I ntnat. J. Math. Math. Si.
(1981)1-37
Vol. 4 No.
MOMENTS OF PROBABILITY MEASURES
ON A GROUP
HERBERT HEYER
Mathematisches Institut der Universitt
Auf der Morgenstelle i0
D-7400 Tbingen, W. Germany
(Received October 28, 1980)
ABSTRACT.
New developments and results in the theory of expectatiors
and variances for random variables with range in a topological group
are presented in the following order
notions (3) The three series theorem in
(i) Introduction (2) Basic
Banach spaces (4) Moment
Conditions (5) Expectations and variances (6) A general three series
theorem (7) The special cases of finite groups and Lie groups (8)The
strong laws of large numbers on a Lie group (9) Further studies on
moments of probability measures.
KEY WORDS AND HRASES. Probability Measures, Lie groups, Expecta-.
ions Variances Smigroup Homomorphism.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
@0B15, @OJ15
2
H. HEYER
1. INTRODUCTION
Moments of probability measures such as expectation and variance are
important tools throughout probability theory and its applications.
Moment conditions appear as helpful and crucial in almost all problems connected with the convergence in various senses of sequences
of independent random variables. We shall restrict our attention to
the almost sure convergence and the scope of the strong law of large
numbers.
The aim of this paper is to report on new developments and results
in the theory of expectations and variances for random variables
taking their values in a topological group. The groups under discussion will be either the additive groups of topological vector spaces
or locally compact groups. While the framework of a topological
vector space.admits a very natural extension of the notion of moments,
some new ideas are needed in the case of an arbitrary locally compact
group. Our exposition will be twofold" it will indicate the imitation
procedures and at the same time describe the innovation available.
We start with the classical set up. Let
on a probability space
(,(7,
X
be a real random variable
P). Its expectation and variance are
defined by
E (X)"
[XdP
V (X)"
(X
[xdP x
E(X))2dp
and
(x
E(X))2Px(dx)
3
MOMENTS OF PROBABILITY MEASURES ON A GROUP
resp.
In 1939 P. Lvy [18] systematically extended the notions of expectation and variance to spherical random variables
,
:
the torus
X
.
taking values in
instead of the real line
Lvy pro-
posed the definition
V (X)
mum in the definition of
X. Thus
E(X)
a
;(x-a) 2 Px(dx)
X, and every number
for the variance of
of
inf
V(X)
such that the infi-
a
is attained he called an expectation
is defined as any
a
f(X-ao )2 Px(dx)
.<
satisfying the ine-
quality
for all
a 6
(x-a) 2 Px(dx)
.
In the following, generalizations of Kolmogorov’s three-series theorem will be of central interest. For real random variables this
theorem states that for any sequence
series
Xj
converges a.s.
of random variables the
(Xj)j 1
X.] <
a.s. for abbreviation)
iff the following condition holds"
There exists a number
c
> o
such that
> C]
<
H. HEYER
4
(b)
Z
3
(e)
Z v
(x. i
3
j>.l
,
<
E (X. i
j.>l
[IXjl
-< c]
[IXjl
-< c]
and
In the case of spherical random variables the necessary and suffiX. <
cient condition for
a.s.
turns out to be
jl
(c’)
E
V (X)
<
E (X)
<
J
j.<l
(b’)
Z
J
j>l
and
E (X.)
where (b’) has to be read in the sense that
3
j>.l
fqr ,any choice of expectations
choice
of E
(Xj).
E
(Xj)
of
Xj
or for
< =
.s.om.e
,suitable
See L6vy [18] and Bartfai [1].
It is the nonuniqueness of the expectation in this case which yields
the appropriate extension of the concept to arbitrary compact groups.
2. BASIC NOTIONS
Before entering the subject proper a review of some basic notions
from probability theory on a topological group seems to be in order.
For details in the locally compact case the reader is referred to
Heyer [14].
5
MOMENTS OF PROBABILITY MEASURES ON A GROUP
Let
be a (completely regular) topological group with unit ele-
G
j%tb(G)
e. By
ment
G
on
we denote the set of all bounded Radon measures
in the sense that
b(G)
is a linear functional on the space
lowing property" For every
C
of
Igl
(
G
of bounded continuous functions on
G
1
l(g)l
such that
g(C
and
> 0
e
possessing the fol-
there exists a compact subset
for all
b(G)
g e
with
O. Clearly, every Radon measure on
G
can
be considered as a bounded regular sorel measure on the Borel--
(G)
algebra
of
Ab(G)
G. In
In fact, the mapping (,)
b(G)
G
from
*
G
m-
mG
on
into
is separable
G. This measure is unique up to a posi-
is compact. On the subsemigroup
sures on
R.(G)
b(G)
+
w
is separable and metrizable.
tive multiplicative constant, m is
gY
x
b(G)
is a locally compact group, then there exists a left (or right)
Haar measure
G
tb(G)
is continuous on norm bounded subsets
and metrizable iff
If
we introduce the weak topology
G
the weak topology
o-finite, and it is bounded iff
ALl(G)
w
of all probability mea-
coincides with the vague topolo-
defined as the topology of pointwise convergence on the space
v
.i(G)
of continuous functions on
g
with compact support.
is compact or locally compact iff
G
is compact. For any compact
subgroup
H
of
G,
mH
denotes the normed Haar measure of
fined as the (both sided) H-invariant measure in
CI(G)
H
de-
having
as its support. The normed Haar measures of compact subgroups are
b
and also of
exactly the idempotents of
I(G)
]b+(G).
H
H. HEYER
6
For every
G
x
the symbol
ex
The set of all Dirac measures on
denotes the Dirac measure in x.
G
will be abbreviated by
(G).
THE THREE-SERIES THEOREM IN BANACH SPACES
We are given a Bernoulli sequence on some probability space (n,O,P),
that is a sequence
P [c.
such that
Let
E
(j)j.>l
-I]
3
of independent random variables
cj
1
for all j >- I.
P [c. = +i]
3
p [I, 2]
be a Banach space,
q [2,].
and
We start with the following
E
DEFINITION.
is said to be of
there exists a constant
K > 0
n
.
Analoguously,
J J
k > 0
I"
and all sequences {x I,
(S,
E.
in
cot
q if theme
n
..., x n}
Clearly, every Banach space is of type
measure space
x n}
such that
n
n >.
...,
{x I,
is said to b of (Rademacher)
exists a constant
for all
J
j=l
and all sequences
E
if
n
j=l
1
p
such that
n
for all
(Rademacher) type
,
)
and
p
[1,2]
1
in
E.
.
LP(s, ,
and of cotype
the space
For any
u) is
MOMENTS OF PROBABILITY MEASURES ON A GROUP
of type 2
^
7
2 v p.
p and of cotype
We are now considering sequences
(Xj)j>.I
of independent random var-
iables taking values in a separable Banach space E. The norm closed
unit ball of
E
will be denoted by
variable and
C
a set in the Borel--algebra
defined to be equal to
E(X)
P-integrable,
Given a sequence
X
C
on
X
B. If
and to
is an E-valued random
(E),
of independent
E-valued random variables
,
we shall study the a.s. convergence of the series
X..
3
jl
Let
p 6 [i,[.
DEFINITION. A separable Banach space
Kolmogor, ov prope, rt.y of order p
(KPp)
E
is said to admit the
if for every sequence
,
of independent E-valued random variables, the series
($1)
(S2)
>l. P
j>.l
.
j>.l
(Xj.1 cB
E
(fiX j" 1 cB
is said to admit the
(SKPp)
< c]
E
j.>l
($3)
[1IX J li
X
(Xj)j.>I
con-
j>.l J
such that
c > o
verges a.s. whenever there exists a constant
E
is
X.
will denote the expectation of
(Xj)j>.l
X
otherwise. If
0
is
X.1 C
then
<
E, and
is summable in
-E (X j" 1 cB )P)
<
s,,t.rgng K,o, lmogoro.v property of.. orde_r p
if for every sequence
(Xj)j>.I
of independent E-valued random
variables, the a.s. convergence of the series
,
j>.l
X
J
is equivalent
Ho HEYER
8
($1)
to
($3)
through
c > O.
for some
THEOREM 3.1. A separable Banach space
p [I,2]
iff
E
is of type
E
admits
(KPp)
for
p.
Concerning the PROOF of the theorem we note that the implication
""
is essentially known. See Woyczyski [35], p. 431 for the special
2. The inverse implication follows directly from Hoffmann-
case p
J6rgensen [15], p. i16.
E
THEOREM 3.2. A separable Banach space
admits (SKP 2
iff
E
is a Hilbert space.
The PROOF of this result is based on a famous theorem of Kwapie’s
on the type-cotype characterization of Hilbert spaces. See Kwapie
[].
MOMENT CONDITIONS IN THE CASE OF AN ABELIAN LOCALLY COMPACT GROUP
.
We start by restating the classical three-series theorem in temms of
characters of the additive group of the real line
For any k-valued random variable
and any character X of
some
where
of the
y
Yo
for some
on a probability space
of the form
we define the
c
X
X(X) =
x
Xy(X):
A-valued random variable
c > o, and
Z
P)
e iyx for
yo Z,
denotes the principal branch
-valued random variable exp( i X)
Noting that with this definition of
Y:
=
(,0,
exp(i
Yo
Z, [X
#
X)
X
exp(i ).
Z] = [-c <X .< c], we
MOMENTS OF PROBABILITY MEASURES ON A GROUP
9
arrive at the following version of Kolmogorov’s result.
THEOREM 4.1. For any sequence
(Xj)j>l
of independent R-valued
random variables the following statements are equivalent"
(i)
(ii)
.
<
X.
a,s,
There exists a neighborhood
U:
]-c, c]
of o
such that
(a)
E
P[X
J
E(Y
J
V(Y
J
j.>l
(b)
.
.
j>-I
(c)
j>.l
<
U]
E
<
and
<
Using the above device the convergence conditions (a) through (c) of
(ii) can be restated as
(a’)
E
P [X
CU]
<
(b’)
E (log
oXj)
< ==, and
(c’)
V (log
xoXj)
<
the latter two holding for every character
of
Here the l-valued random variable of the form log
.
oX
is defined as
i0
iZ
H. HEYER
Z denotes the principal branch of the -random variable
where
exp( ixoX).
Now, let
G
be an Abelian locally compact group which we write mul-
tiplicatively with unit element
e. We suppose that
countable. If we denote the character group of
G
G
by
is second
G
we have
the following result of R.P. Pakshirajan [27].
THEOREM 4.2. For every sequence
(Xj)j>,l
of independent G-valued
random variables the following statements are equivalent"
(i)
<
H X.
a.s.
j.>i
(a)
(ii)
.
.
Given any (compact) neighborhood
P
j>.l
(b)
[Xj
E (log
j>,l
(c)
>3
U]
oX J
V (log oX)
log
oXj
of e, the series
converges.
<
<
the latter two conditions holding for every
dom variables
U
X
G A, and the
-ran-
defined as indicated above.
The PROOF of this theorem relies on a few auxiliary results.
LEMMA I. For every
valent:
X 6 G" the following statements are equi-
MOMENTS OF PROBABILITY MEASURES ON A GROUP
(i)
<
xoXj
a.s.
E (xoX)
(ii)
<
J
j
LEMMA 2. For every
tions
ape
X
GA the following two sets of condi-
equivalent
E(oX_.)
(i)
<
F.
.
and
j.
(ii)
Ii
V(oX
j.>l
E(Z
j>.l
<
J
V(Z
and
j>.l
J
J
<
<
LEMMA 3. The following statements are equivalent"
X.
(i)
j>.l
(ii)
]
<
a,s,
() There exists a (compact) neighborhood
that
j.<
xoXj
of
e
such
J
j>.l
(8)
U
<
a.s.
for all
X 6 G ^.
A more condensed version of Kolmogorov’s theorem reads as follows.
THEOREM 4.3. Let
G
be a second countable Abelian locally
compact group. Then for any sequence
(Xj)j>.I
of independent G-valued
random variables the following conditions are equivalent:
12
H. HEYER
H
(i)
<
X.
a.s.
oX.
(ii)
<
G
for every
a.s.
For the PROOF as given in Pakshirajan [27] one notes this"
(ii) follows directly from the continuity of
The implication (i)
G.
the characters X
(i) has to be shown for compact and for dis-
The implication (ii)
crete Abelian groups first, before the structure theorem for locally
compact Abelian groups can be applied.
5. AN AXIOMATIC APPROACH TO EXPECTATIONS AND VARIANCES
Let
G
.
be a compact group and let
such that
(G) c
DEFINITION.
J"
"
be a subsemigroup of
.admit .an
is said to
continuous semigroup homomorphismC
G
e.xpectation
E
if
J%L 1
E
is a
satisfying the following
conditions
(E i)
E (c x)
x
for all
(E 2)
E ()
e
for all symmetric measures
REMARK. If
admits an expectation, then
any nontrivial idempotent of
tent of
ALl (G)
G.
x
of the form
LI(G).
mH
"
_In fact, let
does not contain
u
be an idempo-
for some compact subgroup
H
MOMENTS OF PROBABILITY MEASURES ON A GROUP
of
G
.
such that u
E(x *H
E(H)
E(e x)
x
E(
x
Then
E(
e, which shows
e
mH
H)
H
13
for all
x*mH
x E
H
whence
xE( ), and consequently
H
{e}.
From this remark it becomes clear that for the subsequent discussion
we shall assume that the subsemigroups
borhoods of
in
ee
J
(G)
with
=
are neigh-
I(G).
REMARK. Under this additional assumption
Lie group (of dimension
G
is necessarily a
o).
p
For this one has to show that there exists a neighborhood of
without any nontrivial subgroup of
borhood of
e
pact subgroup
thus
G. Suppose that for every neigh-
there exists a nontrivial and hence a nontrivial com-
K. Then
mK
is a neighborhood of
and consequently
mK
e
K
{e}. But this contradicts the
assumption.
DEFINITION. A compact group
exists a subsemigroup
borhood of
e
in
ALl(G)
of
I(G)
such that
G
admits
(G)
with
."
an
expectation if there
=
which is a neigh-.
admits an expectation.
The following result is due to V.M. Maksimov [22], [24].
THEOREM 5.1. For any compact group
G
the following statements
are equivalent
(i)
G admits an expectation.
(ii) There exist integers p, q, r >. o such that G
=
SU(2) p x
qlq x
r
2,
H. HEYER
14
SU(2)
where
denotes the special linear rouD of dimension 2,
the 1-dimensional torus and
2
the 2-element roup.
Concerning the PROOF of this existence theorem we shall sketch the
ideas of both implications.
(1) Compact groups admitting expectations are necessarily of the
form
q
(q)"
Let
SU(2, p) x’’(q) x
G
:
2(r)-
and
2(r)
SU(2)
SU (2,p):
with
p,
2"
denote the system of irreducible (unitary contin-
F.(G)
G. For every
uous) representations of
u
bl(G)
and each
the Fourier transform of u
corresponding to the class
()
is defined by
resentative D
)t
Dg
(p):
-i’D (()(x)p(dx)
E
with rep-
M (d(g), ()
G
()) is the
’],(D ()) of
d(D
where d(o)"
space
()"
For any subset #C of
ALl(G)
dimension of the representing (Hilbert)
or
D ().
we consider the subset
D
()
(]) of
M(d(o), {). By the Peter-Weyl theorem the set
ol(G)"
is dense in
{u
tl(G)" D()(u)
[LI(G).
’
o for finitely many
It is therefore quite reasonable to deal with
representations of the form
15
MOMENTS OF PROBABILITY MEASURES ON A GROUP
N
DN
D(Oi
i=l
We note that
DN(f)
is a subsemigroup of
M(d(DN),
is a subsemigroup of
LI(G).
STEP I. Every semigroup homomorphism
G1
) whenever
induces a semigroup homomorphism
from
from
oN"
DN(F)
into some group
into G
1.
DN
nN
For every
and
generated by the set
D ()
let
{D(e)(x):
x
r(D (
))
G}. Then
be the matrix algebra
..(D (e))
is simple and
thus isomorphic to one of the algebras GL(d(o), <), GL(d(o), )
GL(d(o), 4). Analoguously one defines
(DN).
STEP II. Every semigroup homomorphism
group
G1
from
can be extended to a group homomorphism
group (generated by)
(D N)
into
G i"
o
DN(J)
into a
from the
H. HEYER
16
DN()
Inclusion
(D N
STEP III. It turns out that the compact group
product of homomOrphic images of the groups
GL(d,) into
G
is a direct
GL(d,), GL(d,) or
G. But such homomorphisms are known from the general
theory of determinants over fields.
(2)
Determination of expectations on the groups
SU(2),
and
2"
(2a) We first consider the group
SU(2)
I’1 2
"
+
I1 2
}
and define the semigroup
"
{
I(su(2))"
det
For every u
"
E(U).
k(u)
0 }.
we introduce
I--.(p)
where
d
S (2
S(2)(s
">
8
is determined by the condition det E(U)
is the unique expectation of
SU(2).
1. Then
E
17
MOMENTS OF PROBABILITY MEASURES ON A GROUP
(2b) For every i
j(1) 6
{1,
be a character of
again for every
u
u 6
{1,
i
...,
exp i j(1) with
tl()
Given a measure
N}
we obtain,
the l-th Fourier coefficient of
in the form
DI()
Let
.
D I()"
N} let
"
"
r
i I)
with
rI 6
+,(I)
I(). nl(u), ..., nl(u #
{u
E
we define an expectation
<
E()"
where
I exp
(1
(,(I),
(2c) The group
(N)), (YI’
denotes any vector of
YN
< (j(1),
...,
of
...,
j(N)),
0}. Then for every
by
..., yN)>
N
(mod
2),
satisfying
(Yl’ ’’’’YN )>
{e, x}
2
[0,2[
I.
admits a unique expectation
E
defined by
if
u({e}) > U({x})
EU):
otherwise
whenever
1(7 2).
Now we proceed to second moments.
DEFINITION.
"
is said to admit a variance
V
if
V
is a non-
H. HEYER
18
+
constant continuous semigroup homomorphism
satisfying the
following conditions(v)
v(F)
[O,1]
(v2)
v()
1
(G).
iff v
+
Any nonconstant continuous semigroup homomorphism
a weak variance on
J.
Variances and weak variances on
on
G
I(G)
G. In the first case
are called variances and weak variances for
we also say that
"
is called
admits a variance if there exists a variance
JLl (G)
REMARK. A weak variance
V
for all nontrivial idempotents
F is a
u of jLI(G)
variance iff
on
in
V(v)
0
J
Variances have been first defined by V.M. Maksimov in [19] and [20].
For a detailed presentation of the theory we recommend Heyer [14],
2.4.
THEOREM 5.2. Let
representatives
G
D (1)
be a compact group, o 1, ..., o N
..., D ( N and k 1, ..., kN 6 ,
with
+.
Then the mapping
N
(o i)
k.
m
i=l
I(G)
from fC
into
+
is a weak variance for
G. Moreover, the
following statements are equivalent"
(i)
V
(ii)
The smallest set
is a variance for
G.
[Ol, "’’’ ON
closed under products and
MOMENTS OF PROBABILITY MEASURES ON A GROUP
conjugates and containing
.
N(
DN
(iii)
D
(.)
{1’ ..., N}
19
equals
.
is faithful.
1
i=1
THEOREM ,5.3. For any compact group
G
the following statements
are equivalent"
admits a variance.
G
(ii)
is a Lie group (finite or infinite).
The PROOF of this result involves the general form of weak variances
Lb(G).
on
STEP I. Let
A
with unit element
be a simple algebra over
a nonconstant continuous homomorphism from the multiplicative
and
A
semigroup
and a
into
+*
k
JR+.
of
A
such that
o(a)
for all
D
Then there exist a representation
det
D(a)l k
A.
a
STEP II. Let
) A.
I
A:
be the Hilbert sum of a family
iI
(A.).
1
lI
with unit and let
of simple algebras over
constant continuous semigroup homomorphism form
there exist a finite subset
sentation
D I.
of
A.I
I
of
as well as a
A
I
and for every
k.l
E
+*
be a non-
into
such that
+.
iI’o
Then
a repre-
20
H. HEYER
(a)
for all
Idet Di(a i)
A.
(ai)iEl
a"
H
iEI o
A"
Application of this fact to the algebra
pact group
G
(and its Haar measure )
(G,) for our com-
which by the Peter-Weyl
theorem admits the decomposition
.
(G, )
where for every
.,
o
L
is the simple ideal
L
with
X
()
tr
D
"
{X
for
L
D
(G,)}
*f- f
"
E
,
yields
STEP III. Any nonconstant continuous semigroup homomorphism
form
L b (G)
into
+
is of the form
N
()
i-1
for all
b(G),
where
Idet
D
(
i
()I k i
k 1,
...,
k
N
+.
MOMENTS OF PROBABILITY MEASURES ON A GROUP
6. A GENERAL THREE-SERIES THEOREM
Let
be a compact group. For any sequence
G
consider the corresponding sequences
tion products
*
k+l*
k,n
DEFINITION. The sequence
vergent if for every
(j)j>.l
(k
Un"
n n>k
L1 (G)
in
we
of partial convolu-
is said to be
the sequence
ko
(uj)j>.l
(k,n)n>k
co.mposition c.o,n.-.
converges. In the
(k)
for every
case of composition convergence one has lim k
,n
n-=
for some compact subgroup H of G.
ko, and lim k)"
H
k
H is called the basis of (uj)j 1. In fact, H is the maximal com(k)
(k)
for all
pact subgroup of G with the property
*
(k)
()
x E H or
o.
* H whenever k
)"
ex
THEOREM 6.1. Let
tion
E
and a variance
every sequence
(j)j>.l
be a compact group
G
V
in
lent"
(i)
(uj)j>.l
(ii)
(a)
H
j.>]
(b)
E
admitinz
=n exreca-
on some subsemigroup
"
the following statements are equiva-
is composition convergent with basis {e}.
E ()
3
(i- V
<
())
<
For the PROOF of the theorem see Maksimov [24].
H. HEYER
22
DISCUSSION. The preceding result generalizes the classical
three-series theorem to compact (Lie)groups, since in
known equivalence theorem states that for any sequence
.
.
the well-
(Xj)j>.I
of
independent random variables, a.s. convergence and convergence in
n
of n-th pari e of the sequence (
X
distribution of
jl J’
tial sums, are equivalent.
Xj)n)l
This equivalence holds for random variables taking values in a io-
cally compact group
(See
G
iff
G
has no nontrivial compact subgroups.
[14 ], 2.2) Therefore, in the case of a compact group
X by the
j)l j
gence of the sequence ()
n n >-i of n-th partial products n
motivated to replace the a.s. convergence of
G
one is
W
-conver-
o,n
In the case of a separable Banach space the equivalence theorem
holds without any restriction. One recalls the
quoted in Woyczyski
It’Nisio theorem as
[35], p. 274.
7. THE SPECIAL CASES OF FINITE GROUPS AND LIE GROUPS
Here we shall discuss versions of the three-series theorem for general Lie groups including finite groups.
First of all we take up the
the form
G
{Xl, ..., Xp},
case of
a
where
xI
finite group G
of order p of
denotes the neutral element
e of G.
Let
(Xj)j>.l
be a sequence of independent G-valued mandom variables
on a probability space (,O, P). As befome we form
MOMENTS OF PROBABILITY MEASURES ON A GROUP
(Yn)n>l
the corresponding sequence
Y
n
X1
Clearly
an
23
of G-valued random variables
Xn
(Yn)n.>l
j(m) >. 1
converges a.s. if for P_ a.a. m
such that
X.(m)
for all
e
]
j
there exists
>- j(m). Thus,
for the
set
U N
C"
[X.
e]
]
j.>l
l.>1
(Yn)n >-1 we get P(C) 0 or 1. It
convergence of (Yn)n->l is equivalent to the
of points of convergence of
fol-
lows that the a.s.
ine-
quality
P( [Xj
J’>Jo
e])
>
Since for every j.>l the distribution
form
Pj
(Yn)n.>l
1
(j)
xI
+
+
0
p
PX.3
pj
(j)
Xp
Jo
for some
of
X
"> 1.
is of the
3
the a.s. convergence of
is in fact equivalent to the inequality
(j)
J>’Jo
With the notation
(j)
>
max (
0
for some
1
"’’’
THEOREM 7.1. For any sequence
(Xj)j>I
Jo >
p
1.
for all
j >. 1,
V.M. Maksimov in [19] proves the
variables
X.
]
with distribution
.
of G-valued random
of the form
24
H. HEYER
xI
1
j
+ c,
+
xP
the following statements are equivalent"
(i)
E
V(u )) <
($
v(.)
>
in the sense that
0
for some
J o >-
1
o
e(J)
(ii)
>
O.
j.>
As a direct consequence of this result one oDtains the three-series
THEOREM 7.2. For any sequence
random variables
(Xj)j>I
with distribution
Xj
uj
of independent G-valued
the following state-
ments are equivalent.
(i)
(Yn)n>i
.
(ii) (a)
(b)
converges
a.s.
(I- V( ))
<
j.
J
(Uj)j>1
converges to
e"
DISCUSSION (Comparison with the classical situation).
Condition (ii) (a) corresponds to the classical condition
(A)
j->l’
V(Xj IU
:
E
[Xjlu
-E
(Xjlu)]2
<
25
MOMENTS OF PROBABILITY MEASURES ON A GROUP
V
for the variances
(Xj 1 U)
of truncated variables
Xjlu,
U denoting
a neighborhood of 0.
Condition (ii) (b) corresponds to the classical condition
(B)
E (X
j>-I
since from
(A)
<
J i U)
(B)
and
follows that the variables
Xjl U
are
concentrated at 0.
It turns out that the third classical condition
(C)
,
P
[Xj
[U]
<
G. The corresponding
is redundant in the case of a finite group
condition (in terms of a nemghborhood U of e) can be deduced from
conditions (ii) (a) and (b). In fact, these conditions together with
the above THEOREM yield
(j)
>
O.
J>’Jo
But
E
j>.l
P
[Xj [U]
Ix J
whence condition (C).
P [X.
1
(1
j>.
e]
P [X
implies
(1
el)
j>.
ssJ)
6
H. HEYER
26
We proceed to the case_of_a..Lie group G.
G
Let
p >. 1
be a Lie group of dimension
X.
sequence of independent G-valued random variables
j. We consider
with distribution
Yn
partial products
on
]
(Yn)n>.l
the sequence
Xn
X1
(Xj)j>.I
and let
be a
(,0%, P)
of n-th
If for some neighborhood
U
of
e the series
.
P
diverges, then
.
[U]
Ix
(Yn)n>-I
U
([U)
(1
(U))
diverges a.s. Since we are interested in
studying the a.s. convergence of
(Yn)n>,l
we may assume without loss
of generality that
X.(n)
3
where
U
U
c
j
for all
-> 1,
can be chosen as a (local) coordinate neighborhood defin-
ing a local coordinate system of G. Such a coordinate neighborhood
generates a connected component of G, whence we may also assume that
G is connected.
Let
{Xl, ..., Xp}
denote a fixed local coordinate system with coor
dinate neighborhood
U
e. For every
of
x
x will denote the coordinate vector (x (x),
I
respect to the system
{Xl, ..., Xp}.
continuous bijection from
U
the G-valued random variable
U
...,
Xp(X))
The correspondence x
onto its image in
Xj
the symbol
induces an
P.
of x with
x is a
For every jl
P-valued
random varia-
.
ble
27
MOMENTS OF PROBABILITY MEASURES ON A GROUP
(on(,(,
3
P)).
The following significant result of V.M. Maksimov [23] is basic for
his further studies (also in [21]) in establishing a three-series
theorem for Lie group-valued random variables.
THEOREM 7.3. Let
p
>-
1 and
G
{Xl, ..., Xp}
U
neighborhood
be a connected Lie group of dimension
a local coordinate system with coordinate
of e. Let
(Xj)j.>I
be a sequence of independent
G-valued random variables on (fl,O, P) and
(Yn)n>.l
the corresponding
sequence of n-th partial products. Then
(Yn)n>-I
(i)
(a)
E
converges a.s. if
(j)
0
for all
j>.l, and
j>:l.
(Yn)n.>l diverges a.s. if
(a) E(Xj)
for all
0
(ii)
j.>l, and
r. IIv x J
j.>i
V(j)
Here,
of
j,
and
denotes the vector of the variances of the components
ii. Ii
stands for the Euclidean norm in
DISCUSSION. Let
(Xn)n.>l
be a sequence in
U
P.
.
and let
(Xn)n>l
denote the correspollding sequence of canonical coordinates. Then,
for Abelian groups
G, the product
H x. and the sum
j->i 3
j.>i ]
s imul-
taneously converge or diverge. Moreover, it has been shown that in
28
H. HEYER
G
the case of a non-Abelian group
.
this conclusion does not hold
for any local coordinate system. Therefore we cannot expect the simultaneous convergence of
X.
[
and
j>’l 3
a probabilistic version of this fact-
If
,
.
E(j)
jl 3
product
-
.
j>’l
3
There is, however,
v( )II <
then the sum
J
j>.l
converges a.s. On the other hand, by the THEOREM above, the
for all
0
j>’l
and
I
X. must also converge a.s. An analoguous statement
j>’l 3
holds true for simultaneous divergence a.s.
THEOREM 7.4. (One sided three-series theorem). Let
connected Lie group of dimension p >, 1 and
x 1,
coordinate system with coordinate neighborhood
...,
U
x
a local
P
of e. Let
be a sequence of independent G-valued random variables on
and
(Yn)n>.l
sequence
(a)
De a
G
(Xj)j>.I
(,OL, P)
the corresponding sequence of n-th partial products. The
(Yn)n>’l
Xj(n)
=
j.>
J
j>-i
J
converges a.s. if
U
for all
MAKSIMOV’S CONJECTURE. Let
a coordinate neighborhood of
j >.1.
G
be a connected Li,_ grcup and
e. For every sequence
)j>’l
U
of inde-
pendent G-valued random variables and the correspondil.- sequence
(Y)
of n-th partial products the following statei nus are equivn n>’l
alent"
29
MOMENTS OF PROBABILITY MEASURES ON A GROUP
(i)
(Yn)n>. I
converges a.s,
(ii)
(a)
I E( J )II
.
j>.a
J
j>.i
(c)
<
converges for any sequence
x.
j>.
in
(xj)j>.l
G
satisfying
x.
]
E(X.) for all j>.l.
3
REMARK. For Abelian Lie groups Maksimov’s conjecture can be
established with little effort.
8. HIGHER MOMENTS AND THE STRONG LAW OF LARGE NUMBERS ON A LIE GROUP
Moments of higher order of probability measures on a group have been
studied for the first time by Y. Guivarc’h in [11]. They came up in
the study of harmonic functions on locally compact groups and were
applied to versions of the strong law of large numbers for groupvalued random variables.
Let
G
be a locally compact group with a countable basis of its to-
pology. We further assume that
pact neighborhood
closed subgroup [V]
V
G
is (compactly) generated by a com-
of e in the sense that
G
coincides with the
generated by V. For the moment let
V
be fixed.
30
H. HEYER
We introduce a mapping
G-,
V:
V (x)"
for all
V
v(xY)
x 6 V n}
is subadditive, i.e.
v(X)
.<
DEFINITION. A measure
+
*"
+
v(y)
G.
whenever x, y
orde..___r
by
inf {n 6
G. Clearly,
x
+
I(G)
u
is said to admit a moment of
if there exists a neighborhood
V
of e with [V]
G
such that
It can be shown that this definition is independent of the particular
choice of the neighborhood
If
JLI(G)
V.
admits a moment of order I and if
then the integral
(x) (dx)
exists. This observation motivates the following
DEFINITION.
u
6LI(G)
is called centered if
(d)
U admits a moment of order
1,
(b)
Jkdu
(G,)
o
for all
k 6 Hom
Hom(G,),
31
MOMENTS OF PROBABILITY MEASURES ON A GROUP
The subsequent DISCUSSION shows that the notion of centering defined
above generalizes the classical one.
Let
G
.
and G’ the closed commutator
Hom (G,{)
subgroup of
G. Then
G*
ly
G*
(
*
K
.
onto a closed subgroup of
such that
G
[P
x
q]-
of the form
It is easy to see that for any
u
CG*) ^.
x
Let
which maps
into
P
I(G)
is centered iff its image
der 1,
x
G/G’. Consequent-
is finite-dimensional and so is its dual ["
denote the canonical mapping from
x
x
(G/G’) /
denotes the maximal compact subgroup of
K
where
(G/G’) *
q
G
p, q >. o,
with
admitting a moment of or-
under the canonical mapping
is centered (in the classical sense).
As an application we cite the following profound
THEOREM 8.1. (Strong law of large numbers). Let
nable, connected Lie group,
a measure in
of all orders, centered and satisfying
compact neighborhood
V
$1(G)
[supp(z)]-
of e and for any sequence
G
be an ame-
admitting moments
G. Then for any
(Xj)j.>I
of inde-
pendent G-valued random variables (on a probability space (,0, P))
with common distribution
products
Yn
X1
bers in the sense that
B, the sequence
"X n
(Yn)n>.l
of n-th partial
satisfies the strong law of large num-
32
H. HEYER
1
n
lim
n-
V
Yn
o
[P].
9. FURTHER STUDIES ON MOMENTS OF PROBABILITY MEASURES
Measures of dispersion such as variances for probabilities on the
one-dimensional torus
Z
have been first introduced by P. L6vy
[18],
and they have been applied to the study of limiting distributions
also by P. Bartfai [1].
Expectations and variances for probabilities on linear spaces are
naturally defined as in the Euclidean space. First versions of the
three-series theorem for Hilbert spaces or for the more general
G-spaces are due to E. Mourier [25], [26]. For an expository presentation of the subject until 1970 see P. Ressel [28]. A discussion of
the case of Hilbert space is also contained in the book [10] of
I.I. Gihman and A.V. Skorohod.
Dispersions of probabilities on a sphere appear at an early stage in
the work [7] of R.A. Fisher. Applications to problems of statistical
estimation have been discussed by W. Uhlmann [32] and H. Vogt [34].
In the framework of the hyperbolic plane and space., expectations and
variances can be introduced via derivatives of Fourier transforms of
measures. A starting point was set by F.I. Karpelevich, V.N. Tutuba-
lin and M.G. Shur in [16]. A different but similar approach is due
MOMENTS OF PROBABILITY MEASURES ON A GROUP
33
to V.N. Tutubalin [31]. Applications to the central limit theorem
indicated already in [16] have been made precise 5y J. Faraut [6].
They rely on the general theory of spherical functions (S. Helgason
[12], N.J. Vilenkin [33]), especially for the Gelfand pair (SL (2,),
SO (2, )), as discussed particularly in [29] by M. Sugiura, and
their impact to probability theory on certain symmetric spaces
(R. Gangolli
[8], [9]).
The central limit theorem envisaged can be extended to all hyperSolic
spaces. Its generalization to arbitrary Gelfand pairs seems to be a
realistic goal.
There are various other approaches to dispersion measures of proDaDi-
lities in a general setting.
The axiomatics of Section 5 extends to semigroups (see T. Byczkowski
et al. in [2] and [3]). The semigroup
+
is the Dasic structure of
an approach to the notion of variance via generalized quadratic
forms within the framework of hypergroups and generalized translation spaces (H.
Chbli [4], [5],
K. Trimche 30]). Finally, we only
mention the generalization of the Khintchine functional as a measure
of dispersion and its importance to delphic theory.
References and a few details are contained in the author’s note [13]
and in his monograph [14].
H. HEYER
34
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