Page - 329 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 421 so that,bydefinitionofA: ∀m∈N0,∀a∈A, E(X(m)a(X))=μmE(a(X+m)), (9) equalityholdingtriviallywhenm=0. Inparticular, taking a=1∈A—that is, a(x)=1 (x∈N0)—(9) recovers,atonce, thePoissonfactorialmoments: ∀m∈N0, E(X(m))=μm whence, in furtherparticular,wealsorecover: E(X)=μ, E(X2)=μ2+μandE(X3)=μ3+3μ2+μ. (10) WearereadynowtoproveTheorem2. ProofofTheorem2. Inviewof (1)and(2), it suffices toshowthat thefirst twomomentsofS∗ remain boundedasπmin→0.BytheCauchy–Schwarz inequality, this in turn isadirectconsequenceof the followingresult. Lemma 1. Let X ∼ Po(μ) (μ > 0), and put Xμ := X log(X/μ), with 0log0 := 0. Then, there exist b(1),b(2) : (0,∞)→ (0,∞) such that: (a)0≤E(Xμ)≤ b(1)(μ) and 0≤E(X2μ)≤ b(2)(μ),while: (b) for i=1,2 : b(i)(μ)→0asμ→0+. Proof. By(6), a(1)0 (ξ) := log(ξ/μ)∈A0. Takingm= 1and a∈Abasedon a(1)0 in (9), andusing (7), givesatonce the statedboundsonE(Xμ)with b(1)(μ)=μ(μ− logμ),whichdoes indeed tend to0 asμ→0+. Further, let a(2)0 (ξ) := ξ(log(ξ/μ)) 2. Takingm=1and aas therestrictionof a(2)0 toN0 in (9)gives E(X2μ)=μE(a(2)(X+1)).Notingthat {x∈N0 : log((x+1)/μ)<0}= { ∅ (μ≤1) {0,...,μ−2} (μ>1) , inwhichμdenotes thesmallest integergreater thanorequal toμ, andputting B(μ) := { 0 (μ≤1) μ∑ μ−2 x=0a (2)(x+1)p(x) (μ>1) , (7), (10), andl’Hôpital’s rulegive thestatedboundsonE(X2μ),with b(2)(μ)=B(μ)+μ∑∞x=0(x+1)(x− logμ)2p(x) =B(μ)+μE{X3+X2(1−2logμ)+X((logμ)2−2logμ)+(logμ)2} =B(μ)+μ4+4μ3+2μ2+μ(logμ)2+(μlogμ)2−2μ(μ+2)(μlogμ) which, indeed, tends to0asμ→0+. AsaresultofTheorem2, thedistributionof thedeviance is stable in this limit. Further,asnoted in [2], eachofν,τ, andρcanbeeasilyandaccuratelyapproximatedbystandardtruncateandbound methods in the limitasπmin→0. Thesearedetailed inAppendixB. 329