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From moments convergence to weak convergence

Let P be the set of probability measures μ on R such that R[X]L1(μ). Let us consider the equivalent relation on P given by μ1μ2 if and only if for every PR[X],

Pdμ1=Pdμ2

(i.e. μ1 and μ2 share the same sequence of moments). We say that μP is characterized by its moments when its equivalent class is a singleton. Every compactly supported probability measure on R belongs to P and is indeed characterized by its moments thanks to the Weierstrass density theorem. Beyond compactly supported probability measures, one may use the Carleman criterion. The following classical lemma is useful when using the moments method, for instance when proving the Wigner semicircle theorem.

Lemma 1 (From moments convergence to weak convergence) If μ,μ1,μ2, belong to P with

limnPdμn=Pdμ

for every PR[X] and if μ is characterized by its moments then for every fCb(R,R)

limnfdμn=fdμ.

Proof: The convergence assumption implies that for every polynomial P

CP:=supn1Pdμn<.

Thus, by Markov's inequality, for every real R>0,

μn([R,R]c)CX2R2

and therefore (μn)n1 is tight. As a consequence, by Prokhorov's theorem, it suffices now to show that if (μnk)k1 converges weakly to ν then ν=μ. Recall that the weak convergence here is also known as the narrow convergence and corresponds to the convergence for continuous bounded functions.

Let us show that μ=ν. Let us fix some PR[X] and a real number R>0. Let φR:R[0,1] be continuous with 1[R,R]φR1[R1,R+1]. We start from the decomposition

Pdμnk=φRPdμnk+(1φR)Pdμnk.

Since (μnk)k1 tends weakly to ν we have

limkφRPdμnk=φRPdν.

Additionally, by Cauchy-Schwarz's and Markov's inequalities,

|(1φR)Pdμnk|2μnk([R,R]c)P2dμnkCX2CP2R2.

On the other hand, we know that

limkPdμnk=Pdμ.

Therefore, we obtain

limRφRPdν=Pdμ.

Using this for P2 we obtain by monotone convergence that PL2(ν)L1(ν) and then by dominated convergence that

Pdν=Pdμ.

Since P is arbitrary and μ is characterized by its moments, it follows that μ=ν. ◻

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