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