FACIAL SOFT BIOMETRICS - Library of Ph.D. Theses | EURASIP

107– We are now left withN −F = 2 subjects, that can be associated freely to any of the F = 3specific picked categories. Hence the third term is F N−F = 3 2 = 9 corresponding to thecardinality of {1,2,··· ,F} N−F .Proof of Lemma 5 Recall from (3.22) thatand note thatP(F) =F N−F(ρ−F)!(N −F)! ∑ Ni=1 iN−i( (N −i)!(ρ−i)! ) −1 , (A.6)∑ Ni=1 iN−i( (N −i)!(ρ−i)! ) −1 ≥ (ρ−N)!corresponding to theNth summand (i = N), and corresponding to the fact that all summands arenon-negative. As a resultP(F) ≤F N−F(ρ−F)!(N −F)!(ρ−N)! .Using Stirling’s approximation [AS02] that holds in the asymptotically high ρ setting of interest,we haveF N−FP(F) ˙≤(ρ−F) ρ−F (N −F) N−F (ρ−N) ρ−N eand as a result−(2ρ−2F),(A.7)P(f)˙≤(frρ) rρ(1−f)(ρ−frρ) ρ−frρ (rρ−frρ) rρ−frρ (ρ−rρ) ρ−rρ e 2ρ(1+fr)=ρ rρ(1−f) ρ −ρ(1−fr)(fr) −rρ(1−f) (1−fr) ρ(1−fr)·ρ −ρr(1−f) ρ −ρ(1−r)(r −fr) ρr(1−f) (1−r) ρ(1−r) e2ρ(1+fr). (A.8)In the above we use . = to denote exponential equality, wheref . = ρ −ρB ⇐⇒ − limρ→∞logfρlogρ = B,(A.9)with ˙≤, ˙≥ being similarly defined. The result immediately follows.