*3.22. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,4b* × (R\[0, 1])

By recalling PSP,4b := { (*β*A, *β*H, *α*A, *α*H) ∈ PSP : *α*<sup>A</sup> 6= *α*<sup>H</sup> > 0, *β*<sup>A</sup> = *β*<sup>H</sup> ∈ [1, ∞[ } (cf.(49)), the assertions preceding Proposition 13 remain valid. However, the proof of Proposition 13 in Appendix A.1 contains details which explain why it cannot be carried over to the current case PSP,4b. Thus, the generally valid lower bound *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1 cannot be improved with our methods.

#### *3.23. Concluding Remarks on Alternative Lower Bounds for all Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*) × (R\[0, 1])

To achieve the Goals (G10 ) to (G30 ), in the above-mentioned investigations about lower bounds of the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*), *λ* ∈ R\[0, 1], we have mainly focused on parameters *p L λ* , *q L λ* which satisfy (35) and additionally (56). Nevertheless, Theorem 1 (b) gives lower bounds *B L λ*,*X*0,*n whenever* (35) is fulfilled. However, this lower bound can be the trivial one, *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≡ 1. Let us remark here that for the parameter constellations (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ PSP,2 × R [0, 1] ∪ ISP,2 ∪ PSP,3a × R [0, 1] ∪ ISP,3a ∪ PSP,3b × R [0, 1] ∪ ISP,3b one can prove that there exist *p L λ* , *q L λ* which satisfy (35) for all *x* ∈ N<sup>0</sup> as well as the condition (generalizing (56))

> *p L <sup>λ</sup>* ≥ *α<sup>λ</sup>* , *q L <sup>λ</sup>* ≥ *β<sup>λ</sup>* , (where at least one of the inequalities is strict) ,

and that for such *p L λ* , *q L λ* one gets the validity of *Hλ*(*P*A,*n*||*P*H,*n*) ≥ *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = *B*e (*p L λ* ,*q L λ* ) *<sup>λ</sup>*,*X*0,*<sup>n</sup>* > <sup>1</sup> for all *X*<sup>0</sup> ∈ N and all *n* ∈ N; consequently, Goal (G1<sup>0</sup> ) is achieved. However, in these parameter constellations it can unpleasantly happen that *n* 7→ *B L λ*,*X*0,*n* is oscillating (in contrast to the monotone behaviour in the Propositions 11 (b), 12 (b)).

As a final general remark, let us mention that the functions *φ* tan *λ*,*y* (·), *φ* sec *λ*,*k* (·), *φ* hor *λ* (·), *φ*f*λ*(·) –defined in (52)–(54) and Properties 3 (P20)–constitute linear lower bounds for *φλ*(·) on the domain N<sup>0</sup> in the case *λ* ∈ R\[0, 1]. Their parameters *p L λ* ∈ n *p* tan *λ*,*y* , *p* sec *λ*,*y* , *p* hor *λ*,*y* , f*pλ* o and *q L λ* ∈ n *q* tan *λ*,*y* , *q* sec *λ*,*y* , *q* hor *λ*,*y* , *q*e*λ* o lead to lower bounds *B L λ*,*X*0,*n* of the Hellinger integrals that may or may not be consistent with Goals (G10 ) to (G30 ), and which may be possibly better respectively weaker respectively incomparable with the previous lower bounds when adding some relaxation of (G10 ), such as e.g., the validity of *Hλ*(*P*A,*n*||*P*H,*n*) > 1 for all but finitely many *n* ∈ N.
