Proof of Prop Sample Clauses

Proof of Prop. 2 R = log

Proof of Prop. 1a): Due to Definition 1, for any rate point (r1, r2) ∈ Rcom(ǫ), there is a pair of transmit covariance matrices (Ψ1, Ψ2) such that com U stat(r1, r2, Ψ1, Ψ2) ≥ 1 − ǫ. (3.65) This implies that U stat(ri, Ψ1, Ψ2) ≥ 1 − ǫ, i = 1, 2. Hence, (r1, r2) ∈ Rind (ǫ1, ǫ2). i stat □ inst com Proof of Prop. 1b): For any rate point (r1, r2) ∈ Rcom(ǫ), we have U inst(r1, r2) = i Pr{B} ≥ 1 − ǫ. From (3.50) and (3.51), we get that U inst(r1, r2, t) ≥ 1 − ǫ = 1 − ǫi, inst since Pr{Ci} ≥ 0 and Pr{Di} ≥ 0. Hence, it follows that (r1, r2) ∈ Rind (ǫ1, ǫ2). □ This statement is quite intuitive; the definition of common outage says that the entire IC is in outage when at least one of the links is in outage. Individual outage is less restrictive since it allows one link to be in outage while the other is not.

Proof of Prop