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:

  1. Convergence in $L^2$ implies convergence in expectation and second moment.
  2. 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:

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

  1. LTI
  2. $D(f) = i 2\pi f$
  3. not stable
  4. causal

Integrability

Notes on Integrability

🏷 Category=Probability