-
Comments
To Chapter 1
1.1 .• Definitions 1.1.1 and 1.1.2 and Theorem 1.1.1 are due
to
Klebanov (1981), Theorems 1.1.2, 1.1.5 were obtained in the
report by
Klebanov, Mkrtcjan (1982). Theorem 1.1.3 was obtained in the
paper
by Klebanov, Melamed (1983), where it was applied to the
investigati-
on of relsvationtype equations. By its ideas this theorem is
similar
to Theorem 4.4 from the monograph by Krasnosel'ski (1966).
Theorems
1.1.4 and 1.1.6 are the new results.
1.3. As we have remarked above the proof of validity of
property
2 from definition of the strong Sposi tiveness in Example 1.3.
1repeats the corresponding result by Yu.V.Linnik (1960). The same
ide-
as are utilized in Example 1.3.2.
Note also that the examples of strongly gposi tive families
canbe obtained by consideration of Hardy field (see Bourbaki
(1965». So-
me theorems on extension of Hardy fields and their applications
are
given in the paper by Klebanov (1971).
To Chapter 2
2.1. Polya's (1923) theorem has been somewhat simplified and
rep-
roved in various situations by Arnold and Isaakson (1978), Eaton
(1966),
Laba, Lukacs (1965). Definition 2.1.1 and Theorems 2.1.1, 2.1.2
are
due to Klebanov (1981); Klebanov, Mkrtcyan (1982) investigated
the
stability of this characterization problem.
2.2, As it was noted above the results connected with
characteri-
zation of the normal distribution by a property of identical
distri-
bution of a monomial and a random linear form were studied by
Shimizu
(1968), (1978), Shimizu, Davies (1979), Avksentiev (1982),
Klebanov,
Melamed and Zinger (1982). Theorem 2.2.1 was obtained by
Klebanav,
-
162
(1982). Theorem 2.2.3 was announced in the report by Kleba-
nov, Melamed, Zinger (1982).
2.3. The role of stable laws in the theory of summation of
in
dependent random variables is well known, thus the receipt of
various
characterizations of these laws is of considerable interest.
Theorem
2.3.1 was obtained in the report by Klebanov, (1982). The
identical distribution of a monomial and a random linear form in
con
nection with the characterization of stable distributions was
conside-
red by Shimizu and Davies (1979), (1981§), (1981E). Theorem
2.3.2 ex-tends some of these papers. Theorems 2.3.2 2.3.4 are
the
new results.
2.4. Limit theorems for the sums of a random amount of random
va
riables were considered by many authors. The results similar to
those
given in this section one can find in Szantai (1971), Kovalenko
(1965),
Gnedenko (1982). Characterizations of the exponential
distribution by
the property of identical distribution of a monomial and a
random sum
were obtained in the papers by Arnold (1973), (1975), Azlarov,
Dzamir-
zaev, Sultanova (1972), Azlarov (1979). However, as far as we
know,
characterizations of the Laplace distribution and those of
distributi
ons with the c.f.'s of the form of 1/(1+ by the property of
identical distribution of a monomial and a random sum have not
been obtained earlier, except Theorems 2.4.1 and 2.4.5,
announced in
the report by Klebanov, Melamed, Zinger (1982). The rest results
of
this section are stated here for the first time.
2.5 and 2.6. A highly detailed account of characterizations
of
distributions by the properties of zero regression of a linear
statis-
tic on another one in the case of identically distributed
variables
and determinated coefficients of the forms is given in the
monograph
by Kagan, Linnik, Rao (1973). From the point of view of
functional
equations in which result these problems, they do not very much
differ
from those of characterization by the property of identical
distribu
-
163
tion of linear forms. However for the forms with random
coefficients
such effect appears only in some special cases, which are
considered
here. Consideration of the forms of a general form is associated
with
sizable difficulties and for the present there are no somewhat
general
in this direction.
To Chapter 3
3.1. As it has been already noted, characterizations of the
expo-
nential distribution by the property of identical distribution
of a
monomial and an order statistic investigated by many
authors.
One can find rather detailed bibliographical references on these
que-
stions in the books by Galambos, Kotz (1978), Azlarov, Volodin
(1982).
Definition 3.1.1 in a somewhat different context was given in
the pa-
per by Klebanov (1978). Theorems 3.1.1 and 3.1.2 are the new
results,
characterizing the Weibulldistribution. Of course, with the help
of a
monotone transformation they can be reduced to characterizations
of
the exponential distribution, but it does not result in any
simplifi-
cation of the statements or the proofs.
Theorem 3.1.3 is a new result, showing the importance of
random-
ness of the number of variables by which the correspondent order
sta-
tistic is constructed. As it has been shown in Theorem 3.1.1,
the re-
sulting distributions have a meaning of the limit ones. Theorem
3.1.1
can be obtained from the results by Gnedenko (1982), too.
Theorems 3.1.4 and 3.1.5 present characterizations of the
exponen-
tial distribution by combined properties of the sums and extreme
sta-
tistics. As far as we know, characterization results of such
type we-
re not obtained earlier.
3.2. A survey of results connected with the reconstruction of
di-
stribution of a sample by a distribution of statistics is given
in the
monographs by Kagan, Linnik, Rao (1913) and Galambos, Kotz
(1978). The-
orem 3.2.1 is stated here merely as an illustration of the
method. Ap
-
164
parently it is well known for specialists. One can easily
suggest
another proof of this theorem, consisting in calculation of
moments
of variables by moments of the form L and hence by momentsof the
variables X. . The existence of moments can be easily dedu-
Jced from Yu.V.Linnik's theorem on (see Linnik
(1960».
Theorem 3.2.2 is a new result.
Theorems 3.2.3 and 3.2.4 are extensions of Linnik's (1956)
results
and of the corresponding theorems from the book by Kagan,
Linnik, Rao
(1973). The deduction of the basic functional equation (3.230)
has be-
en copied by us from the book by Kagan, Linnik, Rao (1973).
To Chapter 4
4.2. The results of this section were obtained by a somewhat
dif-
ferent method in a paper by Klebanov (1978). They turn out to be
con-
nected with the problem of identical of statistics Xand a Xj :
11 ,where a = COl18t .
4.3. The definition of relevation of distributions F and Gis due
to Krakowski (1973). Characterization of the exponential dist-
ribution by relevation-type equations was obtained in the paper
by
Grosswald, Kotz, Johnson (1980). The extension of their results
is gi-
ven in Klebanov, Melamed (1983). Theorem 4.3.1 is a
strengthening of
these results. Johnson, Kotz (1979) extended a scheme of
replacements,
considered by Krakowski. And they introduced a notion of
e-releva-tion of distributions. Theorems 4.3.2-4.3.4 are the new
reSUlts, Un-
doubtedly it is interesting to extend Theorem 4.3.) to the case
of
e -relevation and investigate other properties of e-relevationof
distributions.
4.4. The averaged lack of memory property has been
investigated
and used by Ahsanullah (1976), (1978), Grosswald, Kotz (1978),
Huang
(1981), Klebanov, Melamed (1983) (see also Galambos, Kotz
Azla-
-
165
rov, Volodin (1982». One can use here methods of papers by
Shimizu
(1979), Klebanov (1980), Davies (1981) too. The methods of
in-let of randomness in the lack of memory property, considered
in
4.4, are due to the authors. The corresponding results are
new.
4.5. by properties Qf the records are conside-
red in detail in the monograph by Galambos, Kotz (1978). For
receipt
of the new results when investigating the phenomenon of
identical dis-
tribution of R1 and Rj - Rj-1 one can use methods of papersby
Huang (1981), Shimizu Klebanov (1980), Klebanov, Melamed
(1983). These methods, as a rule, are connected with the
convolution-
type equations. And the results, obtained in 4.5, are more
directed
to the method of intensively monotone operators. They are stated
here
for the first time. There are possible also other statements of
the
problems of characterization of distributions by properties of
the re-
cords.
4.6. Both the statement of the problem and the results are
new.
An analogue of Theorem 4.6.1 for the case of linear forms is
Theorem
5.6.1 (see Chapter 5).
To Chapter 5
5.2. There is a considerable number of characterizations of
the
normal distribution in Hilbert space by the properties of
identical
distribution of linear forms. Note among them the papers by Rao
(1969),
(1975), Eaton, Pathak (1968), Voikovic (1981). Theorems 5.2.1
and
5.2.2 in a sense are similar to Rao's (1975) paper.
5.3. There are similar results on characterization of the
normal
distribution in Euclidean spaces in the papers by Eaton (1966),
Laha,
Lukacs (1965), Ghurye, Olkin (1973), Rao (1975), Goodman, Pathak
(1975),
Klebanov (1970), (1974), Voikovib (1981).
5.4. Such introduction of the Laplace distribution in Hilbert
spa-
ce, apparently, appears here for the first time. The theorems of
this
-
166
section are the new results.
5.5. Marshall and Olkin's distribution was introduced and
inves-
tigated in the papers by Marshall, Olkin The lack
of memory property has served as a base for its untroduction.
The-
orem 5.5.1 is a new result, using a bivariate version of the
averaged
lack of memory property. In the papers by Paulson (1973) and
Arnold
(1975) there were introduced multivariate versions of the
exponential
distribution by utilizing the properties of linear statistics.
Theorem
5.5.2 is related to the result by Paulson (1973).
Different methods of introduction of the multivariate
exponential
distribution are discussed in the papers by Pickands, 3.111
(1976),
Ban, Pergel (1982).
5.6. Both the statement of the problem and the results are due
to
the authors and are published here for the first time.
-
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AvksentieY, D.I. (1982) On identical distribution of random
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Characteriza-tion properties of the exponential distribution and
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-
SUBJECT INDEX
A
additive type of distribution 84
argument of a characteristic function 87
B
-conditionally independent random variables 31
-intensively monotone operator 7
bivariate exponential distribution 149
bivariate lack of memory property 149
bivariate integrated lack of memory property 150
Borel- 5 -algebra 145
C
Carleman's condition 98
characteristic function 16
functional 134
property 23
class {J)} 92
m 921ft, 93
Completely monotone function 100
completely symmetric probability density function 97
component of a probability distribution 23
stable law 37
conditional independence 31
conditionally independent random variables 29
convolution of distributions 109
convolution-type equation 165
Cramer's theorem 23
-
172
D
device (technical) 101
dimension of a tUbular statistic 91
distribution function 19
of records 125
reconstructed by moments 19
with a monotone hazard rate 22
E
eigen-function 5
eigen-value 5
element (of a technical device) 102
in storage 108
equicontinuity 11
exponential distribution 22
F
function of connection between distribution fucntions 72
characterisric functions 25
G
Gaussian measure in Hilbert space 145
geometric distribution 43
random variable 43
H
hazard rate function 22
Hilbert space 133
I
integrated lack of memory property 118
intensively monotone operator 1
-
146
146
173
L
lack of ageing 101
lack of memory property 101
Laplace distribution 43
in H11bert space
measure in Hilbert space
transform 19
lifetime distributon 101
limit theorem 44
linear functional 134
operator 5
statistic 24
Linnik's theorem 19. 49
on c:I:. -decompositions 58
location parameter 84
logistic distribution 82
lower semi-continuity 8
M
Marshall and Olkin distribuU:on 150
Minlos and Sazonov's theorem 146
moment problem 19
multivariate normal distribution 138
N
normal distribution 23
in Hilbert space 134
nuclear operator 134
o
order statistic 71
-
174
P
Polya's theorem 23. 48
positive operator 5
solution 2
positively definite function 146
proper subset 4
property of ageing 101
R
random coefficients 28
parameter 46
variable 16
reconstruction of a distribution 84
an additive type of distribution 84
records 125
reflexive Banacn space 7
relevation 109
e -relevation IIIreliability of an element.s system
s
102. 102
eervicing element (of a technical device) 108
set c_ 35
C.... 51
space of continuous functions 1
weakly continuous fucntions 7
spherical coordinates 94
spherically sYmmetric 143
stable distribution 33
laws 33
star-shaped statistic 92
strictly positive nuclear oper-ate r 134
-
175
Remark In K1 (R» we have generators (A,a) where a is an
isomorphism of A and relations given by split exact sequences,
and in
we divide out by (A,O) and (A,1 A) if i > 0 but only by(A, 0)
if i = 0 •
The basic ingredient in the proof of theorem 1.1 is the
Bass-Heller-
Swan homomorphisms which are described as follows: Let (A,a)
represent
an element of K_, (R) , so A is an object of 1 (R) . Adjoining
in-1+_1determinates t,t-1 we obtain A[t,t- 1] an object of (R[t,t
J)
in the obvious way, and we may also think of a as an isomorphism
of
A[t,t- 1J . Define the isomorphism of A[t,t-1J on
homogeneous
elements by
={x if s-degree of x is < 0s
Pt(x)t·x if s-degree of is > 0x
The commutator will be the identity of A[t,t- 1J except for
a
band -k js k where k is a bound for a and we may think of
this commutator as a ;;zi -graded isomorphism over R[t,t -1 J .
In [1]s -1we show this gives a welldefined monomorphism A : K . (R)
->- K . +1 (R [t,t J).
-1 -1
In [1J we do not discuss the dependency of s, so it seems
appropriate
to do that here: Let 9 ( . One easily sees that regrading
;;zi+1 by g sends bounded isomorphisms to bounded isomorphisms
so
GZ (i+1 ,;;Z) acts on K_i (R)
Proposition 1.2 The action of g E on
plica tion by det g
K . (R)-1
is by multi-
Corollary The dependency of s in the Bass-Heller-Swan
monomorphismis given by AS (_1)r-s.Ar.
Proof of corollary: We only consider s=1 and r=2 , the general
casebeing obvious from this. Let g E interchange the first 2
coordinates. Then det 9 = -1 and A2 0g = Ai and the result
follows.
Proof of proposition 1.2: First we show that if 9 is
elementary,
9 = Ers(n) the action is trivial: If A E (R) is regraded by
g
we obtain Ag and we have (A,a) is sent to Ag
where 19 is the identity, if we forget the grading. The problem
is
that 19 is not bounded. Since Ag(j1, ... ,ji) = A(g(j1, ... ,ji»
wessee that 19 preserves all degrees except the r'th degree so Pt
com-