Consider an infinite sequence (Un)n∈N of independent Cauchy random variables, defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scale parameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn≠0 for all n∈N. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn|−1)n∈N is square-summable

Lie, Han Cheng

Sullivan, T. J.

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Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

Han Cheng Lie

T.J. Sullivan

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     warwick.ac.uk/lib-publications      Manuscript version: Published Version The version presented in WRAP is the published version (Version of Record).  Persistent WRAP URL: http://wrap.warwick.ac.uk/135639                              How to cite: The repository item page linked to above, will contain details on accessing citation guidance from the publisher.  Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.   Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  Publisher’s statement: Please refer to the repository item page, publisher’s statement section, for further information.  For more information, please contact the WRAP Team at: wrap@warwick.ac.uk  Electron. Commun. Probab. 23 (2018), no. 8, 1–6.https://doi.org/10.1214/18-ECP113ISSN: 1083-589XELECTRONICCOMMUNICATIONSin PROBABILITYQuasi-invariance of countable products of Cauchy measuresunder non-unitary dilationsHan Cheng Lie* T. J. Sullivan†AbstractConsider an infinite sequence (Un)n∈N of independent Cauchy random variables,defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scaleparameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy randomvariables defined by the same sequence of location parameters and the sequence(σnγn)n∈N of scale parameters, with σn 6= 0 for all n ∈ N. Using a result of Kakutanion equivalence of countably infinite product measures, we show that the laws of(Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn| − 1)n∈N issquare-summable.Keywords: Cauchy distribution; change of measure; equivalence of measure; random sequence.AMS MSC 2010: Primary 60G30, Secondary 60G20; 60E07.Submitted to ECP on November 30, 2016, final version accepted on January 29, 2018.Supersedes arXiv:1611.10289v3.1 IntroductionLet (Un)n∈N be a sequence of independent, Cauchy random variables defined on aprobability space (Ω,F ,P), where each random variable Un has the densityfn(x, δn, γn) =1piγnγ2n(x− δn)2 + γ2n. (1.1)The distribution of Un is parametrised by the location and scale parameters δn andγn respectively. We shall assume throughout that γn is strictly positive for all n ∈ N.For every Un, we may define another Cauchy random variable Wn by multiplicativelyperturbing the scale parameter γn by σn 6= 0, so that the pair (δn)n∈N and (σnγn)n∈Ndetermines the law of (Wn)n∈N. In this note, we prove the following result:Theorem 1.1. Let (Un)n∈N be a sequence of independent Cauchy random variablesdefined by the sequences (δn)n∈N and (γn)n∈N of location and scale parameters, and let(Wn)n∈N be a sequence of independent Cauchy random variables with the sequences(δn)n∈N and (σnγn)n∈N of location and scale parameters, where σn 6= 0 for all n ∈ N.Then the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if (|σn| − 1)n∈N issquare-summable.*Institut für Mathematik, Freie Universität Berlin, Germany. E-mail: hlie@math.fu-berlin.de†Free University of Berlin and Zuse Institute Berlin, Germany.E-mail: sullivan@zib.de http://www.zib.de/sullivanQuasi-invariance of countable products of Cauchy measuresEquivalence of laws under additive perturbations to the sequence of locations ormeans has been studied for Gaussian Un [1, Example 2.7.6], for sequences of randomvariables with finite Fisher information [5], and for stable Radon probability distributionson locally convex spaces [2, Theorem 5.2.1]. In contrast, Theorem 1.1 on multiplicativeperturbations of scale parameters appears to be new.2 Proof of Theorem 1.1We shall use Kakutani’s theorem [4, Theorem 1], which we specialise to our context.Theorem 2.1. Let (Un)n∈N and (Wn)n∈N be given as in Theorem 1.1. Then the lawsof (Un)n∈N and (Wn)n∈N are either equivalent or singular. The laws of (Un)n∈N and(Wn)n∈N are equivalent if and only if both the following conditions hold: the laws P◦U−1nand P ◦W−1n of Un and Wn respectively are equivalent for every n ∈ N, and∑n∈N− logE[√ϕn(Un, |σn|)]<∞ (2.1)where ϕn( · , |σn|) is the Radon–Nikodym derivative of P ◦W−1n with respect to P ◦ U−1n .If σn 6= 0, then the laws of Un and Wn are equivalent for every n ∈ N, withϕn(x, |σn|) = |σn| (x− δn)2 + γ2n(x− δn)2 + σ2nγ2n. (2.2)Let yn := (x− δn)/γn. From (1.1) and (2.2) it follows thatE[√ϕn(Un, |σn|)] =√|σn|pi∫R1√y2n + σ2n√y2n + 1dyn =1pi∫R1√y2n|σn| + |σn|√y2n + 1dyn.(2.3)We use (2.3) in the following lemma.Lemma 2.2. For all |σn| > 0, − logE[√ϕn(Un, |σn|)] is nonnegative, and is strictly posi-tive if and only if |σn| 6= 1. Furthermore, if (2.1) holds, then limn→∞ |σn| = 1.Proof. Using Jensen’s inequality with the concave map x 7→ √x, and using that ϕn is theRadon–Nikodym derivative of P ◦W−1n with respect to P ◦ U−1n , we obtain1 =√E[ϕn(Un, |σn|)] ≥ E[√ϕn(Un, |σn|)],and hence − logE[√ϕn(Un, |σn|)] is nonnegative for all |σn| > 0. Since x 7→ √x isnot affine, equality holds above (and hence − logE[√ϕn(Un, |σn|)] = 0) if and only ifϕn(Un, |σn|) is P-almost surely constant. By (2.3), ϕn(Un, |σn|) is P-almost surely constantif and only if |σn| = 1. This proves the first statement.To prove the second statement, observe that (2.3) implies continuity of the map|σn| 7→ E[√ϕn(Un, |σn|)] on (0,∞). The dominated convergence theorem and the sec-ond equality in (2.3) imply that, if (|σn|)n∈N decreases to zero or increases to infin-ity, then limn→∞E[√ϕn(Un, |σn|)] = 0. Summarising, we have that the map |σn| 7→E[√ϕn(Un, |σn|)] from (0,∞) to [0, 1] is continuous, attains the value 1 if and only if itsargument is 1, and does not increase asymptotically to 1 at either end of its domain.It follows that limn→∞E[√ϕn(Un, |σn|)] = 1 if and only if limn→∞ |σn| = 1. Since (2.1)implies that limn→∞E[√ϕn(Un, |σn|)] = 1, the proof is complete.Remark 2.3. A consequence of Lemma 2.2 that we shall use repeatedly is the following:if (2.1) holds, and if |σn| 6= 1 for all n ∈ N, then, for any 0 < α < 1, there exists someN(α) ∈ N such that0 < ||σn| − 1| < α, ∀n ≥ N(α). (2.4)ECP 23 (2018), paper 8.Page 2/6http://www.imstat.org/ecp/Quasi-invariance of countable products of Cauchy measuresTo simplify the subsequent presentation, we shall assume that α is sufficiently small forour purposes, and that N(α) = 1. If (2.1) holds, then by Lemma 2.2 we may assume thiswithout any loss of generality, since we may discard finitely many terms in the series(2.1) without losing convergence.Given (2.3), we may define for 0 < α < 1, a function I : (1− α, 1 + α)→ (0,∞) byI(|σ|) := 1pi∫R1√y2 + σ21√y2 + 1dy. (2.5)Therefore, I(|σ|) = 1 if and only if |σ| = 1. Furthermore, for some 0 < α < 1 (see Remark2.3), it follows from (2.4) that the integrand on the right-hand side of (2.5) is dominatedby an integrable function that depends on α but not on σ. Thus, by the dominatedconvergence theorem, we may interchange integration and differentiation to computethe derivative of I with respect to |σ|. By computing the higher-order derivatives of(y2 + σ2)−1/2 with respect to |σ| and employing the same argument, the assertion holdsfor higher-order derivatives of I as well. The Taylor expansion of I(|σ|) about |σ| = 1 upto second order isI(|σ|) = 1 + a1(|σ| − 1) + a2(|σ| − 1)2 +O((|σ| − 1)3), (2.6a)am =1m!1pi∫R1√y2 + 1(∂∂ |σ|)m1√y2 + |σ|2∣∣∣∣∣∣|σ|=1dy. (2.6b)Note that the am are independent of |σ|.We shall use the following lemma to calculate the values of am for m = 1, 2.Lemma 2.4. Let r, s ∈ N0 with r ≤ s− 1. Then∫Rx2r(x2 + 1)sdx = pi(2r)!(2(s− r − 1))!4s−1r!(s− r − 1)!(s− 1)!Proof. Letting y := x2, we have dx = 2y−1/2dy, so∫Rx2r(x2 + 1)sdx =∫ ∞0yr−1/2(y + 1)sdy =∫ ∞0y(r+1/2)−1(y + 1)sdy.Using [6, Equation (2)] and properties of the gamma function Γ(t) =∫∞0xt−1e−xdx (forRe t > 0), we have, for 0 < Re (r + 1/2) < Re s,∫ ∞0y(r+1/2)−1(y + 1)sdy =Γ(r + 12)Γ(s− r − 12)Γ(s)=(2r)!4rr!√pi(2(s− r − 1))!4s−r−1(s− r − 1)!√pi1(s− 1)! .Simplifying the right-hand side yields the desired conclusion.Thus we havea1 =1pi∫R1√y2n + 1(−1)(y2n + 1)3/2dyn = − 1pi(pi0! 2!4 · 1! · 1!)= −12, (2.7a)a2 =12pi∫R1√y2n + 12− y2n(y2n + 1)5/2dyn =12pi(9pi8− pi2)=516. (2.7b)Given the function I, a sequence (σn)n∈N satisfying (2.4), and (2.7b), definen :=1|σn| − 1 (I(|σn|)− 1− a1(|σn| − 1)) , ∀n ∈ N. (2.8)ECP 23 (2018), paper 8.Page 3/6http://www.imstat.org/ecp/Quasi-invariance of countable products of Cauchy measuresLemma 2.5. Let n be as in (2.8). If (2.1) holds, then for some 0 < α < 1 and someC > 0 that do not depend on n,c ||σn| − 1| ≤ |n| ≤ C ||σn| − 1| , ∀n ∈ N. (2.9)Proof. First, note that by Remark 2.3, we may bound the constant in the O(||σn| − 1|3)term in (2.5) by some C > 0 that may depend on α, but not on |σ|, i.e.I(|σ|) = 1 +2∑m=1am(|σ| − 1)m + C(|σ| − 1)3, ∀ |σ| ∈ (1− α, 1 + α).Thus by (2.8), the triangle inequality, and (2.4), we obtain|n| =∣∣a2(|σn| − 1) + C(|σn| − 1)2∣∣ ≤ max{C, a2} ||σn| − 1| (1 + ||σn| − 1|) ,which yields the upper bound in (2.9) since ||σn| − 1| < α by (2.4). For the lower bound,we have for the same C > 0 as above that|n| ≥ |a2(|σn| − 1)| −∣∣C(|σn| − 1)2∣∣ = ||σn| − 1| (|a2| − C ||σn| − 1|) .and for sufficiently small α, we can make |a2| − C ||σn| − 1| strictly positive, by (2.4).Again by making α small enough, we have from (2.8) and (2.7a) thatlog I(|σn|) =∑m∈N(−1)m−1m(I(|σn|)− 1)m =∑m∈N(−1)m−1m(n − 12)m(|σn|−1)m. (2.10)Proposition 2.6. If the laws of (Un)n∈N and (Wn)n∈N are equivalent, then (|σn| − 1)n∈Nis square-summable.Proof. If the laws of (Un)n∈N and (Wn)n∈N are equivalent, then by Theorem 2.1, (2.1)holds. Without loss of generality, we may assume that |σn| 6= 1 for all n ∈ N, sinceotherwise the corresponding summand vanishes, by Lemma 2.2. Furthermore, (seeRemark 2.3), we may also assume that (2.4) holds for some 0 < α < 1 that we can set assmall as needed for (2.10) to hold and such that we can rewrite log |σn| as a Mercatorseries in |σn| − 1, for all n ∈ N. Using these conditions, (2.3), (2.5), and (2.10), we obtain− logE[√ϕn(Un, |σn|)] = −∑m∈N(−1)m−1m[12+(n − 12)m](|σn| − 1)m. (2.11)The summand for m = 1 is, by (2.8) and (2.5),−n(|σn| − 1) = (a1(|σn| − 1) + 1− I(|σn|)) = −a2(|σn| − 1)2 + C(|σn| − 1)3,where C > 0 may depend on α but not on |σn|; see the proof of Lemma 2.5. Expandingthe coefficient of (|σn|−1)2 for the summand in (2.11) corresponding to m = 2, we obtainthat its value is12[12+(2n − n +14)]=38+12(2n − n).By Lemma 2.5, 2n − n converges to zero linearly in |σn| − 1. Therefore, combining thisobservation with the preceding two equations, we obtain from (2.11) and (2.7b) that, forsome C ′ independent of |σn|,− logE[√ϕn(Un, |σn|)] =(− 516+38)(|σn| − 1)2 + C ′((|σn| − 1)3)=116(|σn| − 1)2 + C ′((|σn| − 1)3).ECP 23 (2018), paper 8.Page 4/6http://www.imstat.org/ecp/Quasi-invariance of countable products of Cauchy measuresTherefore, by making α sufficiently small, we may ensure that for any 0 < δ < 1/16,1(|σn| − 1)2∣∣∣∣− logE[√ϕn(Un, |σn|)]− 116(|σn| − 1)2∣∣∣∣ = |C ′| ||σn| − 1| < δ.Thus, for any 0 < δ < 1/16,0 <(116− δ)(|σn| − 1)2 < 116(|σn| − 1)2 −∣∣∣∣− logE[√ϕn(Un, |σn|)]− 116(|σn| − 1)2∣∣∣∣ .Since − |x| < x if x > 0, and − |x| = x if x < 0, it follows that the right-hand side of theinequality above is less than or equal to − logE[√ϕn(Un, |σn|)]. Summing over n yields0 <(116− δ)∑n∈N(|σn| − 1)2 <∑n∈N− logE[√ϕn(Un, |σn|)] <∞,and therefore the summability of Kakutani’s series implies that (|σn| − 1)n∈N is square-summable.Proposition 2.7. If (|σn| − 1)n∈N is square-summable, then the laws of (Un)n∈N and(Wn)n∈N are equivalent.Proof. Without loss of generality, we may assume that the `2 norm ‖(|σn| − 1)n∈N‖2 of(|σn|−1)n∈N is strictly less than 1. Then, for 1 < p ≤ q ≤ +∞, the `p-norm of (|σn|−1)n∈Nis larger than or equal to the `q-norm of (|σn| − 1)n∈N. Using this fact, the sum formulafor a geometric progression, and the hypothesis on the `2 norm of (|σn| − 1)n∈N, we have∑m≥2∑n∈N||σn| − 1|m ≤∑m≥2(∑n∈N||σn| − 1|2)m/2=‖(|σn| − 1)n∈N‖221− ‖(|σn| − 1)n∈N‖1/22<∞. (2.12)Since (|σn| − 1)n∈N converges to zero, we may choose α in (2.4) so small that, for ndefined in (2.4), it holds that |n| < 1/8 for all n ∈ N, by Lemma 2.5. Since∣∣∣∣12 +(n − 12)m∣∣∣∣ ≤{|n| m = 112 +(|n|+ 12)m m ≥ 2,the condition that |n| < 1/8 implies that we may bound the left-hand side by 1 for allm ∈ N. Using this observation and the triangle inequality, we may bound the right-handside of (2.11) according to−∑m∈N(−1)m−1m[12+(n − 12)m](|σn| − 1)m ≤∑m∈N||σn| − 1|m ,and hence from (2.11) it follows that − logE[√ϕn(Un, |σn|)] ≤ Σm∈N ||σn| − 1|m. Sum-ming this inequality over n yields∑n∈N− logE[√ϕn(Un, |σn|)] ≤∑n∈N∑m≥2||σn| − 1|m ,where the right-hand side is finite, by changing the order of summation in (2.12).Therefore, (2.1) holds, and by Theorem 2.1, the laws of (Un)n∈N and (Wn)n∈N areequivalent.Remark 2.8. An anonymous referee pointed out that one can express the functionI defined in (2.5) in terms of complete elliptic integrals (see, e.g., [3, Section 3.152,Formula 1] and [3, Section 8.112, Formula 1]), and that the series representationsof these elliptic integrals in [3, Section 8.113] may provide an alternative method forobtaining the results that we presented above.ECP 23 (2018), paper 8.Page 5/6http://www.imstat.org/ecp/Quasi-invariance of countable products of Cauchy measuresReferences[1] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, AmericanMathematical Society, Providence, RI, 1998. MR-1642391[2] V. I. Bogachev, Differentiable measures and the Malliavin calculus, Mathematical Surveys andMonographs, vol. 164, American Mathematical Society, Providence, RI, 2010. MR-2663405[3] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, eighth ed., El-sevier/Academic Press, Amsterdam, 2015, Translated from the Russian, Translation editedand with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition[MR2360010]. MR-3307944[4] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224.MR-0023331[5] L. A. Shepp, Distingunishing a sequence of random variables from a translate of itself, Ann.Math. Statist. 36 (1965), 1107–1112. MR-0176509[6] F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J.Math. 1 (1951), 133–142. MR-0043948Acknowledgments. The authors are supported by the Freie Universität Berlin withinthe Excellence Initiative of the German Research Foundation. The authors thank theeditors and referees who read the article for their helpful comments.ECP 23 (2018), paper 8.Page 6/6http://www.imstat.org/ecp/

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Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

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