Partly Divisible Probability Measures on Locally Compact Abelian Groups

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<ul><li><p>Math. Nachr. 213 (2000), 5 15</p><p>Partly Divisible Probability Measures on Locally Compact</p><p>Abelian Groups</p><p>By Sergio Albeverio of Bonn, Hanno Gottschalk of Roma and Jiang {Lun Wu of</p><p>Bochum</p><p>(Received October 15, 1997)</p><p>(Revised Version June 10, 1999)</p><p>Abstract. A notion of admissible probability measures on a locally compact Abelian group</p><p>(LCA {group) G with connected dual group bG = IRd Tn is dened. To such a measure , a closedsemigroup () (0,) can be associated, such that, for t (), the Fourier transform to thepower t,</p><p>(^t</p><p>, is a characteristic function. We prove that the existence of roots for non admissible</p><p>probability measures underlies some restrictions, which do not hold in the admissible case. As we</p><p>show for the example ZZ2, in the case of LCA{ groups with non connected dual group, there is no</p><p>canonical denition of the set ().</p><p>1. Introduction</p><p>Let (G;+) be a locally compact Abelian group (LCA group) and X a G valuedrandom variable (measurable with respect to the Borel algebra on G) on an ar-bitrarily given probability space with probability measure P . As in the real randomvariable case, it is interesting to know whether X can be decomposed into a sum of nindependent identically distributed G valued random variables X1; : : : ; Xn, say,</p><p>X = X1 + +Xn ; P a. e.</p><p>Alternatively, in terms of = P X1, the image (probability) measure of P underX, the above equation becomes the following n fold convolution equation</p><p> = 1n 1</p><p>n| {z }n times</p><p>;(1.1)</p><p>1991 Mathematics Subject Classication. Primary: 60B15, 60E10; Secondary 43A05.Keywords and phrases. Locally compact Abelian groups, admissible probability measures, partly</p><p>divisible probability measures.</p></li><li><p>6 Math. Nachr. 213 (2000)</p><p>where 1n</p><p>= P X11 is the image (probability) measure of P under X1. Furthermore,Equation (1.1) can be written in terms of characteristic functions (via the Fouriertransform) as follows</p><p> =c 1</p><p>n</p><p>n:</p><p>We may generalize this problem and ask whether for q = mn</p><p>with n;m 2 IN beingmutually prime (abbr. m. p.) there exists a characteristic function (c. f.) dm</p><p>nsuch</p><p>that()m =</p><p>(dmn</p><p>n:(1.2)</p><p>Let M1(G) denote the space of probability measures on G. Given a measure 2 M1(G), we may thus define a set of positive rational numbers alg() as fol-lows:</p><p>alg() =nq =</p><p>m</p><p>n: n;m 2 IN m.p. and 9 c. f. dm</p><p>nsuch that ()m =</p><p>(dmn</p><p>no:</p><p>This set is clearly designed to contain the information about the (algebraic) divisibilityof .</p><p>In the case where is a real function with 0 &lt; 1, we can present a candidatefor the solutions of Equation (1.2) as follows</p><p>dmn</p><p>= ()mn</p><p>which leaves open the question whether mn is a number such that ()mn is a character-</p><p>istic function.So the (general) problem which arises in connection with the above observation can</p><p>be roughly formulated as follows: for which kind of probability measures is the set</p><p>() =t 2 (0;1) : ()t is a characteristic function}</p><p>well defined and what kind of subsets of (0;1) do possess the property of being() sets for some probability measure ?</p><p>We have studied this question, which in the case G = IR goes back to D. Dugue[7], in a previous note for this special case [1], where also concrete examples have beendiscussed. More examples can be found in [4, 7]. All these examples together indicatethat there exist a great variety of sets S (0;1) such that a measure with () = Scan be found.</p><p>In this paper we extend these considerations to the framework of locally compactAbelian groups. In particular, in Section 2 we give a proper definition of the set ()for the case of a LCA group with connected dual group bG. This leads to the notionof admissible probability measures on G. In Section 3 we show that the algebraicdivisibility given by alg() for non admissible probability measures is restrictedby some rather general considerations, which do not apply to the admissible case. InSection 4 we discuss the situation G = ZZ2 as the simplest example of a LCA groupwith non connected dual group and we show that there is no canonical definition ofthe set () in this case.</p></li><li><p>Albeverio/Gottschalk/Wu, Partly Divisible Measures on LCAGroups 7</p><p>2. Admissible probability measures on LCA groupswith connected dual groups</p><p>In this section we extend the notion of admissible probability distributions on IRgiven in [1] to the case of probability measures on a LCA group (G;+) with arcwiseconnected dual group</p><p>( bG; . Recall that M1(G) is the set of probability measures onG. We say 2M1(G) is admissible, if there is some continuous function : bG! Cwith (1) = 0 such that = e . We shall denote the set of all admissible probabilitymeasures on G by Ma1(G).</p><p>The function is uniquely determined. In fact, if 1 and 2 fulfil the above condi-tions, then</p><p> 1() 2() = 2i k() with k() 2 ZZ for all 2 bG:Since the left hand side of the above equation is continuous, the right hand sidemust also be continuous and is thus constant, since a topological space X is connectedif and only if every continuous map from X to ZZ is constant. Evaluation of the aboveequation at = 1 now yields that 1 = 2. For 2 Ma1(G) we call the function : bG ! C, which is uniquely determined by the conditions specified above, thesecond characteristic associated with .</p><p>We now give the main definition of this article in analogy to [1]:</p><p>Definition 2.1. Let G be a LCA group with arcwise connected dual group bG.For 2 Ma1(G) with second characteristic we define the set () of positive realnumbers as</p><p>() =t &gt; 0 : et is a characteristic functiong :</p><p>Remark 2.2. (i) By Theore`me 4 of [6], bG is arcwise connected ) bG = IRd Tn ,G = IRd ZZn, where Tn stands for the n dimensional torus. In particular, arcwiseconnected LCA groups are also locally arcwise connected.</p><p>(ii) Let L(G) denote the complex measures on G which have a logarithm in theBanach algebra of complex measures with the convolution as multiplication and L1(G)the probability measures in L(G). Then L1(G) Ma1(G) where equality holds if bGis compact (i. e. G = ZZn) and the inclusion is proper if bG is not compact (G =IRd ZZn; d 6= 0, since e. g. any Gaussian measure supported on one of the copiesof IR has unbounded second characteristic and thus does not belong to L1(G). Forresults on the characterization of L(G) see e. g. [8, 9, 11].</p><p>(iii) () (0;1) is a closed semigroup under addition with IN (). Either() = (0;1) or there exists &gt; 0 such that () [;1). If the interior of () isnot empty, then there exists &gt; 0 such that () [;1), cf. Prop. 2.1 and 2.2 of[1].</p><p>(iv) 2 Ma1 (G) is infinitely divisible if and only if () = (0;1) (Schoenbergstheorem [3]). If () 6= (0;1), we say is partly divisible and if () = IN we call minimally divisible (see [1] for examples).</p></li><li><p>8 Math. Nachr. 213 (2000)</p><p>We now want to give a topological characterization of the Fourier transform ofadmissible probability measures.</p><p>The assumptions on bG allow us to apply the theory of covering spaces to characterizethe Fourier transforms of admissible probability measures.</p><p>Let us first recall some basic notions of algebraic topology following [12]. Let X; Ybe topological spaces. Then : (X; x0) ! (Y; y0) denotes a continuous map from X toY which maps x0 to y0. The fundamental group of X (resp. Y ) based at x0 (resp. y0)is denoted by (X; x0) (resp. (Y; y0)). Then : (X; x0) ! (Y; y0) is the grouphomomorphism induced by .</p><p>A covering space(X; x0; </p><p>of a topological space (X; x0) consists of a topological</p><p>space(X; x0</p><p>and a map :</p><p>(X; x0</p><p> ! (X; x0) such that for every x 2 X thereexists an arcwise connected open x neighborhood U X such that each arcwiseconnected component of 1(U) is homeomorphic to U under the restriction of tothis component. A lifting of the map : (Y; y0) ! (X; x0) to</p><p>(X; x0; </p><p>is a map</p><p> : (Y; y0)!(X; x0</p><p>such that the following diagram commutes:</p><p>(Y; y0) (X; x0)</p><p>(X; x0</p><p>?</p><p>&gt;</p><p>-</p><p>If (Y; y0) is arcwise connected and locally arcwise connected and (X; x0) has a cov-ering space</p><p>(X; x0; </p><p>, then by a well known theorem (c. f. Theorem 5.1 [12] p.</p><p>128) a map : (Y; y0) ! (X; x0) has a lifting : (Y; y0) !(X; x0</p><p>if and only if</p><p>((Y; y0)) ((X; x0</p><p>holds. If such a lifting exists, it is unique.</p><p>Theorem 2.3. Let G be a LCA group with arcwise connected dual group bG andC = C f0g. Then a probability measure on G is admissible if and only if thefollowing two conditions hold:</p><p>1. () 6= 0; for all 2 bG, i. e., can be seen as a map : ( bG; 1 ! (C ; 1);2. </p><p>(( bG; 1 = f0g holds for : ( bG; 1 ! (C ; 1).</p><p>Proof . By Remark 2.2 (i), bG is also locally arcwise connected.Let R = (0;1) IR be the Riemannian surface of the logarithm. We shortly denote</p><p>the element (1; 0) 2 R by 1. We set : (r; ) 2 R 7! rei 2 C </p><p>then it is clear that (R; 1; ) is a covering space of (C ; 1). Furthermore, we noticethat</p><p>logR : (r; ) 2 R 7! log r + i 2 Cmaps (R; 1) homeomorphically to (C ; 0), where the symbol log on the right handside of the mapping stands for the real logarithm. Thus by using the lifting argumentagain, we derive the following commutative diagram:</p></li><li><p>Albeverio/Gottschalk/Wu, Partly Divisible Measures on LCAGroups 9</p><p>( bG; 1 (C ; 0)(R; 1)</p><p>(C ; 1)</p><p>- </p><p>@@</p><p>@@R</p><p>?</p><p>@@</p><p>@R</p><p>(logR)1</p><p>@@</p><p>@@IlogR</p><p>exp</p><p>Thus, a probability measure on the LCA group G has a Fourier transform whichcan be represented in the form = e with :</p><p>( bG; 1 ! (C ; 0) if and only if () 6= 0,for all 2 bG, and : ( bG; 1 ! (C ; 0) can be lifted to (R; 1). Taking into account thetopological properties of bG and (R; 1) = 0 (and consequently ((R; 1)) = f0g), anapplication of the theorem on the existence of liftings given above then concludes theproof. 2</p><p>Remark 2.4. (i) Of course, the second condition of Theorem 2.3 is trivial if( bG; 1 = f0g. This is true for G = bG = IRd. In this special case Theorem 1.2</p><p>can be obtained without using homotopy, see e. g. [1] for d=1 or [2] p. 220 223for d 2 IN. Therefore, the notion of admissible probability measures on LCAgroupspresented here really extends the notion of admissibility given in [1] in the case thatG = IR</p><p>(and also extends the discussion presented in [2] for G = IRd; d 2 IN.</p><p>(ii) If the LCA group G has arcwise connected and locally arcwise connected dualgroup bG with nontrivial fundamental group, there exist non admissible probabilitymeasures 2M1(G) with () 6= 0 for all 2 bG. Notice that the simplest example forsuch a LCA group G is ZZ with dual group T1, hence let us take G = ZZ as an exampleto elucidate this point. In this case, we have</p><p>( bG; 1 = (T; 0) = [0; 2]=02. Then( bG; 1 = (T; 0) = ZZ. Furthermore we identify (C ; 1) with ZZ. Let n 2 ZZ f0g</p><p>and n be the Dirac measure at n. For s 2 T we get that n(s) = eisn 6= 0. Butn : ZZ ! ZZ obviously acts as multiplication by n and thus n</p><p>(( bG; 1 = nZZ 6= f0g.</p><p>Thus, n is not admissible. Since a copy of T is contained in every such bG, the aboveargument carries over to the general case.</p><p>(iii) The above example also shows that for any LCA group GV = IRd+n witharcwise connected and locally arcwise connected dual group bG (therefore one hasG 6=V !), in general, Ma1(G) 6 Ma1(V ): We have for 2 M1(G) \Ma1(V ) with V second characteristic that 2 Ma1(G) , (x; y) = (x; y + z) for all x 2 IRd; y 2IRn; z 2 2ZZn.</p><p>Remark 2.4 (ii) gives rise to the question, whether such non admissible 2M1(G)with () 6= 0 for all 2 bG can occur as integer roots of some admissible probabilitymeasure 2Ma1(G)? The following theorem gives a negative answer.</p><p>Theorem 2.5. Let G be a LCA group with locally arcwise and arcwise connecteddual group bG. For 2Ma1(G) and n 2 IN, we assume that 1n 2 M1(G) is a solutionto the problem</p></li><li><p>10 Math. Nachr. 213 (2000)</p><p> = 1n 1</p><p>n| {z }n times</p><p>:(2.1)</p><p>Then 1n</p><p>is admissible. Furthermore, 1n</p><p>is the only probability measure on G whichsolves the above problem.</p><p>Proof . The condition that c 1n() 6= 0 for all 2 bG follows immediately by taking</p><p>the Fourier transformation of Equation (2.1). It remains to show that condition 2. ofTheorem 2.3 holds.</p><p>Since 1n</p><p>solves the above problem, the following diagram commutes:</p><p>( bG; 1 (C ; 1)</p><p>(C ; 1)</p><p>?</p><p>z 7! zn</p><p>&gt;c 1</p><p>n</p><p>-</p><p>By identifying (C ; 1) with (ZZ; 0), we get that the following diagram commutes:</p><p>( bG; 1 (ZZ; 0)</p><p>(ZZ; 0)</p><p>?</p><p>m 7! nm</p><p>&gt;d 1</p><p>n</p><p>-</p><p>Since (( bG; 1 = f0g, this is only possible if d 1</p><p>n(( bG; 1 = f0g. Thus, 1</p><p>nis</p><p>admissible.Let be the second characteristic of and 1</p><p>nthe second characteristic of 1</p><p>n.</p><p>Since =( c 1</p><p>n</p><p>n = en 1n and n 1n</p><p>is continuous and fulfils n 1n(1) = 0 we get that</p><p> = n 1n. But this determines 1</p><p>nuniquely and thus 1</p><p>nis also uniquely determined.</p><p>2</p><p>In other words, this theorem says that there exists a solution to Equation (2.1) ifand only if 1n 2 (). If such a solution exists, then its Fourier transform equals toe</p><p>1n , where is the second characteristic of . Since the convolution of an admissible</p><p>measure with itself is again admissible, this proves that () really measures thedivisibility of , i. e. alg() = () \Q.</p><p>Remark 2.6. The uniqueness of the roots can also be obtained using the imbeddingof G into a vector space V and applying the result of Bauer [2] p. 220 223 on theuniqueness of roots of probability measures on (finite dimensional) vector spaces. G admissibility of the roots then follows from Remark 2.4 (iii).</p><p>Remark 2.7. (i) The considerations of this section and Theorem 3.1 below canbe extended to the case of LCAgroups with only connected (not arcwise connected)</p></li><li><p>Albeverio/Gottschalk/Wu, Partly Divisible Measures on LCAGroups 11</p><p>dual group bG as follows: By [10] Theorem N (p. 15), any such bG can be obtainedas projective limit of arcwise connected LCA groups bG = IRd Tn. Then isadmissible if and only if there exists a projective family of functions :</p><p>( bG; 1 !(C ; 1) s. t. (the projective limit) p lim = and there exists s. t. fulfils theconditions of Theorem 2.3.</p><p>(ii) Likewise, if G is a LCAgroup with dual group bG which can be written asinductive limit of arcwise connected LCA groups bG, then a measure on G isadmissible if and only if every projection to bG fulfils the conditions of Theorem2.3.</p><p>3. On the divisibility of non admissible probability measures</p><p>Let G be a LCA group with arcwise connected dual group bG. In this sectionwe investigate what happens if for a given probability measure on G either of therequirements of Theorem 2.3 is not fulfiled. This leads to restrictions on alg(). Firstwe consider this for the case that condition 1. of Theorem 2.3 hold and condition 2.is violated:</p><p>Theorem 3.1. Let 2 M1(G) Ma1(G) such that () 6= 0 holds for all 2 bG.We identify (C ; 1) with ZZ and n j </p><p>(( bG; 1 means that for all z 2 (( bG; 1</p><p>there exists l 2 ZZ such that z = nl. Then(i)</p><p>alg() [</p><p>n2IN : nj((bG;1))1n</p><p>IN :</p><p>(ii)</p><p>alg() \</p><p>z2((bG;1))f0g1jzj IN :</p><p>(iii) alg() is bounded from below by 1=minnjzj : z 2 </p><p>(( bG; 1 f0go.</p><p>Proof . (i) By a diagram analogous to the second diagram in the proof of Theorem2.5 we get for q 2 alg(); q = mn ; (m; n 2 IN mutually prime) that for a solution ofEquation (1.2)</p><p>n mn(( bG; 1 = m(( bG; 1 :(3.1)</p><p>Since n and m are mutually prime, this can only be true if for z 2 (( bG; 1 there</p><p>exists l 2 ZZ s. t. z = nl holds.(ii) Let q; m; n be as above. By Eq....</p></li></ul>