Page - 328 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 whereD∗ is defined implicitly and0log0 := 0. The terms ν, τ, and ρ aredefinedby thefirst two momentsofS∗via thevectors( N ν ) :=E(S∗)= ( N ∑ki=0E(n ∗ i log ( n∗i/μi ) ) ) , (1) ( N ρτ √ N · τ2 ) :=Cov(S∗)= ( N ∑ki=0Ci · ∑ki=0Vi ) , (2) whereCi :=Cov(n∗i ,n ∗ i log(n ∗ i/μi))andVi :=Var(n ∗ i log(n ∗ i/μi)). Theorem2. Eachof the termsν,τ, andρ remainsboundedasπmin→0. Westartwithsomepreliminaryremarks.Weuse the followingnotation:N :={1,2,...}denotes the natural numbers, whileN0 := {0}∪N . Throughout,X ∼ Po(μ)denotes a Poisson random variablehavingpositivemeanμ—that is,X isdiscretewithsupportN0 andprobabilitymass function p :N0→ (0,1)givenby: p(x) := e−μμx/x! (μ>0). (3) Putting: ∀m∈N0, F[m](μ) :=Pr(X≤m)=∑mx=0p(x)∈ (0,1), (4) for givenμ, {1−F[m](μ)} is strictlydecreasingwithm, vanishing asm→∞. For all (x,m)∈N20 , wedefinex(m)by: x(0) :=1; x(m) := x(x−1)...(x−(m−1)) (m∈N) (5) so that, ifx≥m,x(m)= x!/(x−m)!. ThesetA0 comprisesall functions a0 : (0,∞)→Rsuchthat, asξ→0+: (i) a0(ξ) tends toan infinite limit a0(0+)∈{−∞,+∞},while: (ii)ξa0(ξ)→0. Ofparticular interesthere,by l’Hôspital’s rule, ∀m∈N, (log)m∈A0, (6) where (log)m : ξ→ (logξ)m (ξ>0). Foreach a0∈A0, a0denotes its continuousextensionfrom (0,∞) to [0,∞)—that is: a0(0) := a0(0+); a0(ξ) := a0(ξ) (ξ> 0)—while, appealing to continuity,wealso define0a0(0) :=0.Overall,denotingtheextendedrealsbyR :=R∪{−∞}∪{+∞}, andputting A :={a :N0→Rsuchthat0a(0)=0} wehavethatAcontains thedisjointunion: {all functions a :N0→R}∪{a0|N0 : a0∈A0}. Werefer to a0|N0 as thememberofAbasedona0∈A0. Wemakerepeateduseof twosimple facts. First: ∀x∈N0, 0≤ log(x+1)≤ x, (7) equalityholding inbothplaces if, andonly if,x=0. Second, (3)and(5)give: ∀(x,m)∈N20 withx≥m, x(m)p(x)=μmp(x−m) (8) 328