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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Arnold, B.C. (1971). Two characterizations of the exponential distri-bution using order statistics, unpublished manuscript. Iowa StateUnfv ,

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Arnold, B.C. (1975). A characterization of the exponential distribu-tion by multivariate geometric compounding, Sankhya, A, 37, 164-173.

Arnold, B.C., Isaacson, D.L. (1978). On normal characterizations bythe distribution of linear forms, assuming finite variance, Sto-chastic Processes and Their Appl., 7, 227230.

AvksentieY, D.I. (1982) On identical distribution of random linearforms, Mat.Zametki, 31, 6, 947948 (in Russian).

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de Finetti, B. (1937) La prevision: ses lois logiques, ses sourcessubjectives, 7, 168.

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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-