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- The following topics will be covered in this lecture:
- Extending the concept of random vectors – stochastic processes
- Continiuity in probability
- Separability
- Gaussian stochastic processes
- Wiener Processes

We are now familiar with the

**Gaussian model for a random state**, and its**centrality due to the central limit theorem**.- Likewise, we have developed a variety of tools to extend the Gaussian model, and to efficiently simulate it numerically.

We will now make a fundamental leap that will allow us to model time-varying random processes – i.e., we will introduce

**stochastic processes**.A

**stochastic process**is an**infinite collection of random variables**.We will ultimately use this object to model uncertain signals based on mechanistic processes;

- however, we want to emphasize from the beginning that a
**stochastic process is just a generalization of a random variable / vector**.

- however, we want to emphasize from the beginning that a
However, unlike a random vector \( \pmb{x}\in\mathbb{R}^{N_x} \), a stochastic process requires considerably more care.

- This is due to the fact that a stochastic process \( \pmb{x}_t \) is the
**generalization of a random vector into infinite dimensions**.

- This is due to the fact that a stochastic process \( \pmb{x}_t \) is the

Stochastic Processes

Astochastic processis a family of random variables, \( X(\omega, t) \), indexed by a real parameter \( t \in T \) and defined on a common probability space \( (\Omega, A, \mathcal{P} ) \).

The “real parameter \( t \)” in the above is almost always taken a time variable.

Thus, a random process can be thought of as a

**function that takes both the sample point and time as arguments**.Hence, we will often use the notation \( X(\omega, t) \), or \( X_t \) for short, to denote a random variable depending on time.

Note, in many cases we will actually

**consider time to be discrete in nature**;- this is a consequence, in part, of the fact that we
**often have measurements discretely and not continuously in time**.

- this is a consequence, in part, of the fact that we
If the parameter set \( T = \mathbb{N} \), the natural numbers, then \( X_t \) is often called a random sequence to reflect the discrete nature of the time parameter.

- In this case, we will often use a notation such as \( X_k \) corresponding to a sequence of times \( \{t_0, t_1, \cdots, t_k, \cdots\} \) to denote the discrete nature of the time variable.

On the other hand, if \( T = \mathbb{R} \), then \( X_t \) is called either a random function or process.

We should also note that the values of \( X_t \), for a fixed value of \( t \),

**can also be a discrete or continuous random variable**.If we look at random processes in a certain way, we can see that they are a

**natural extension of the idea of random vectors to infinite collections**:

\[ \begin{align} &\begin{matrix}\text{Random variable} & X \end{matrix} & \begin{matrix}\text{Random vector} & & \pmb{x}^\top:= \begin{pmatrix} X_1 & \cdots & X_{N_x}\end{pmatrix}\end{matrix}\\\\ &\begin{matrix}\text{Random sequence} & \begin{pmatrix} \cdots & X_k & \cdots \end{pmatrix}\end{matrix} & \begin{matrix} \text{Random process} & \begin{pmatrix} \cdots & X_t & \cdots \end{pmatrix}\end{matrix} \end{align} \]

- The key notion to take away is that a
**random process is a set of time functions**, each of which is one possible outcome, \( \omega \), out of the set of all possible outcomes, \( \Omega \). - A random variable returns a real number, and a random process
**returns a continuously indexed collection of real numbers or a real-valued function**.

As a quick example, consider a sine wave with an amplitude given as a random variable taking on values from −1 to 1 with a uniform distribution:

\[ \begin{align} X(\omega, t) := A(\omega) \sin(t): \Omega \times \mathbb{R} \rightarrow \mathbb{R} & & A(\omega) \sim U[-1, 1] \end{align} \]

Again, we emphasize that the random variable \( X \) takes \( \omega \) and returns a function.

- In particular, if we know which sample point, \( \omega \), we have selected, we know the plot of the function \( X_t \) for all \( t\in \mathbb{R} \).

We can consider this in Python as follows, where

`A`

below is a uniform random variable on \( [-1,1] \)

```
import matplotlib.pyplot as plt
import numpy as np
np.random.seed(123)
A = np.random.uniform(low=-1, high=1)
A
```

```
0.3929383711957233
```

Having determined the particular outcome for \( \omega \), we now define a function in time given by

\[ \begin{align} 0.3929383711957233 * \sin(t). \end{align} \]

- We plot the function defined on the last slide for 5 cycles of sine:

```
time_points = np.linspace(start=0, stop=10*np.pi, num=10000)
plt.subplots(figsize=(24, 6))
```

```
(<Figure size 2400x600 with 1 Axes>, <matplotlib.axes._subplots.AxesSubplot object at 0x7fec2c398940>)
```

```
plt.plot(time_points, A*np.sin(time_points))
```

```
[<matplotlib.lines.Line2D object at 0x7fec2bdd3ba8>]
```

```
plt.show()
```

We just showed how

**fixing \( \omega \)**yields an**outcome for a random function defined for all time**.- This particular function, defined over the time variable \( t \), is known as a
**sample path or a realization of the random process**.

- This particular function, defined over the time variable \( t \), is known as a
On the other hand, if we fix \( t \) and let \( X \) vary over \( \omega \), i.e.,

\[ \begin{align} X_{t_0}(\omega) := X(\omega,t)|_{t=t_0}, \end{align} \]

we get a random variable \( X_{t_0} \) defined on \( (\Omega, \mathcal{A}, \mathcal{P}) \) with

\[ \begin{align} \{\omega : X_{t_0} (\omega) \leq x \} \in \mathcal{A}, \end{align} \] i.e., these form the

**generating sets of the observable events**for the probability space.This is what is meant that the parameters \( t \) indexes a family of random variables.

We can consider random processes as infinite-dimensional extensions of the idea of random vectors.

However, when we extend our ideas to

**infinite dimensions**,**new complications arise**.With random vectors, we can completely define the probability of the vector as a collection of random variables by the joint probability,

\[ \begin{align} P(x_1, \cdots, x_{N_x}) := \mathcal{P}\left( X_1(\omega) \leq x_1 , \cdots, X_{N_x}(\omega) \leq x_{N_x}\right) \end{align} \]

However, when \( T \) is an infinite set, we have to be more careful.

If \( T \) is a

**countably infinite set**, then we can**characterize the process**with sets of the form\[ \begin{align} \{\omega: X(\omega,t_k) \leq a, \quad k= 1,\cdots\} = \cap_{k=1}^\infty \{\omega : X_{t_k} \leq a\}; \end{align} \]

- in the measure-theoretic approach, this type of countably-infinite intersection still generates the observable events of the process.

However, trouble arises with the

**uncountable infinity of the continuum**for such constructions, where\[ \begin{align} \{\omega: X(\omega,t_k) \leq a, \quad k= 1,\cdots\} = \cap_{t\in\mathbb{R}} \{\omega : X_{t_k} \leq a\} \end{align} \] an

**intersection over a continuum can lead to problems**with our construction of observable events.- In particular, if we follow the above alone, the
**probability of**\( \mathcal{P}\left(X_t \geq 0 : \forall t\in[0,1]\right) \)**may not actually be defined**.

- In particular, if we follow the above alone, the

- To get around the problems which might arise from continuous-time stochastic processes, we need to add two more conditions to our process.

Continuous in probability

A stochastic process \( X(\omega, t) \) is said to becontinuous in probabilityat \( t \) if \[ \begin{align} \lim_{s\rightarrow t} \mathcal{P}\left(\vert X(\omega, s) - X(\omega, t) \vert \geq \epsilon\right) =0 \end{align} \] for all \( \epsilon > 0 \).

Note that in the above, the probability we are referring to is over all possible outcomes \( \omega\in \Omega \).

This means there is

**such a small set of possible \( \omega \)**for which we would see a discontinuity, that the**probability of observing such an event is zero**.Being continuous in probability means then that the

**possibility of seeing a nonzero jump in zero time**has**probability zero**.

- Our second condition is known as separability, which is described as follows.

Separability

A stochastic process \( X(\omega, t) \) is said to beseparableif there exists a countable, dense set \( S \subset T \) such that for any closed set \( K \subset [-\infty, \infty] \) the two sets \[ \begin{align} A_1 = \{\omega : X(\omega, t) \in K, \forall t \in T \} & & A_2 =\{\omega: X(\omega, t)\in K, \forall s \in S\} \end{align} \] differ by a set \( A_0 \) such that \( \mathcal{P}\left(A_0\right)=0 \).

Separability means that for a

**continuous-parameter random process**,**it is possible to analyze it like a discrete-parameter one**.When we say that \( S \) is dense in \( T \), we mean that \( S \) must contain enough points of \( T \) so as to provide

**essentially a complete representation**of \( T \).That is, if we plotted out the set \( \{X(\omega, t) : t \in S\} \) versus \( t \), it would look indistinguishable from \( \{X(\omega, t), t \in T \} \).

Lets suppose that \( T = \mathbb{R} \), the real line, or perhaps the positive real line \( \mathbb{R}^+ \).

The simplest way to get a separating set of the continuous-time interval \( T \) is to simply choose \( S \) to be the set of rational numbers \( \mathbb{Q} \cap T \).

Mathematically, what this means is that for any point \( t \in T \), there exists a discrete sequence \( t_k \in S \) such that \( t_k \rightarrow t \).

Thus, if \( X(t, \omega) \) is a separable process, it is such that \( X(t_k , \omega) \rightarrow X(t, \omega) \) except on the zero probability set mentioned in the definition.

We, therefore, have

**well-defined discretized limits in time**with probability one.

Just like we extended the Gaussian to multiple variables, we can now

**extend the (multivariate) Gaussian**to a**family of Gaussian random (vectors) variables parameterized by time**.We will need to start with identifying the notion of independence properly for a stochastic process.

Independent increments

Let \( X \) be a random process defined on the time interval, \( T \). Let \( t_0 < t_1 < \cdots < t_n \) be a partition of the time interval, \( T \). If the increments, \( X(t_k) − X(t_{k−1}) \), are mutually independent random variables for any partition of \( T \), then \( X \) is said to be aprocess with independent increments.

In the above, this is to say that if we take arbitrary, discrete time points \( t_{k-1} < t_{k} < t_{k+1} \in T \), the following holds

\[ \begin{align} \mathcal{P}\left( X_{k+1} - X_{k} \leq a | X_{k} - X_{k-1} = b\right) = \mathcal{P}\left( X_{k+1} - X_{k} \leq a\right), \end{align} \] and vice-versa in the time indices.

In particular, the size of any

**past difference in realizations****doesn't affect the probability of any future difference**in realizations.

- We now introduce the notion of a
**Gaussian process**, and two of its quintessential applications. - A commonly understood Gaussian process is actually
**“white noise”**, visualized to the right.

Gaussian processes

We say that a random process, \( X_t \), is aGaussian processif for every finite collection, \( X_{1}, X_{2}, \cdots , X_{N_x} \), the corresponding density function, \[ \begin{align} p(x_1 , . . . , x_n ), \end{align} \] is a Gaussian density function, for the joint, multivariate Gaussian in \( \pmb{x}^\top:= \begin{pmatrix}X_1, & \cdots, & X_{N_x} \end{pmatrix} \).