# 34 Gaussian Processes and Brownian Motion

- A stochastic process \(X(t)\) is a
**Gaussian process**if for any time points \(t_1, \ldots, t_n\) the process values \(X(t_1), \ldots, X(t_n)\) have a joint Gaussian (a.k.a. Multivariate Normal) distribution. - That is, a stochastic process \(X(t)\) is a Gaussian process if for any \(n\) and any time points \(t_1, \ldots, t_n\) any linear combination of the process values \(X(t_1), \ldots, X(t_n)\) has a Normal (Gaussian) distribution.
- In particular, at any time \(t\) the process value \(X(t)\) has a Normal (Gaussian) distribution.
- In particular, at any pair of times \(t\) and \(s\) the pair of process values \((X(t), X(s))\) has a Bivariate Normal (Bivariate Gaussian) distribution.
- The distributional properties of a Gaussian process are completely specified by its:
- mean function \(\mu_X(t)\), and
- autocovariance function \(C_{X}(s,t)\) (or autocorrelation function \(R_{X}(s,t)\)).

- If a Gaussian process is WSS then it is strict-sence stationary.
- Remember, WSS does not imply strict-sense stationary in general.

**Example 34.1 **The noise in a signal \(X(t)\) is modeled as a Gaussian process with mean 0.9 and autocovariance function \(C_{X}(t,t+\tau)=0.04e^{-0.1|\tau|}\).

Is \(X(t)\) a WSS process?

Specify the distribution of \(X(3)\).

Compute \(\text{P}(X(3) > 0.6)\).

Without doing any further calculations, specify the distribution of \(X(t)\), and \(\text{P}(X(t) > 0.6)\), for any \(t\). Explain.

Specify the joint distribution of \(X(3)\) and \(X(8)\).

Determine the probability that the noise at \(t = 3\) seconds is more than 0.5 above the noise at \(s= 8\) seconds.

Compute \(\text{P}(X(3) > 0.6 | X(8) = 0.5)\).

Without doing any calculations, find \(\text{P}(X(3) > 0.3 | X(100) = 0.6)\).

- A stochastic process \(\{B(t), t\ge 0\}\) is a
**Brownian motion**(process) (a.k.a. Wiener process) with scale parameter \(\alpha\) if:- \(B(0)=0\)
- For all \(t,\tau\ge 0\), \(B(t+\tau)-B(t) \sim N(0,\alpha\sqrt{\tau})\)
- So Brownian motion has
*stationary increments*

- So Brownian motion has
- For all \(t,\tau\ge 0\), \(B(t+\tau)-B(t)\) is independent of \(\{B(s), 0 \le s \le t\}\)
- So Brownian motion has
*independent increments*

- So Brownian motion has
- For each outcome, the sample path \(t\mapsto B(t)\) is a continuous
^{1}function of \(t\).

- A Brownian motion process is a Gaussian process with \[\begin{align*} \text{Mean function:} & & \text{E}(B(t)) & = 0\\ \text{Variance function:} & & \text{Var}(B(t)) & = \alpha^2 t\\ \text{Autocovariance function:} & & \text{Cov}(B(t), B(s)) & = \alpha^2 \min(t, s)\\ \end{align*}\]

*The stochastic process represented by Figure 29.1 and Figure 29.2 is a Brownian motion process with scale parameter \(\alpha=1\).*

**Example 34.2 **Brownian motion is often used to model the behavior of charge carriers (electrons and holes) in semiconductors. Suppose that for one particular semiconductor material, the motion of holes is a given direction (say, in microns per second) is Brownian motion with \(\alpha = 100\).

What does \(B(3)\) represent?

What does \(B(5)-B(2)\) represent?

Do the symbols \(B(3)\) and \(B(5)-B(2)\) denote the same random variable?

Does \(B(3)\) have the same distribution as \(B(5)\)? If not, state a quantity involving \(B(5)\) that does have the same distribution as \(B(3)\).

Is \(B(3)\) independent of \(B(5)\)? If not, state a quantity involving \(B(5)\) that is independent of \(B(3)\).

Is \(B(3)\) independent of \(B(5) - B(2)\)? If not, state a quantity involving \(B(5)\) that is independent of \(B(3)\).

Compute the probability that a carrier is more than 200 microns from its initial position, in either direction, after 3 seconds.

Compute the probability that at time 5 seconds a carrier is more than 200 microns away, in either direction, from its position at time 2 seconds.

- Let \(X_1,X_2, \ldots\), be independent with \(\text{P}(X_i=+1)=1/2\) and \(\text{P}(X_i=-1)=1/2\)
- Define \(S_0=0\) and \[ S_n = \sum_{i=1}^n X_i \]
- The discrete-time stochastic process \(\{S_n, n=0,1,2,\ldots\}\) is a
*symmetric simple random walk (RW)*. - Properties of a symmetric simple random walk \[\begin{align*} \text{Mean function:} & & \text{E}(S_n) & = 0\\ \text{Variance function:} & & \text{Var}(S_n) & = n\\ \text{Autocovariance function:} & & \text{Cov}(S_n, S_m) & = \min(n, m)\\ \end{align*}\]
- A Brownian motion process is a continuous — in time and in state — limit of a simple symmetric random walk.
- Continuous path: define for all \(t\ge0\) by “connecting the dots”
- Continuous in time: speed up time between steps by factor of \(n\). (Take a step at times \(1/n,2/n,\ldots\) instead of \(1,2,\ldots)\)
- Continuous in state: scale down size of step by factor
^{2}of \(\sqrt{n}\). (Size of step is \(\pm 1/\sqrt{n}\) instead of \(\pm 1\).) - Take limit as \(n\to\infty\) to get “continuous random walk”, which can be shown to be a Brownian motion process \[ \lim_{n\to\infty}\frac{1}{\sqrt{n}}S_{nt} = B(t) \]

Fun fact: while every path of Brownian motion is a continuous function of \(t\), it can be shown that every path of Brownian motion is not differentiable for any \(t\).↩︎

Note: \(\text{E}(|S_n|)=c\sqrt{n}\) so the magnitude of the value of the SSRW after \(n\) steps is on the order of \(\sqrt{n}\). The square root scaling is “just right”; with some other factor the process would either blow up or collapse to 0 in the limit.↩︎