Research Article | Open Access

Wilfried Hazod, "The Concentration Function Problem for Locally Compact Groups Revisited: Nondissipating Space-Time Random Walks, -Decomposable Laws, and Their Continuous Time Analogues", *Journal of Mathematics*, vol. 2013, Article ID 540471, 15 pages, 2013. https://doi.org/10.1155/2013/540471

# The Concentration Function Problem for Locally Compact Groups Revisited: Nondissipating Space-Time Random Walks, -Decomposable Laws, and Their Continuous Time Analogues

**Academic Editor:**Ralf Meyer

#### Abstract

The concentration function problem for locally compact groups is concerned with the structure of groups admitting adapted nondissipating random walks. It is closely related to discrete relatively compact M- or skew convolution semigroups and corresponding space-time random walks, and to -decomposable laws, respectively, where denotes an automorphism. Analogous results are obtained in the case of continuous time: nondissipating Lévy processes are related to relatively compact distributions of generalized Ornstein-Uhlenbeck processes and corresponding space-time processes and to -decomposable laws, respectively with denoting a continuous group of automorphisms acting as contracting mod. a compact subgroup.

#### 1. Introduction

Let be a locally compact group, an (w.l.o.g.) adapted probability measure. , more precisely, the random walk , is called nondissipating (also nonscattering) if for some compact subset the (right) concentration functions fail to converge to , with time . To avoid trivialities, throughout is supposed to be noncompact, else any random walk would be nondissipating.

If the random walk is nondissipating, is relatively (right) shift compact, equivalently, is relatively compact [1].

Furthermore, if denotes the smallest closed normal subgroup of containing the support , then . Hence there exists such that with .

Denoting the restriction of the inner automorphism to by , we obtain , and is represented as , with hence the random walk is representable as , for all (with , .)

The existence of nondissipating adapted random walks has a strong impact on the structure of : it is shown in [1] that , where is a compact -invariant subgroup of and (for ), respectively, denote the contractible, respectively, the -contractible subgroups. As easily seen, for a random walk , is representable as , , (“” denoting convolution on ) and hence satisfies the relation , . We call a sequence satisfying such a relation “discrete time M-(or Mehler-)-semigroup” (in fact, in analogy to the continuous time case, the (discrete) semigroup of transition kernels is called discrete Mehler semigroup.) This M-(*convolution*)-*semigroup* is also called *skew convolution semigroup* or -*semigroup*. Moreover, if is nondissipating, the M-semigroup is relatively compact, a crucial property in the sequel. Conversely, discrete M-semigroups define *space-time* random walks on the *space-time building* . So, nondissipating adapted random walks (on ) correspond in a 1-1-manner to relatively compact M-semigroups (on ), cf. Theorem 6 later. We furthermore show that exists and is -*decomposable*, that is, with relatively compact M-semigroups of *cofactors* ; furthermore, is right -invariant. Conversely, for any such measure with cofactors , we have . Thus -decomposable measures , respectively, and the cofactors generate nondissipating random walks and vice versa.

The first part of the paper may be considered as an overview of more or less known results around the concentration function problem and its relations to M-semigroups and -decomposability. These results are needed as motivation and as a tool box for proofs in the second part which is concerned with the continuous time analogues. The random walk is replaced by a continuous convolution semigroup , the distributions of a Lévy process. At first (Propositions 12 and 13), we show that is nondissipating and adapted iff some (hence all) *skeleton random walk * is nondissipating; it is adapted on , the group generated by . Furthermore, we show that the subgroup is independent of and is a normal subgroup in (not only within the group ), and we observe . Thus the results for discrete times apply easily to the continuous time setup. Furthermore, , and there exists a continuous one-parameter group such that with , . Moreover, in that case, there exists a homogeneous group (i.e., a contractible Lie group) and a compact -invariant subgroup , such that , .

Theorems 20 and 22 contain the continuous time analogues of Theorem 6, in particular, also a characterization of relative compactness of M-semigroups in terms of logarithmic, respectively, first order moments of the corresponding Lévy measure (on resp., .)

One of the aims of the paper is to point out that different branches of investigations lead to the same or equivalent objects: (1) nondissipating random walks, respectively, continuous convolution semigroups, (2) -decomposable laws and the subclasses of (semi-) stable and self-decomposable laws, and (3) (relatively compact) skew (discrete or continuous) convolution semigroups. While in the past in the cases (2) and (3) most investigations were concerned with continuous time models, the emphasis in the first case was on discrete time models.

All these investigations—for discrete and continuous times—are related to the structure of contractible, respectively, -contractible subgroups , respectively, of locally compact groups. See for example, [2–5], [6, Ch. III], [1, 7–15], and the literature mentioned there.

For the history of the concentration function problem for random walks on locally compact groups the reader is referred to the survey of Jaworski [1] showing previous developments and a recent state of investigations: beginning with the pioneer works [16–20] to the investigations [1, 11, 13].

Continuous time models (for the cases (2) and (3)) had been investigated in the past in different papers. Beginning with the pioneer work [21] (with slightly different representations) to [22, 23], [6, § 2.14]. See also [24] for Mehler hemigroups and embedding of discrete time models into continuous time ones. In [22] the reader will find some more hints to the literature, in particular also to vector spaces, beyond the locally compact group case.

#### 2. Discrete Time: Nondissipating Random Walks

Recall the following notations: denotes a locally compact group with unit . denotes the filter basis of Borel neighbourhoods of the unit. For , the set of probabilities, let be the image of under the inverse mapping . Denotes convolution on , the th convolution power, , where denotes the point measure in . w.l.o.g. is supposed to be adapted; that is, is the closed group generated by the support . denotes the smallest closed normal subgroup containing . The concentration function of the random walk is defined as for compact . The probability measure , or more precisely, the random walk is called nondissipating if fails to converge to with , for some compact . Recall that we defined afore for and a compact -invariant subgroup : and . is contracting if . And if a contracting automorphism exists, is called contractible. Analogously we define -contracting automorphisms, respectively, -contractible groups, where denotes a compact -invariant subgroup. will throughout denote the set of accumulation points, and denotes the closure of a set .

We collect some properties of adapted random walks:

*Fact 1. *(a) is nondissipating iff is relatively (right) shift compact; that is, for some , is relatively compact. Equivalently, iff is relatively compact. Indeed, then exists.

In that case, if is noncompact, the following assertions (b)–(f) hold:

(b) , hence the shifts can be chosen as for some .

(c) The restriction to of the inner automorphism , , is considered as automorphism of , and hence has a canonical representation (with product for , where .)

(d) There exists a compact subgroup such that . If is compact then .

(e) and are representable as , is identified with a probability of . Hence by induction, , , with , , , and is identified with . Furthermore, is relatively compact. (Indeed, as is convergent, exists.) To distinguish, in the sequel “” will denote convolution on while convolution on is denoted by “”. (Thus we have .)

(f) Put . Then is the smallest closed -invariant normal subgroup of containing . (The closed group generated by may be smaller than .)

(a) See [1, Corollary 3.2] and the literature mentioned there. In particular, [25], [12, Theorem 1]. For the convergence of , see [26, 27].

(b), (c) See [1, Theorem 3.4], [25]. (d) See [1, Theorems 3.5 and 3.9]. If is compact, then, as is compactly contracting, follows.

(e) The representation of is immediately verified, and relative compactness follows by [1, Theorem 3.9].

(f) is adapted, hence is generated by . , hence is -invariant and . By definition, is the smallest closed normal subgroup with these properties. It is easily shown by examples that may be larger than the group generated by

As already mentioned, is always supposed to be noncompact, else any random walk would be nondissipating. If is compact then (by Fact 1(d)), and any “space-time” random walk , , is nondissipating.

We have to recall some more notations and facts: let be a locally compact group. A function is subadditive (resp., a “gauge”) if (resp., for some constant ). is submultiplicative if . As immediately seen, if is subadditive (and ≥0), then is submultiplicative (and ≥1), and if is submultiplicative (and ≥1), then is subadditive (and ≥0). Thus is subadditive and is submultiplicative. We will always assume tacitly that is Borel measurable. Therefore, is locally bounded (see [28, Proposition 1]), hence there exist such that . To any we fix such an (and a neighbourhood ).

Let be compactly generated, for some symmetric compact . Then we define a subadditive function . According to [28, Proposition 1], any gauge, in particular any subadditive function , is dominated by , that is,

*Definition 1. *(a) Let . A subadditive function is called -function (cf. [1, Definition 2.6]) if for some , , we have , . is a regular -function if in addition , is relatively compact for some (hence all) and iff . In fact, is continuous and -bi-invariant (see [1, Propositions 2.8 and 2.10].)

Let , and let denote a nonnegative measure on such that is a bounded measure outside any neighbourhood of the unit.

(b) , respectively, possess finite first-order moments if , respectively, for all subadditive Borel functions and are defined previously.

(c) Analogously, we define for a group : , respectively, possess finite logarithmic moments if , respectively, for all regular -functions , as in (c).

In fact, if the integrals in (b), respectively, in (c) are finite for some (as above), they are finite for all . And if the integrals are finite for some relatively compact , they are finite for all .

*Remark 2. *The definition of logarithmic moments depends on -functions and hence on and on the particular automorphism . If and is a regular -function, and if we define , , then is generated by and (cf. [1, Proposition 2.14].) If is an other relatively compact symmetric neighbourhood generating , then (see [28, Remarque 3].) Hence possesses finite first-order moments if , equivalently, is integrable. Hence possesses finite first-order moments iff (see [1, Proposition 2.14, Corollary 2.15].)

Note that for contractible groups logarithmic moments are frequently defined in terms of “group-norms” or “homogeneous norms” (e.g., in [29, 2.6, 2.11], [6, 2.7.26 (f)]), [30]), generalizing the situation of vector spaces. We show in Lemma A.2 and Theorem 22 (cf. Remark 21) that on homogeneous groups this definition is equivalent to Definition 1. In fact, if is a contracting automorphism, then there exist and a continuous group such that (cf., e.g., [6, 2.8.14]), and therefore, Lemma A.2, respectively, Theorem 22 apply also for discrete times.

*Fact 2. *Let be a locally compact group; , , and, as before, . Let . ( denotes the normalized Haar measure on the compact subgroup .) Then the following assertions are equivalent:(i) is weakly convergent ; that is, exists.(ii) possesses finite logarithmic moments.(iii) possesses finite first-order moments.

The equivalence of the assertions “(i)-(ii)” is shown in the context of invariant metrics on , respectively, of regular -functions, in [1, Theorem 3.9, Propositions 4.2 and 4.3]. For “” see [1, Corollary 2.15].

The equivalence of the conditions “(i)-(ii)” (as well as equivalence with stochastic and almost sure convergence of corresponding random variables) is folklore for vector spaces (cf. [31], or e.g., the monograph [32, Lemma 3.6.5]), for homogeneous groups [29], [6, 2.14.24], for general contractible groups [30] (in the context of “group norms”.) As long as almost sure convergence of corresponding random variables is considered, Lévy's equivalence theorem is involved, and hence the underlying group is usually supposed to be second countable. In the situation here a superfluous condition as is metrizable, cf. [1, Theorems 2.5, Proof of Theorem 3.9]. Furthermore, in most cases, metrizability may be supposed without loss of generality.

*Remark 3. *The groups and , generated by the supports of the random walk, respectively, the M-semigroup (cf. Definition 4 below) are -compact. Therefore, and are representable as projective limits of second countable locally compact groups (cf. [33, page 101, Example 11] or [34, Theorem 5.2]): , respectively, , where are compact, -invariant subgroups and . Therefore in many investigations second countability may be assumed w.l.o.g. For example, is convergent, respectively, is relatively compact, respectively, satisfies (2) below iff for all the projections to the quotients share this property.

*Definition 4. *(a) A sequence in satisfying
is called discrete M-(convolution)-semigroup (also called Mehler semigroup, -semigroup, skew convolution semigroup, etc.)

Note that we are mainly interested in the behaviour of the random walk, respectively, the M-semigroup for large times ; hence the definition of is of minor importance. We could replace by an idempotent such that , that is, . denoting the Haar measure on a compact subgroup .

(b) is called -decomposable, if for some cofactor we have . Then by induction, . denotes the set of cofactors of with respect to . As immediately seen, one can choose for ; hence the cofactors can be chosen as (discrete) M-semigroup .

(c) is called right -invariant if (for some compact subgroup ).

(d) For short, is --decomposable if is -decomposable and right -invariant.

is closed, hence according to the shift-compactness theorem (cf. [35, III, Theorems 2.1 and 2.2] (for metrizable groups), [36, Theorem 1.2.21]), is compact in . If is relatively compact, in particular, if , then, again by the shift-compactness theorem, is relatively compact.

*Remark 5. *In the following, we distinguish carefully between adapted and nonadapted space-time random walks. Let be a nondissipating space-time random walk on with relatively compact M-semigroup . If the random walk is not adapted let denote the closed subgroup generated by . Hence, on , is nondissipating and adapted. Therefore there exists a closed subgroup in , normal in , containing the supports , and , such that with there exists a relatively compact M-semigroup (with respect to ) , such that for . Let ; hence with , . Then iff . Note that since .

Furthermore, , , and . Hence , analogously, , . And thus .

is the smallest closed normal subgroup of containing , and ; hence it follows that a nondissipating space-time random walk is adapted on iff for any closed subgroup and , with normalizing and satisfying

In this sense we may w.l.o.g. assume that a nondissipating random walk on a locally compact group is adapted, respectively, a relatively compact M-semigroup on a locally compact group (with respect to ) satisfies condition (3). (If necessary replacing by , , as previously mentioned.)

Relations between nondissipative random walks, M-semigroups, and -decomposable laws are collected in the following.

Theorem 6. *(a) Let be a locally compact group and a nondissipating adapted random walk. Then there exist a closed normal subgroup and , such that with , one has , for a compact, -invariant subgroup . Furthermore, the random walk is representable as , where is a relatively compact M-semigroup (with respect to ), and condition (3) is satisfied.**(b) Conversely, let be a locally compact group, , and let be a relatively compact M-semigroup in , and assume condition (3) to be satisfied. Then the space-time random walk is nondissipating and adapted. ( denotes ). Hence (a) applies. In particular, for some compact -invariant subgroup .**(c) Non-dissipative random walks, respectively, relatively compact M-semigroups define -decomposable laws.**In fact, in case (a) or (b) we obtains that exists and is --decomposable.**(d) Conversely, -decomposable laws define nondissipating random walks, respectively, relatively compact M-semigroups.**Precisely, assume that . Let be --decomposable. Then there exists a M-semigroup of cofactors of such that . In particular, is relatively compact, and then and hence (a), respectively, (b) apply.**(e) Let , . An adapted space-time random walk is nondissipating, respectively, a M-semigroup satisfying (3) is relatively compact iff possesses finite logarithmic moments, respectively, iff possesses finite first-order moments.*

*Proof. *For (a), (b) see Fact 1 and the discussion in Remark 5.

(c) Let be a relatively compact M-semigroup on . Since is uniformly -contracting on compact subsets, the accumulation points of are supported by . Assume, for some subnets, that and and, w.l.o.g., for all . Then on the one hand, with on the other. Hence follows. That is, exists and is obviously right -invariant.

Furthermore, yields --decomposability of .

(d) Conversely, assume to be --decomposable. is relatively compact as is -contracting, and all accumulation points are supported by . Right -invariance of implies that . As afore mentioned, we can choose , in , hence as an M-semigroup of cofactors. Hence and yield relative compactness of and (2) according to the shift-compactness theorem.

(e) Follows by Fact 2, Remark 3, and [1, Corollary 2.15, resp., Proposition 4.3].

More generally we have the following.

*Remark 7. *(a) Let be a locally compact group, , as previously mentioned. Let , be a M-semigroup, respectively, the corresponding space-time random walk. As immediately seen, if the M-semigroup is only supposed to be relatively shift compact, then the random walk is nondissipating. And if the random walk is not adapted, Remark 5 applies.

(b) Let be a locally compact group, , and let be -decomposable, with as M-semigroup of cofactors. Then , , yields that is relatively shift compact and therefore (a) applies.

(c) If in (b) in addition is assumed to be relatively compact (e.g., if as in Theorem 6), then, again by the shift-compactness theorem, the M-semigroup of cofactors is already relatively compact.

Note that in Theorem 6 and Remark 7, if and , will not in general exist. See for example, Example 3.16 in [1], with compact and an infinite number of accumulation points. Further types of examples are obtained in the following way.

*Example 8. *(a) Let be a contractible locally compact group, with contracting , hence . Let be a relatively compact M-semigroup, . As , exists (Theorem 6(c) with ). Let be a finite cyclic group, let generating with , and assume for some that (e.g., ). Put , define as , and put finally . Then is a relatively compact M-semigroup (with respect to ) in , where . But and , both infinitely often; hence is not convergent. In fact, . But we have .

(b) A trivial case: let be a monothetic compact group with dense subgroup (for some ). Put , obviously -contracting; hence . Then is a M-semigroup (with respect to ), which is relatively compact in . But is not convergent, in fact, . Here we have .

In [12, Theorem 1], it is shown for a nondissipating random walk that there exists a measurable dense -invariant subgroup on which a sequence acts as contracting mod. a compact subgroup , and are concentrated on cosets of . Hence is weakly contracting mod. on (cf. the definition in [12]), and thus , for some -invariant subgroup [12, Theorem 5]. In interesting examples, for example, in the context of (semi-) stable laws, we have for a compact -invariant subgroup . In fact, investigations of the structure of contractible and -contractible sub-groups had also been pushed forward in connection with investigations of (semi-) stable laws. See for example, [4–6, 10] and the literature mentioned there. However, concentration functions were not used as an essential tool (except in connection with random time substitutions and *geometric* (semi-) stability, cf. [6, 37]). Nevertheless it is worth to point out that semistable laws provide interesting examples of relatively compact M-semigroups, hence of nondissipating random walks.

*Example 9 (cf., e.g., [6, § 3.5]). *Let be a continuous convolution semigroup in , and let be a locally compact group. Let and . is (strictly) -semistable if for all , . The idempotent is a normalized Haar measure on a compact -invariant subgroup . If is second countable, the contraction subgroups and are Borel sets, and we have for all . Hence we assume that , and therefore (as mentioned previously).

Let now . Then . Hence is --decomposable with cofactors . In particular, (and hence all ) possess finite logarithmic moments. (This could also be proved by direct calculation.) In that example, are right -invariant and exists. (And thus trivially also ).

If is closed and , then is closed, hence and . is known to be closed if there exist contracting continuous one-parameter groups of automorphisms [5] moreover for -adic Lie groups [38] and more generally for totally disconnected groups if is a “tidy” automorphism [14]. And we have , at least in the case of totally disconnected groups or of Lie groups. See [11]; see also [13–15] for previous results. For Lie groups see [5], [6, Theorem 3.2.13]. For more information concerning the decomposition property , cf. [39, 40]. However, will not in general be closed, not even on a 2-dimensional torus (cf., e.g., [5], [6, Example ]). Then, as mentioned previously if w.l.o.g. , is weakly contracting on , and hence for some compact, -invariant subgroup . Note that, if is not closed, additional conditions are needed to guarantee existence of (cf. Example 8 above).

The following result will explain in more detail the interplay between limit behaviour of relatively compact M-semigroups and -decomposability.

Proposition 10. *Let . Let be a relatively compact M-semigroup. To avoid measurability problems, is supposed to be second countable. Put and .**Then one has the following*(a)* (for any **).*(b).(c)*If at least some **and ** belong to a commutative **-sub-semigroup ** of **, then any ** is **-decomposable with ** for some *, .(d)*If ** for some compact **-invariant subgroup ** and **, for all **, and if one assumes in addition that **, then ** exists, and ** is *-*-decomposable with cofactors * .

*Proof. *By assumption, is compactly contracting on .(a)Let , and . Assume that that w.l.o.g. for all (else pass to subsequences). Then , . Hence , and analogously, follows.(b)Assume that . Then ; hence , and analogously follows.(c)Let ; hence for some . Let . w.l.o.g. assume that , else replace in the following by . Then, for all , and (along a subsequence) for some . Hence . That is, . Whence the assertion with .(d)Since by assumption are right -invariant, the accumulation points are right -invariant and supported by ; thus . Whence and thus .

The following example will illustrate Proposition 10.

*Example 11. *Let as before.

(a) Let be a -semistable continuous convolution semigroups (as in Example 9), and let be -decomposable with cofactor , with denoting the homothetic transformation on . Then the subordinated measure is -decomposable with cofactor . The set is a commutative sub-semigroup of containing and the cofactors (cf. Proposition 10(c)).

Let for some compact -invariant subgroup . Assume for all , then also and for all .

(b) Let be -decomposable with cofactors , . e.g., let be -semistable continuous convolution semigroups and Assume that and commute. e.g., for semistable laws, assume that commute for all Then is -decomposable with cofactor . But note that if are semistable and if is in general not -semistable for

If , for -invariant compact subgroups , then . If and commute, then is a compact -invariant subgroup and , the subgroup generated by , .

(c) In general, without commutativity assumptions, in (b) will not in general be -decomposable. In particular, for point measures , we obtain: is in general not -decomposable. In fact, without commutativity assumption, the left shifted measure, , is, as easily seen, -decomposable with cofactors . Representing as we see that is -decomposable, with , but in general not -decomposable.

(d) To simplify notations we assume that . Let and assume that now . Then and (in (c)) are concentrated on cosets of compactly contractible subgroups of : and . Hence, with we have: and . Furthermore, , where , .

(e) In case (c), (d) we determine the sets of accumulation points and (Proposition 10):

, hence , where . fact, obviously . On the other hand, assume and , w.l.o.g. . Then and

Furthermore, . Thus .

#### 3. Continuous Time: Nondissipating Continuous Convolution Semigroups

Next we replace the random walk by a continuous convolution semigroup (the distribution of a Lévy process on , if is metrizable). In contrast to the discrete time case, now the idempotent is uniquely determined.

W.l.o.g. we assume that is generated by . For short, is called adapted then. is nondissipating if the concentration functions do not converge to 0 for some compact (for ).

For any the random walk is called *skeleton random walk*. Note that a skeleton random walk do not need to be adapted on hence we introduce for the subgroups as smallest closed subgroup containing . Hence is adapted on . We define to be the smallest closed normal subgroup of containing . Hence and . (For later use we identify with .)

Again, to avoid trivialities, throughout in the sequel, is assumed to be noncompact (else any continuous convolution semigroup would be nondissipating). Hence also the closed subgroups are noncompact and the results of Section 1 apply. In fact, if for some , is compact, then is uniformly tight. But then is uniformly tight. Therefore (and all groups ) are compact.

First we compare the behaviour of concentration functions of continuous convolution semigroups and of their skeleton random walks.

Proposition 12. *Let be an adapted continuous convolution semigroup. Then the following assertions are equivalent.*(i) *is nondissipating.*(ii)*For all [(**) some]* *, the skeleton random walk* *is nondissipating (and adapted on* *).*(iii) *is relatively (right) shift compact.*(iv) *is relatively compact. (Then * *exists.)*(v)*For all [(**) some]* , * (considered as measure on* *) has finite first-order moments.*(vi)*For all [(**) some]* , * (considered as measure on **) has finite logarithmic moments. That is, for all regular* *-functions* *on* *one has* .

The precise formulations of conditions (v) (resp., (vi)) are: for all subadditive functions on (resp., for all regular -functions on ) we have (resp., ). Note that for a subadditive function on the restriction is subadditive. However, the sets of subadditive functions defining moments on , might have no subadditive extension to . Analogously, for , the sets of - and -functions may be different (even if , as will be shown in Proposition 13 later).

*Proof. * for all probabilities , hence is decreasing. Hence iff . Thus, as immediately seen, “”. And obviously we have “” and “”, “”. The equivalences “” (for any fixed ) follow by Theorem 6, and hence we also have . Convergence of follows by Fact 1(a): For all exists. Hence for commensurable , in particular, for all rational . Furthermore, for , is uniformly continuous. Whence for all real easily follows.

“”. By Fact 1(a), is equivalent to shift compactness of . Let such that is relatively compact. Then, being a continuous convolution semigroup, the set is relatively compact (where , if ). Whence follows.

“”. Let be relatively compact. Hence for any there exists a compact such that for all , . Therefore, fails to converge to .

A priori , the smallest closed normal subgroup of containing , might not be normal in and might depend on . In order to apply the results of Section 1 we have to overcome these difficulties.

Proposition 13. *Let be an adapted continuous convolution semigroup. Then one has for all , and .*

*Proof. *Obviously, and for all , for some (all) . First we consider dyadic numbers and then proceed to real applying continuity of .

Let . Then obviously since . For all it follows, , hence

In fact, for all , and hence for all . Therefore,
Put , and hence, by (5)*Claim 1.* We have

By (6) it suffices to show that . According to (4) and (5) we have
Analogously we have . Consequently, yields . Thus, as normalizes , we obtain .*Claim 2. *. is a closed normal subgroup of such that . But is minimal with this property. Whence , thus .*Claim 3. *. yields (as ). And ; therefore (since ). Hence, according to the definition, follows. Together we obtain , whence the assertion follows

By induction, we obtain for all , and .*Claim 4. * for all .

At first we consider dyadic .(1), whence . Hence .(2). Thus .

Hence for all dyadic . Continuity of easily yields , and