On Stein’s Method for Infinitely Divisible Laws with Finite First Moment by Benjamin Arras & Christian Houdré

Author:Benjamin Arras & Christian Houdré
Language: eng
Format: epub
ISBN: 9783030150174
Publisher: Springer International Publishing

Proof

The proof is similar to the proof of Theorem 4.10 and so is only sketched. Clearly, , for all and so the sequence converges in distribution toward X. Then, with the previous notations,

Moreover, thanks to (4.32),

since

Thus,

which implies that

and therefore,

(4.34)

By the very definition of R, Proposition 3.​8, and straightforward computations,

Hence,

Moreover, further straightforward computations lead to

(4.35)

Combining (4.31) and (4.34)–(4.35) gives

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