Combining (29) and (33) yields that which is integrable. Therefore, the integral ∫ ∞ −∞ φ ( y − z ) log p$_{Y}$ ( y ; µ )d y is uniformly convergent for all z ∈ U . Moreover, φ ( y − z ) log p$_{Y}$ ( y ; µ ) is a holomorphic function of z on U for each y ∈$_{R}$ . According to the differentiation lemma [18], ξ ( z ) is a holomorphic function of z on U . It then follows that it is holomorphic on the whole complex plane$_{C}$. Lemma 2 is thus established. Let F ( µ ) be a real-valued function defined on the convex set Λ( q ) and µ$_{0}$ ∈ Λ( q ) . Define the weak derivative of F ( µ ) at µ$_{0}$ as whenever the limit exists. The following result, which finds its parallel in [6], [9], [10] gives the weak derivative of the mutual information function I ( µ ) . Lemma 3: Let µ$_{0}$, µ ∈ Λ( q ) , the weak derivative of the mutual information function I ( µ ) at µ$_{0}$ is Proof: Define µ$_{θ}$ = (1 − θ ) µ$_{0}$ + θµ for all θ ∈ (0 , 1] . It can be shown that Therefore, it suffices to show that In the remainder of this proof, we find a function independent of θ that dominates the integrand so that dominated convergence theorem can be used to establish (39) by exchanging the order of the limit and the integral therein. 12 lim θ → 0 + ∫ ∞ −∞ 1 θ p$_{Y}$ ( y ; µ$_{θ}$ ) log p$_{Y}$ ( y ; µ$_{θ}$ ) p$_{Y}$ ( y ; µ$_{0}$ ) d y = 0 . (39) F ′ $_{µ$_{0}$}$( µ ) = lim θ → 0 + F ((1 − θ ) µ$_{0}$ + θµ ) − F ( µ$_{0}$ ) θ (35) | φ ( y − z ) log p$_{Y}$ ( y ; µ ) | ≤ e r 2 √ 2 π $^{[}$e − 1 $_{2}$( y − r ) 2 + e − 1 $_{2}$( y + r ) 2 ] $^{(}$1 2 y 2 + ay + b$^{)}$ , (34) I ′ $_{µ$_{0}$}$( µ ) = ∫ d ( x ; µ$_{0}$ ) µ (d x ) − I ( µ$_{0}$ ) . (36) 1 θ ( I ( µ$_{θ}$ ) − I ( µ$_{0}$ )) = 1 θ ∫ ( d ( x ; µ$_{θ}$ ) − d ( x ; µ$_{0}$ )) µ$_{θ}$ (d x ) + 1 θ ( ∫ d ( x ; µ$_{0}$ ) µ$_{θ}$ (d x ) − I ( µ$_{0}$ )$^{)}$ (37) = − 1 θ ∫ ∞ −∞ p$_{Y}$ ( y ; µ$_{θ}$ ) log p$_{Y}$ ( y ; µ$_{θ}$ ) p$_{Y}$ ( y ; µ$_{0}$ ) d y + ∫ d ( x ; µ$_{0}$ ) µ (d x ) − I ( µ$_{0}$ ) . (38)