depends only on the difference $t_1 - t_2$, not on the absolute times. Hence $R_X(t_1, t_2) = R_X(t_1 - t_2)$.

SSS is a very strong condition: it requires all finite-dimensional distributions to be time-invariant. In practice it is rarely verified directly, and the weaker notion of wide-sense stationarity is used instead.

A stochastic process $\{X(t)\}$ is wide-sense stationary (WSS) if it satisfies two conditions:

1. Constant mean: $$\mu_X(t) = E[X(t)] = \mu \quad \text{for all } t.$$

2. Autocorrelation depends only on the time difference (lag): $$R_X(t_1, t_2) = R_X(t_1 - t_2).$$ Writing $\tau = t_1 - t_2$: $$R_X(\tau) = E[X(t + \tau)\,X^*(t)] \quad \text{for all } t.$$

Key properties of the WSS autocorrelation function $R_X(\tau)$:

• $R_X(0) \geq 0$: Since $R_X(0) = E[|X(t)|^2]$ is the average power of the process.

• Maximum at the origin: $|R_X(\tau)| \leq R_X(0)$ for all $\tau$.

• Hermitian symmetry: $R_X(-\tau) = R_X^*(\tau)$. For real-valued processes this simplifies to $R_X(-\tau) = R_X(\tau)$ (even symmetry).

• Positive semidefiniteness: For any set of times $t_1, \ldots, t_n$ and complex coefficients $a_1, \ldots, a_n$: $$\sum_{i=1}^{n}\sum_{k=1}^{n} a_i\,a_k^*\,R_X(t_i - t_k) \geq 0.$$

Let $\{X(t)\}$ be a WSS process with autocorrelation function $R_X(\tau)$. The power spectral density (PSD) of $X(t)$ is the Fourier transform of $R_X(\tau)$:

$$S_X(f) = \int_{-\infty}^{\infty} R_X(\tau)\,e^{-j2\pi f\tau}\,d\tau.$$

Conversely, the autocorrelation function is recovered by the inverse Fourier transform:

$$R_X(\tau) = \int_{-\infty}^{\infty} S_X(f)\,e^{j2\pi f\tau}\,df.$$

Units: If $X(t)$ has units of volts (V), then $R_X(\tau)$ has units of $\mathrm{V}^2$ and $S_X(f)$ has units of $\mathrm{V}^2/\mathrm{Hz}$ (watts per hertz for a 1-ohm load).

Key properties of the PSD:

• $S_X(f) \geq 0$ for all $f$ (nonnegative).
• For real-valued processes: $S_X(-f) = S_X(f)$ (even symmetry).
• For complex-valued processes: $S_X(-f) = S_X^*(f)$ in general, but if the process is circularly symmetric, $S_X(f)$ is real and nonnegative.

A WSS stochastic process $\{X(t)\}$ is ergodic if time averages computed from a single, infinitely long sample path converge to the corresponding ensemble (statistical) averages.

Mean-ergodic: The process is mean-ergodic if the time-averaged mean converges (in mean square) to the ensemble mean: $$\langle X(t) \rangle_T \;\triangleq\; \frac{1}{2T}\int_{-T}^{T} X(t)\,dt \;\xrightarrow{T \to \infty}\; E[X(t)] = \mu.$$

Autocorrelation-ergodic: The process is autocorrelation-ergodic if $$\frac{1}{2T}\int_{-T}^{T} X(t+\tau)\,X^*(t)\,dt \;\xrightarrow{T \to \infty}\; R_X(\tau) \quad \text{for each } \tau.$$

A process that is both mean-ergodic and autocorrelation-ergodic is simply called ergodic (in the wide sense).

Sufficient condition for mean-ergodicity: $$\lim_{T \to \infty} \frac{1}{2T}\int_{-2T}^{2T} \left(1 - \frac{|\tau|}{2T}\right) C_X(\tau)\,d\tau = 0,$$ where $C_X(\tau) = R_X(\tau) - |\mu|^2$ is the autocovariance. In particular, if $C_X(\tau) \to 0$ as $|\tau| \to \infty$ (i.e., the process "forgets" its past), the process is mean-ergodic.