Analysis of Random Processes in L2
(Ref: Sholtz, Chap.
13.1, 10.2.2, 20.3.4)

## Hilbert Space of Random Variables

For a probability space $(\Omega, \Sigma, P)$,
all random variables with finite second moment forms a **function space** $L^2 (\Omega, \Sigma, P)$.
It is a **Hilbert space** when assigned the inner product $\langle X,Y \rangle = \mathbb{E}[XY]$.

Two random variables are **equivalent** in $L^2 (\Omega, \Sigma, P)$
if they as measurable mappings are equal almost everywhere relative to the probability measure $P$,
denoted as $X=Y \text{ a.e. } P$.

### Convergence in L2

Given space of random variables $L^2 (\Omega, \Sigma, P)$ and metric associated with the $L^2$-norm,
**convergence in $L^2$** is well defined.

Properties:

- Convergence in $L^2$ implies convergence in expectation and second moment.
- The subspace of Gaussian random variables in $L^2 (\Omega, \Sigma, P)$ is also a Hilbert space.
This is called the
**Gaussian Hilbert space**.

Proof of property 2 is depends on property 1.

### Proof of completeness

### Gaussian Hilbert space and other subspaces of $L^2 (\Omega, \Sigma, P)$

## Convergence

Convergence of linear transformation of random sequence

Sufficient conditions for convergence of linear transformation of random sequence:

- $\forall t \in \mathbb{Z}, \forall \varepsilon >0, \exists N \in \mathbb{N}:
\forall m,n>N, \sum_{t'\in S_{m,n}} \sum_{t"\in S_{m,n}} h(t,t') R_X(t',t") h^{*}(t,t") < \varepsilon$
- $\forall t \in \mathbb{Z}, \forall \varepsilon >0, \exists N \in \mathbb{N}:
\sum_{|t'|>N} |h(t,t')| R_X^{\frac{1}{2}} (t',t') < \varepsilon$
- $R_X(t,t)$ is uniformly bounded, $\mathbb{H}$ is LTI, and $h(t)$ is absolutely summable.
- $X(u,t)$ is w.s.s., and $h(t,t')$ is absolutely summable for all t.
- $X(u,t)$ is w.s.s., $\mathbb{H}$ is LTI, and $h(t)$ is absolutely summable.

## Continuity

Notes on Continuity

Random process $X(u,t)$ is continuous at $t=0$, if $R_X(t_1,t_2)$ is continuous at $(t_0,t_0)$.

Random process $X(u,t)$ is uniformly continuous,
if it is w.s.s. and $R_X(t)$ is continuous at the origin $t=0$.

## Differentiability

Notes on Differentiability

Random process $X(u,t)$ is differentiable at $t_0$,
iff $\frac{\partial^2}{\partial t_1 \partial t_2} R_X (t_1,t_2)$ exists at $(t_0,t_0)$

Random process $X(u,t)$ is differentiable $\forall t \in \mathbb{R}$,
if it is w.s.s. and $R_X(t)$ is second-order differentiable at the origin $t=0$.

Properies of differentiator $\mathbb{D}$:

- LTI
- $D(f) = i 2\pi f$
- not stable
- causal

## Integrability

Notes on Integrability

🏷 Category=Probability