Neuenschwander, Daniel

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J Theor Probab (2009) 22: 1058–1060DOI 10.1007/s10959-009-0243-5E R R AT U MErratum to: Uniqueness of Embedding into a GaussianSemigroup and a Poisson Semigroup with DeterminateJump Law on a Simply Connected Nilpotent Lie GroupDaniel NeuenschwanderPublished online: 3 October 2009© Springer Science+Business Media, LLC 2009Erratum to: J Theor Probab (2008) 21: 791–801DOI 10.1007/s10959-008-0175-5There is a change in a step in the proof of the Gaussian part of Theorem 1 in theoriginal article, and the related new proof of a result due to Pap mentioned in thispaper.In Theorem 1 in the original paper, the uniqueness property for the embed-ding c.c.s. among all possible c.c.s. of a Gaussian measure was asserted for sim-ply connected nilpotent Lie groups G. However, the “proof” of this assertion inthe latter paper contains a gap, which we will close here. In fact, the proof for the“(2/r)-exponential tail decrease” of the iterated stochastic integrals under consider-ation in the proof of Theorem 1 in the case of several identical Brownian motions iserroneous. However, due to the skew-symmetry of the Lie brackets, this case does infact not need to be taken into account.The online version of the original article can be found under doi:10.1007/s10959-008-0175-5.D. Neuenschwander ()École des Hautes Etudes Commerciales, Institut de Sciences Actuarielles, Université de Lausanne,1015 Lausanne, Switzerlande-mail: daniel.neuenschwander@bluewin.chD. NeuenschwanderInstitut für mathematische Statistik und Versicherungslehre, Universität Bern, 3012 Bern,SwitzerlandD. NeuenschwanderInstitut de Science Financière et d’Assurances, Université de Lyon, Université Claude BernardLyon 1, 50 avenue Tony Garnier, 69007 Lyon, FranceJ Theor Probab (2009) 22: 1058–1060 1059Specifically, the text in the original articleLet Z = ∫ 10∫ sm0 . . .∫ s20 dB1(s1)dB2(s2) . . . dBm(sm) be an iterated stochastic integral. . . Using Lemma 3 and the just proved uniform (2/r)-exponential tail decreaseshould be replaced as follows:Assume {Bi(t)}t≥0 (i = 1,2, . . . , d) are independent one-dimensional Brownian mo-tions such that for every i it holds that {Bi(t)}t≥0 is either centered or degenerate.PutZd(t) :=∫ t0∫ s10. . .∫ sd−10dBd(sd) . . . dB2(s2)dB1(s1).By induction on d we will first prove the assertionP(sup0≤t≤1|Zd(t)| ≥ ρ)≤ exp(−Cρ2/d) (ρ > 0).We will rely on the following (easy to verify) tail estimate for a one-dimensionalstandard normal random variable Y :P(|Y | ≥ ρ) ≤ exp(−Cρ2) (ρ > 0).We find that for an independent standard normal variable Y we haveP(sup0≤t≤1|Zd(t)| ≥ ρ)≤ P(sup0≤t≤1|Zd−1(t)||Y | ≥ Cρ)≤ P(sup0≤t≤1|Zd−1(t)| ≥ ρ(d−1)/d)+ P(|Y | ≥ Cρ1/d)(ρ > 0).Now the before-mentioned assertion follows from the induction hypothesis (ap-plied to sup0≤t≤1 |Zd−1(t)|) and the tail estimate for standard one-dimensional nor-mal random variables (applied to Y ). Denote by {W(t)}t≥0 the Lévy process on Gwith c.c.s. {μ(1)t }t≥0. Since every centered Brownian motion on the vector space(Rd ,+) can be written as the image of a linear endomorphism of standard Brown-ian motion on (Rd,+), it follows that {W(t)}t≥0 is a finite sum of processes ofthe type {Zd(t)}t≥0 (a priori, it would be possible that for certain i < j we havethat the one-dimensional Brownian motions {Bi(t)}t≥0 and {Bj (t)}t≥0 are both non-degenerate with {Bi(t)}t≥0 a.s.= {Bj (t)}t≥0; however, due to the skew-symmetry ofthe Lie bracket [., .], it follows that every term of this type is automatically canceledby the negative counterpart of the same term in every Lie bracket monomial). By thescaling property for the standard one-dimensional Brownian motion {B(t)}t≥0 (whichtells that the processes {B(ut)}t≥0 and {u1/2B(t)}t≥0 coincide in law for u > 0) thisyields that there exists a constant C > 0 such thatη(1)t ({z ∈ G : ||z|| ≥ ρ}) ≤ exp(−Cρ2/d) (ρ > 0; 0 < t ≤ 1).1060 J Theor Probab (2009) 22: 1058–1060By Lemma 3 and this tail estimateRemark 1 Observe that with this change, also the independent proof of the unique-ness of the embedding Gaussian c.c.s. of a Gaussian measure on a simply connectednilpotent Lie group remains true.

Erratum to: Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law ona Simply Connected Nilpotent Lie Group

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Erratum to: Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law ona Simply Connected Nilpotent Lie Group

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