Ferkans — Interactive Telecom Tutor

Definition:
Standard Wiener Process (Brownian Motion)

The standard Wiener process (or standard Brownian motion) $\{W(t) : t \geq 0\}$ is a real-valued stochastic process satisfying:

$W(0) = 0$
Independent increments: For $0 \leq t_1 < t_2 < \cdots < t_N$ , the increments $W(t_2) - W(t_1), W(t_3) - W(t_2), \ldots, W(t_N) - W(t_{N-1})$ are mutually independent.
Gaussian increments: $W(t) - W(s) \sim \mathcal{N}(0, t - s)$ for $0 \leq s < t$ .
Continuous paths: $t \mapsto W(t)$ is continuous (almost surely).

Properties 1--3 uniquely determine the finite-dimensional distributions. Property 4 (path continuity) is an additional requirement that ensures the process has well-behaved sample paths. Its proof requires delicate measure-theoretic arguments (the Kolmogorov continuity criterion).

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Theorem: Covariance Function of the Wiener Process

The Wiener process $\{W(t)\}$ is a zero-mean Gaussian process with covariance function $C_W(s, t) = \mathbb{E}[W(s)W(t)] = \min(s, t), \quad s, t \geq 0.$

Write $W(s) = W(\min(s,t)) + [W(s) - W(\min(s,t))]$ . If $s \leq t$ , the second term is the increment $W(s) - W(s) = 0$ , and $\mathbb{E}[W(s)W(t)] = \mathbb{E}[W(s)(W(s) + (W(t)-W(s)))] = \mathbb{E}[W(s)^2] = s = \min(s,t)$ , since the increment $W(t)-W(s)$ is independent of $W(s)$ .

Proof

Assume $s \leq t$ without loss of generality

Decompose $W(t) = W(s) + (W(t) - W(s))$ . The increment $W(t) - W(s)$ is independent of $W(s)$ (by the independent increments property) and has zero mean.

Compute the covariance

$\mathbb{E}[W(s)W(t)] = \mathbb{E}[W(s)(W(s) + (W(t)-W(s)))]KATEXPLACEHOLDER0END= \mathbb{E}[W(s)^2] + \mathbb{E}[W(s)]\,\mathbb{E}[W(t)-W(s)]KATEXPLACEHOLDER1END= s + 0 = s = \min(s,t).$ $

Symmetry

If $t \leq s$ , the same argument gives $\min(s,t) = t$ . Since $\mathbb{E}[W(s)W(t)] = \mathbb{E}[W(t)W(s)]$ , the formula $\min(s,t)$ holds for all $s, t \geq 0$ . $\blacksquare$

The Wiener Process Is Not WSS

The variance $\text{Var}(W(t)) = \mathbb{E}[W(t)^2] = t$ grows linearly with time. Since the variance is not constant, the Wiener process is not wide-sense stationary. Equivalently, $C_W(s,t) = \min(s,t)$ depends on both $s$ and $t$ , not just on $s - t$ .

However, the increment process $\Delta W(t) = W(t + \Delta) - W(t)$ is stationary: its distribution depends only on $\Delta$ , not on $t$ .

Theorem: The Wiener Process Is Nowhere Differentiable

Almost surely, the sample paths of the Wiener process $W(t)$ are continuous but nowhere differentiable: for every $t \geq 0$ , $\limsup_{h \to 0} \frac{|W(t+h) - W(t)|}{|h|} = \infty \quad \text{a.s.}$

The increment $W(t+h) - W(t) \sim \mathcal{N}(0, h)$ , so its standard deviation is $\sqrt{h}$ . The "derivative" ratio $(W(t+h)-W(t))/h$ has standard deviation $1/\sqrt{h} \to \infty$ as $h \to 0$ . The paths fluctuate too wildly to have a derivative at any point.

Proof

Heuristic scaling argument

$\mathbb{E}\left[\left(\frac{W(t+h)-W(t)}{h}\right)^2\right] = \frac{1}{h} \to \infty$ as $h \to 0$ . While this alone does not prove almost-sure non-differentiability, it shows the "derivative" has infinite mean-square magnitude. The rigorous proof uses the Paley-Wiener-Zygmund theorem or direct analysis with Borel-Cantelli. $\blacksquare$

Example: Random Walk Converges to the Wiener Process

Let $Z_1, Z_2, \ldots$ be i.i.d. with $\mathbb{E}[Z_i] = 0$ and $\text{Var}(Z_i) = 1$ . Define the random walk $S_n = \sum_{i=1}^n Z_i$ and the rescaled process $W_n(t) = S_{\lfloor nt \rfloor}/\sqrt{n}$ (with linear interpolation between integer times). Show that $W_n(t) \to W(t)$ as $n \to \infty$ (Donsker's invariance principle).

Solution

Check finite-dimensional distributions

For fixed $t$ , $W_n(t) = S_{\lfloor nt \rfloor}/\sqrt{n}$ . By the CLT, $S_{\lfloor nt \rfloor}/\sqrt{\lfloor nt \rfloor} \xrightarrow{d} \mathcal{N}(0,1)$ , so $W_n(t) \xrightarrow{d} \mathcal{N}(0, t)$ , matching $W(t)$ .

Independent increments

The increments of $S_n$ over disjoint intervals are sums of independent RVs, hence independent. After rescaling, the increments of $W_n$ over disjoint intervals remain independent, matching the Wiener process.

Convergence of paths (Donsker's theorem)

The full functional convergence $W_n \Rightarrow W$ in $C[0,1]$ (the space of continuous functions with the sup norm) is Donsker's invariance principle (1951). This is the functional central limit theorem: the path-level generalization of the CLT. The proof requires tightness of the sequence $\{W_n\}$ , verified via moment bounds.

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Wiener Process Sample Paths

Visualize sample paths of the Wiener process. Observe the growing variance envelope $\pm c\sqrt{t}$ and the roughness (nowhere differentiability) of the paths.

Parameters

T

(time horizon)10

Number of paths10

Show

\pm 2\sqrt{t}

envelope

\Delta t

(step size)0.01

Random Walk Converging to the Wiener Process

Visualize rescaled random walks $S_{\lfloor nt \rfloor}/\sqrt{n}$ for increasing $n$ and compare with a Wiener process sample. Observe Donsker's invariance principle in action.

Parameters

n

(number of steps)100

Step distribution

Number of paths3

Definition:
Wiener Process with Drift and Diffusion

A Wiener process with drift $\mu$ and diffusion coefficient $\sigma$ is $X(t) = \mu t + \sigma W(t)$ where $W(t)$ is a standard Wiener process. Then:

$\mathbb{E}[X(t)] = \mu t$ (linear drift)
$\text{Var}(X(t)) = \sigma^2 t$
$X(t) - X(s) \sim \mathcal{N}(\mu(t-s), \sigma^2(t-s))$ for $s < t$

This is the simplest stochastic differential equation: $dX(t) = \mu\,dt + \sigma\,dW(t)$ .

Example: First Passage Time of the Wiener Process

For a standard Wiener process, compute the distribution of the first passage time $T_a = \inf\{t > 0 : W(t) = a\}$ for $a > 0$ .

Solution

Use the reflection principle

By the reflection principle for Brownian motion: $\mathbb{P}(T_a \leq t) = \mathbb{P}\left(\max_{0 \leq s \leq t} W(s) \geq a\right) = 2\,\mathbb{P}(W(t) \geq a) = 2\,Q\left(\frac{a}{\sqrt{t}}\right)$ where $Q(\cdot)$ is the Gaussian tail function.

Derive the PDF

Differentiating with respect to $t$ : $f_{T_a}(t) = \frac{a}{\sqrt{2\pi t^3}}\,\exp\left(-\frac{a^2}{2t}\right), \quad t > 0$ This is the inverse Gaussian (or Wald) distribution.

Compute the mean

$\mathbb{E}[T_a] = \int_0^{\infty} t\,f_{T_a}(t)\,dt = \infty$ . The first passage time has infinite mean: while $W(t)$ will almost surely reach $a$ (since $W(t) \to \pm\infty$ a.s.), the expected time to do so is infinite.

Why This Matters: Phase Noise in Oscillators

The local oscillator in a transceiver generates a carrier $\cos(2\pi f_0 t + \Phi(t))$ where the phase $\Phi(t)$ ideally should be deterministic. In practice, $\Phi(t)$ is a Wiener process: $\Phi(t) = \sigma_\Phi W(t)$ , where $\sigma_\Phi^2$ is the phase noise power per unit time. The variance $\text{Var}(\Phi(t)) = \sigma_\Phi^2 t$ grows without bound, causing the oscillator frequency to "wander." This phase noise limits the coherence time of OFDM systems and drives the need for pilot-based phase tracking. In 5G NR, common phase error (CPE) correction using phase-tracking reference signals (PT-RS) directly compensates for the Wiener model of phase noise.

Common Mistake: Do Not Treat the Wiener Process as WSS

Mistake:

Applying WSS tools (PSD, Wiener-Khinchin theorem) directly to the Wiener process $W(t)$ .

Correction:

The Wiener process is not WSS — its variance grows with time. It does not have a well-defined PSD in the standard sense. To use spectral methods, work with the increment process $dW(t)$ (which is white noise in a distributional sense) or analyze the process over finite windows. When the Wiener process models phase noise, the relevant quantity is the PSD of the frequency deviation $d\Phi/dt$ , which is white, not the PSD of $\Phi(t)$ itself.

🔧Engineering Note

Gaussian Process Regression in Channel Prediction

Gaussian process regression (GPR) uses the GP framework for nonparametric Bayesian estimation. Given noisy observations $y_i = f(t_i) + \epsilon_i$ with $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$ and a GP prior $f \sim \text{GP}(0, k)$ with kernel $k$ , the posterior is also a GP with mean and variance: $\mu_*(t) = \mathbf{k}_*^\mathsf{T}(\mathbf{K} + \sigma^2\mathbf{I})^{-1}\mathbf{y}$ $\sigma_*^2(t) = k(t,t) - \mathbf{k}_*^\mathsf{T}(\mathbf{K} + \sigma^2\mathbf{I})^{-1}\mathbf{k}_*$ where $[\mathbf{K}]_{ij} = k(t_i, t_j)$ and $[\mathbf{k}_*]_i = k(t, t_i)$ .

In wireless communications, GPR has been applied to channel prediction: given past channel measurements, predict future channel states by exploiting temporal correlation. The kernel encodes assumptions about the channel's correlation structure.

Historical Note: From Pollen Grains to Financial Markets: The Story of Brownian Motion

1827--1923

Robert Brown observed in 1827 that pollen grains suspended in water exhibit random, jittery motion. Albert Einstein's 1905 paper explained this as the result of molecular bombardment and predicted the $\sqrt{t}$ scaling of displacement. Jean Perrin's experimental verification earned him the 1926 Nobel Prize. Independently, Louis Bachelier used the same mathematical object in his 1900 thesis to model stock prices — predating Einstein by five years. Norbert Wiener provided the rigorous mathematical construction in 1923, proving the existence of continuous-path processes with the required properties. Today, the Wiener process is the foundation of stochastic calculus and appears in fields from physics to finance to communications.

Quick Check

What is $\text{Var}(W(5) - W(2))$ for a standard Wiener process?

$2$

$3$

$5$

$\sqrt{3}$

Correction:

3

By definition, $W(5) - W(2) \sim \mathcal{N}(0, 5-2) = \mathcal{N}(0, 3)$ , so $\text{Var} = 3$ .

Wiener Process (Brownian Motion)

A Gaussian process $\{W(t) : t \geq 0\}$ with $W(0) = 0$ , independent increments, $W(t) - W(s) \sim \mathcal{N}(0, t-s)$ , and continuous sample paths. Covariance: $\mathbb{E}[W(s)W(t)] = \min(s,t)$ .

Related: Gaussian Process, Random Walk

Random Walk

The partial sums $S_n = \sum_{i=1}^n Z_i$ of i.i.d. random variables $Z_i$ . When rescaled as $S_{\lfloor nt \rfloor}/\sqrt{n}$ , converges to the Wiener process (Donsker's invariance principle).

Key Takeaway

The Wiener process is the prototypical Gaussian process with independent increments and continuous paths. Its covariance $\min(s,t)$ and linearly growing variance make it non-stationary but fundamental: it models phase noise in oscillators, stock prices in finance, and diffusion in physics. The random walk limit (Donsker's theorem) shows that the Wiener process is universal — it emerges from any sum of i.i.d. increments, regardless of their distribution.

The Wiener Process (Brownian Motion)

Definition: Standard Wiener Process (Brownian Motion)

Theorem: Covariance Function of the Wiener Process

Assume $s \leq t$ without loss of generality

Compute the covariance

Symmetry

The Wiener Process Is Not WSS

Theorem: The Wiener Process Is Nowhere Differentiable

Heuristic scaling argument

Example: Random Walk Converges to the Wiener Process

Check finite-dimensional distributions

Independent increments

Convergence of paths (Donsker's theorem)

Wiener Process Sample Paths

Parameters

Random Walk Converging to the Wiener Process

Parameters

Definition: Wiener Process with Drift and Diffusion

Example: First Passage Time of the Wiener Process

Use the reflection principle

Derive the PDF

Compute the mean

Why This Matters: Phase Noise in Oscillators

Common Mistake: Do Not Treat the Wiener Process as WSS

Gaussian Process Regression in Channel Prediction

Historical Note: From Pollen Grains to Financial Markets: The Story of Brownian Motion

Quick Check

Wiener Process (Brownian Motion)

Random Walk

Key Takeaway

Definition:
Standard Wiener Process (Brownian Motion)

Definition:
Wiener Process with Drift and Diffusion