Table Of ContentENCYCLOPAEDIA OF MATHEMATICS
Volume 2
ENCYCLOPAEDIA OF MATHEMATICS
Managing Editor
M. Hazewinkel
Scientific Board
J. F. Adamst, S. Albeverio, J. B. Alblas, S. A. Amitsur, I. J. Bakelman, J. W. de Bakker, C.
Bardos, H. Bart, H. Bass, A. Bensoussan, M. Bercovier, M. Berger, E. A. Bergshoeff, L.
Berkovitz, E. Bertint, F. Beukers, A. Beutelspacher, K.-D. Bierstedt, H. P. Boas, J. Bochnak,
H. J. M. Bos, B. L. J. Braaksma, T. P. Branson, D. S. Bridges, A. E. Brouwer, M. G. de Bruin,
R. G. Bums, P. Cameron, H. Capel, P. Cartier, C. Cercignani, J. M. C. Clark, Ph. Clement, A.
M. Cohen, J. W. Cohen, P. Conrad, H. S.M. Coxeter, R. F. Curtain, M. H. A. Davis, M. V.
Dekster, C. Dellacherie, G. van Dijk, H. C. Doets, I. Dolgachev, A. Dress, J. J. Duistermaat, D.
van Dulst, H. van Duyn, H. Dym, A. Dynin, M. L. Eaton, W. Eckhaus, J. Eells, P. van Emde
Boas, H. Engl, G. Eskin, G. Ewald, V. I. Fabrikant, A. Fasano, M. Fliess, R. M. Fossum, B.
Fuchssteinert, G. B. M. van der Geer, R. D. Gill, V. V. Goldberg, J. de Graaf, J. Grasman, P.
A. Griffith, A. W. Grootendorst, L. Gross, P. Gruber, E. J. Hannan, K. P. Hart, G. Heckman,
A. J. Hermans, W. H. Hesselink, C. C. Heyde, K. Hirscht, M. W. Hirsch, K. H. Hofmann, A.
T. de Hoop, P. J. van der Houwen, N. M. Hugenholtz, C. B. Huijsmans, J. R. Isbell, A. Isidori,
E. M. de Jager, D. Johnson, P. T. Johnstone, D. Jungnickel, M.A. Kaashoek, V. Kac, W. L. J.
van der Kallen, D. Kanevsky, Y. Kannai, H. Kaul, M. S. Keane, E. A. de Kerf, W. Klingen-
berg, T. Kloek, J. A. C. Kolk, G. Komen, T. H. Koornwinder, L. Krop, B. Kupershmidt, H. A.
Lauwerier, J. van Leeuwen, J. Lennox, H. W. Lenstra Jr., J. K. Lenstra, H. Lenz, M. Levi, J.
Lindenstrauss, J. H. van Lint, F. Linton, A. Liulevicius, M. Livshits, W. A. J. Luxemburg, R.
M. M. Mattheij, L. G. T. Meertens, P. Mekenkamp, A.R. Meyer, J. van Mill, I. Moerdijk, J.P.
Murre, H. Neunzert, G. Y. Nieuwland, G. J. Olsder, B. 0rsted, F. van Oystaeyen, B. Pareigis,
K. R. Parthasarathy, I. I. Piatetski1-Shapiro, H. G. J. Pijls, N. U. Prabhu, G. B. Preston, E.
Primrose, A. Ramm, C. M. Ringel, J. B. T. M. Roerdink, K. W. Roggenkamp, G. Rozenberg,
W. Rudin, S. N. M. Ruysenaars, A. Salam, A. Salomaa, J.P. M. Schalkwijk, C. L. Scheffer, R.
Schneider, J. A. Schouten, A. Schrijver, F. Schurer, I. A. Segal, J. J. Seidel, A. Shenitzer, V.
Snaith, T. A. Springer, J. H. M. Steenbrink, J.D. Stegeman, F. W. Steutel, P. Stevenhagen, I.
Stewart, R. Stong, L. Streit, K. Stromberg, L. G. Suttorp, D. Tabak, F. Takens, R. J. Takens, N.
M. Temme, S. H. Tijs, B. Trakhtenbrot, L. N. Vaserstein, M. L. J. van de Vel, F. D. Veldkamp,
P: M. B. Vitanyi, N.J. Vlaar, H. A. van der Vorst, J. de Vries, F. Waldhausen, B. Wegner, J. J.
0. 0. Wiegerinck, J. Wiegold, J. C. Willems, J. M. Wills, B. de Wit, S. A. Wouthuysen, S~
Yuzvinski1, L. Zalcman, S.l. Zukhovitzki1
ENCYCLOPAEDIA
OF
MATHEMATICS
Volume 2
Coproduct - Hausdorff- Young Inequalities
An updated and annotated translation of the Soviet
'Mathematical Encyclopaedia'
Springer Science+Business Media, B.V. 1995
This International Edition in 6 volumes is an unabridged reprint of the
original lO-volume hardbound library edition.
(ISBN originallO-volume set: 1-55608-101-7)
ISBN 978-0-7923-2974-9 ISBN 978-1-4899-3795-7 (eBook)
DOI 10.1007/978-1-4899-3795-7
AH Rights Reserved
© 1995 by Springer Science+Business Media Dordrecht
OriginaHy published by Kluwer Academic Publishers in 1995
Softcover reprint ofthe hardcover Ist edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
induding photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner
COPRODUCI' of a family of objects in a category - A [A2] ADAMEK, J.: Theory of mathematical structures, Reidel, 1983.
concept describing the (categorical analogues of the) AMS 1980 Subject Classification: 18AXX
construction of a direct sum of modules or a discrete
union (bouquet) of sets in the language of morphisms.
Let A;, i e/, be an indexed family of objects in a CoRE IN 1HE 1HEORY OF GAMES - The set of
category 9R. An object S, together with morphisms all non-dominated outcomes, that is, the set C of out-
a;:A;~s. is called the coproduct of the family A;, ie/, comes such that a domination s >-Kc cannot hold for
if for any family of morphisms a;:A;~X, ie/, there any outcomes seS, ceC and coalition KE9t;. One
exists a unique morphism a: S ~x such that a;a =a;, defines in this respect:
i e/. The morphisms a; are called the imbeddings of the 1) The core. The set c(v) of imputations that are not
coproduct; the coproduct is denoted by rrz erA· ;(a;), dominated by any other imputation; the core coincides
rr A;, or S=A!*···*An in case /={1, ... ,n}. with the set of imputations satisfying ~; esx; ;;;tv(S) for
zer·
The morphism a figuring in the definition of the copro- any coalition S. If c(v )7!:: 0 and a von
duct is sometimes denoted by rrIE Ja; or (* ); Elai. The Neumann- Morgenstern solution (see Solution in game
coproduct of a family of objects is defined uniquely up theory) exists, then c(v) is contained in any von
to an isomorphism; it is associative and commutative. Neumann- Morgenstern solution.
The coproduct is the dual concept of the product of a
2) The kernel. The set k(v) of individually rational
family of objects in a category.
configurations (x, ~)(see Stability in game meory) such
The coproduct of the empty family of objects is the
that the following inequality holds for any i, 1 E11 e~:
left zero (initial object) of the category. In an Abelian
category, the coproduct is fcequently called the direct [~e(S, x)-p,~e(S, x)Jxj .;;;;; 0,
,, ft
sum of the family A;, i e/, and is denoted by ~;er4i• or
A 1 +···+An in case /={1, ... ,n}. In most where e(S, x)=v(~)- ~kesxk and Tu is the set ot' coal-
categories of structured sets, the coproduct of a family itions containing the player i and not containing the
of objects coincides with the free product of the family, player j. The kernel k(v) is contained in an M\-
and as a rule requires special description. Thus, in the bargaining set.
category of groups, the coproduct is the free product of
3) The nucleolus. The minimal imputation n(v) rela-
groups; in the category of modules it is the direct sum
tive to the quasi-order --<. defined on the set of imputa-
of modules; etc.
tions by: x <:,y if and only if the vector
In a category with null morphisms, if S =I Ter4;(a;)
fJ(x, v)=(81(x, v), ... ,9n(x, v)), where
is a coproduct, there exist uniquely defined m~rphisms
'IT;: S ~A; such that a;'IT; = lA, a;'ITj = 0. In an Abelian 8;(x, v) = max mine(S, x),
category the coproduct and the product of a finite fam- llll=iSeU
ily of objects are one and the same.
lexicographically precedes 9(y, v) . The nucleolus n (v )
References
[1] TSALENKO, M.SH. and SHUL'GEiFER, E.G.: Fundamentals of exists and is unique for any game with a non-empty set
category theory, Moscow, 1974 (in Russian). M Sh T. l k of imputations. In a cooperative game the nucleolus is
. . sa en o
contained in the kernel.
Editorial comments. Also in not necessarily Abelian
categories the coproduct of a family of objects is frequently References
[1] VOROB'Ev, N.N.: 'The present state of the theory of games',
called the sum of a family of objects or the direct sum of a Russian Math. Surveys 15, no. 2 (1970), 77-136. (Uspekhi Mat.
family of objects. Often used notations are II;eA• ~;e1A; Nauk 15, no. 2 (1970), 103-107) A./. Sobolev
and $;e1A;.
Editorijil comments. The Russian word ('yadro') is the
References
[A1] PoPESCU, N.: Abelian categories with applications to rings same for all three notions defined above, but these notions
and modules, Acad. Press, 1973. may be distinguished by prefixing with the corresponding
1
CORE IN THE THEORY OF GAMES
English letterr c-yadro' for core, 'k-yadro' for kernel and 'n- m-1
yadro' for nucleolus). These three notions do not share Z =X+~ Q;(X)t;
i=l
many properties.
See [A1], [A7] for core, [A2] for kernel and [A3] for as t~O better than it approximates F(x, t). If X has
nucleolus. (A4], (A5] are general references. (A6] deals also
zero expectation and unit variance, then the first terms
with mathematical economics and the role of the concept of
of the expansion (1) have the form
the core of a game in that setting.
x = z+(y1h1(z)]+[y2h2(z)+yih3(z)]+ · · ·.
References
[A1] BoNDAREVA, O.N.: 'Certain applications of the methods of
linear programming to the theory of cooperative games', Here Yl =~e3 I "231 2, Y2 =~e.~ I ~et with "r the r-th
Probl. Kibernet 10 (1963), 119-139 (in Russian). cumulant of X, h1(z)=H2(z)l6, h2(z)=H3(z)l24,
[A2] MAscHLER, M. and DAvis, M.: The kernel of a cooperative h
3(z)=-[2H3(z)+HJ(z)]l36, and with Hr(z) the
game', Naval Res. Logist. Quart. 12 (1965), 223-259.
[A3] ScHMEIDLER, D.: 'The nucleolus of a characteristic function Hermite polynomials, defined by the relation
game', SIAM J. Appl. Math. 17 (1969), 1163-1170.
[A4] OwEN, G.: Game theory, Acad. Press, 1982.
[A5] Sz£P, J. and FoRGO, F.: Introduction to the theory of games, #_z)H,(z) = (- IY d;;:> (#_z)=clY (z)).
Reidel, 1985.
[A6] RoSENMOI.LER, J.: Cooperative games and markets, North-
Holland, 1981. Concerning expansions for random variables obeying
[A7] SHAPLEY, L.S.: 'On balanced sets and cores', Naval Res. limit laws from the family of Pearson distributions see
Logist. Quart. 14 (1967), 453-460. [3]. See also Random variables, transfonnations of.
AMS 1980 Subject Classification: 90012
References
[1] CORNISH, E.A. and FISHER, R.A.: 'Moments and cumu1ants in
the specification of distributions', Rev. Jnst. Intemat. Statist. 5
CORNISH- FISHER. EXPANSION- An asymptotic (1937), 307-320.
expansion of the quantiles of a distribution (close to [2] KENDALL, M.G. and STuART, A.: The advanced theory of statis-
the normal standard one) in terms of the corresponding tics. Distribution theory, Griffm, 1969.
[3] BoL'SHEV, L.N.: 'Asymptotically Pearson transformations',
quantiles of the standard normal distribution, in Theor. Probab. Appl. 8 (1963), 121-146. (Teor. Veroyatnost. i
powers of a small parameter. It was studied by E.A. Primen. 8 (1963), 129-155) VI p
. . agurova
Cornish and R.A. Fisher [1]. If F(x, t) is a distribution
function depending on t as a parameter, if ~(x) is the Editorial comments. For the methods of using an Edge-
normal distribution function with parameters (0, 1) , and worth expansion to obtain (2) (see also Edgeworth series),
if F(x, t)~~(x) as t~O, then, subject to certain see also [A1).
assumptions on F(x, t), the Cornish- Fisher expansion
of the function x =p-I [~(z ), t] (where p-I is the References
function inverse to F) has the form [A1] BICKEL, P.J.: 'Edgeworth expansions in non parametric statis-
tics', Ann. Statist. 2 (1974), 1-20.
m-1 · [A2] JoHNsoN, N.L. and Kon, S.: Distributions in statistics. Con-
X = Z + L S,(z )t; + O(tm), (1) tinuous distributions, 1, Houghten Mifflin, 1970.
r=l
AMS 1980 Subject Classification: 60F05, 62E20,
where the S;(z) are certain polynomials in z. Similarly, 60EXX
one defines the Cornish- Fisher expansion of the func-
tion z =~-· [F(x, t)] (~-· being the function inverse to
CoRNU SPIRAL, c/othoid - A transcendental plane
~) in powers of x:
curve (see Fig.) whose natural equation is
m-1
Z = X+ .}: Q;(X )t; + O(tm), (2) a
i=l r =-s'
where the Q;(x) are certain polynomials in x. Formula
where r is the radius of curvature, a =const and s is the
(2) is obtained by expanding ~-I in a Taylor series
arc length. It can be parametrized by the Fresnel
about the point ~(x) and using the Edgeworth expan-
integrals
sion. Formula (1) is the inversion of (2).
If X is a random variable with distribution function I 2 I 2
F(x, t), then the variable Z=Z(X)=~- 1 [F(X, t)] is x = [cos ~ ds, y = [sin ~ ds,
normally distributed with parameters (0, 1) , and, as fol-
lows from (2), ~(x) approximates the distribution func- which are well-known in diffraction theory. The spiral
tion of the variable of Cornu touches the horizontal axis at the origin. The
2
CORRELATION
asymptotic points are M 1( y:;;; I 2, y:;;; I 2) and strictly isotropic if W C K( W). Any strictly isotropic
M2(- y:;; 12,- y:;; 12). subspace is a null subspace. A polarity relative to
which the whole space is a null space is called a null (or
y
symplectic) polarity (see also Polarity).
Let the projective space IIn(K) over a division ring K
be interpreted as the set of linear subspaces of the (left)
linear space Kn + 1 over K. A semi-bilinear form on
Kn + 1 is a mapping f: Kn + 1 X Kn + 1 ~K together with
an anti-automorphism a of K such that
0
f(x +y, z) = f(x, z)+ f(y, z),
f(x,y+z) = f(x,y)+f(x,z),
f(kx,y) = kf(x,y),
The spiral of Cornu is sometimes called the spiral of f(x, ky) = f(x,y)a(k).
Euler after L. Euler, who mentioned it first (1744).
Beginning with the works of A. Cornu (1874), the spiral In particular, if K is a field and a= id, then f is a bi-
of Cornu is widely used in the calculation of diffraction linear form. A semi-bilinear form f is called non-
of light. degenerate provided f(x,y)=O for all x (ally) implies
y =0 (x =0, respectively). Any auto-correlation " of
References
IIn(K) can be represented with the aid of a non-
[1] JAHNKE, E., EMDE, F. and LoscH, F.: Tafeln hiiheren Funk-
tionen, Teubner, 1966. degenerate semi-bilinear form f in the following way:
D.D. Sokolov
for a subspace V of K" + 1 its image is the orthogonal
complement of V with respect to f:
Editorial comments.
References
K(V) = {y EKn + 1: f(x,y)=O for all XE V}
[A1] LAWRENCE, J.D.: A catalog of special plane curve, Dover,
reprint, 1972.
(the Birkhof!- von Neumann theorem, [2]). " is a polar-
AMS 1980 Subject Classification: 53A04 ity if and only iff is reflexive, i.e. if f(x,y)=O implies
f(Y, x)=O. By multiplying fby a suitable element of K
CORRELATION, duality - A bijective mapping " one can bring any reflexive non-degenerate semi-
between projective spaces of the same finite dimension bilinear form f and the corresponding automorphism a
such that SP CSq implies K(Sq)CK(Sp). The image of a in either of the following two forms:
sum of subspaces under a correlation is the intersection 1) a is an involution, i.e. a2 = id, and
of their images and, conversely, the image of an inter-
section is the sum of the images. In particular, the f(y, x) = a(f(x,y)).
image of a point is a hyperplane and vice versa. A
In this case one calls f symmetric if a= id (and hence
necessary and sufficient condition for the existence of a
necessarily K is a field) and Hermitian if a:f=id.
correlation of a projective space Iln(K) over a division
2) a= id (and hence K is a field) and
ring K onto a space IIn(L) over a division ring L is
that there exists an anti-isomorphism a: K~L, i.e. a
f(y, x) = - f(x,y).
bijective mapping for which a(x + y) = a(x) + a(Y ),
a(xy)=a(Y)a(x); in that case IIn(L) is dual to Iln(K).
Such an f is called anti-symmetric.
Examples of spaces with an auto-correlation, i.e. a
A special example of a correlation is the following.
correlation onto itself, are the real projective spaces
Let IIn(K) be a projective space over a division ring K.
(K =R , a= id), the complex projective spaces (K =C ,
Define the opposite division ring K0 as the set of ele-
a: z~Z) and the quaternion projective spaces (K =H,
ments of K with the same addition but with multiplica-
a:z~Z).
tion
A polarity is an auto-correlation K satisfying ,(2 = id. xy = yx.
A projective space IIn(K) over a division ring K admits
a polarity if and only if K admits an involutory anti- a: x~x is an anti-isomorphism from K onto K0 which
automorphism, i.e. an anti-automorphism a with a2 = id. defines the canonical correlation from Iln(K) onto
A subspace W is called a null subspace relative to an IIn(K0 ). The (left) projective space Iln(K0 ), which can
auto-correlation " if PC ~e(P) for any point PEW, and be identified with the right projective space Iln(K)*, i.e.
3
CORRELATION
with the set of linear subspaces of the (n +I)- which is best in the sense that
dimensional right vector space Kn + 1, is the (canonical)
E(X2-X2f = minE(X2-ciXI-c2f;
dual space of IIn(K) (cf. Projective algebra, the con- c.,c2
struction of IIn). see also Regression. As characteristic correlations
Editorial comments. between several random variables there are the partial
References correlation coefficient and the multiple-correlation coef-
[A1] BAER, R.: Linear algebra and projective geometry, Acad. ficient. For methods for testing independence
Press, 1952. hypotheses and using correlation coefficients to study
[A2] BIRKHOFF, G. and NEUMANN, J. VON: 'The logic of quantum
correlation, see Correlation (in statistics).
mechanics', Ann. of Math. 37 (1936), 823-843.
[A3] Dmuoomffi, J.: La geometrie des groupes classiques, A. V. Prokhorov
Springer, 1963.
AMS 1980 Subject Classification: 62JXX
[A4] HuGHES, D.R. and PIPER, F.C.: Projective planes, Springer,
1972.
CORRELATION FUNCTION of a real stochastic pro-
AMS 1980 Subject Classification: 51A10, 10C05,
cess {X(t): teT} - The function in the arguments
15A63
t, seT defined by
B(t, s) = E[X(t)-EX(t)][X(s)-EX(s)].
CORRELATION COEFFICIENT - A numerical
characteristic of the joint distribution of two random For the correlation function to be defined, it must be
variables, expressing a relationship between them. The assumed that the process X(t) has a finite second
correlation coefficient p=p(Xi> X2) for random vari- moment EX(tY for all t e T. The parameter t varies
ables X 1 and X 2 with mathematical expectations here over some subset T of the real line; it is usually
a 1 =EX1 and a2=EX2 and non-zero variances interpreted as 'time', though an entirely analogous
or =D X I and 0~ =ox 2 is defined by definition is possible for the correlation function of a
stochastic field, where T is a subset of a finite-
p(XI> X2) = E(X1 -a1XX2 -a2).
dimensional space. If X(t)=[X1(t), ... ,Xn(t)] is a mul-
O'J 0'2
tivariate stochastic process (stochastic function), then
The correlation coefficient of X 1 and X 2 is simply the its correlation function is defined to be the matrix-
covariance of the normalized variables (X1 -a!) 1 o1 valued function
and (X2 -a2) I o2. The correlation coefficient is sym-
B(t, s) = II Bij(t, s) li7.J=l>
metric with respect to X 1 and X 2 and is invariant where
under change of the origin and scaling. In all cases Bij(t, s) = E[X;(t)-EX;(t)](Xj(s)-EXj(s)]
-1 os;;pos;; 1. The importance of the correlation coeffi-
is the joint co"elation function of the processes X;(t),
cient as one of the possible measures of dependence is
~(t).
determined by its following properties: I) if X 1 and X 2
The correlation function is an important characteris-
are independent, then p(XI> X2)=0 (the converse is not
tic of a stochastic process. If X(t) is a Gaussian process,
necessarily true). Random variables for which p=O are
then its correlation function B(t, s) and its mean value
said to be non-correlated. 2) I p I = I if and only if the
EX(t) (i.e. its first and second moments) uniquely
dependence between the random variables is linear:
determine its finite-dimensional distributions; hence
also the process as a whole. In the general case, the
first two moments are known to be insufficient for a
The difficulty of interpreting p as a measure of depen- full description of a stochastic process. For example,
B(t, s)=e-alt-sl is at one and the same time the
dence is that the equality p=O may be valid for both
independent and dependent random variables; in the correlation function of a stationary Gaussian Markov
general case, a necessary and sufficient condition for process the trajectories of which are continuous, and
independence is that the maximal correlation coefficient also the correlation function of the so-called telegraph
equals zero. Thus, the correlation coefficient does not signal, a stationary Markov point process taking the
exhaust all types of dependence between random vari- two values +I. However, the correlation function does
ables and it is a measure of linear dependence only. determine several important properties of a process: the
The degree of this linear dependence is characterized as so-called second-order properties (i.e. properties
follows: The random variable expressed in terms of second moments). In view of this,
and also because of their relative simplicity, correlation
A O"z
X2 = p-(X1-a1)+a2 methods are frequently employed both in the theory of
O'J
stochastic processes and in its statistical applications
gives a linear representation of X 2 in terms of X 1 (see Correlogram).
4
CORRELATION FUNCTION IN STATISTICAL MECHANICS
The rate and nature of decrease of the correlations as [3] GIKHMAN, 1.1. and SKOROKHOD, A.V.: Introduction to the
I t - s I~ oo provides an idea of the ergodic properties theory of stochastic processes, Saunders, 1969 (translated from
the Russian).
of a process. Conditions relating to the rate of A.S. Kholevo
decrease of correlations, in some form or another, AMS 1980 Subject Classification: 62JXX
appear in limit theorems for stochastic processes. Local
second-order properties, such as mean-square
CORRELATION FUNCDON IN SfATISTICAL
continuity and differentiability, provide a useful - MECHANICS - A function describing the influence of
though extremely crude - characteristic of the local particles or groups of particles on one another and the
behaviour of a process. The properties of the trajec- effects due to the interaction of subsystems of the sys-
tories in terms of the correlation function have been tem under consideration.
investigated to a considerable degree in the Gaussian In classical statistical mechanics, the correlation
case (see Sample function). One of the most complete functions G
2(1, 2), G3(1, 2, 3), ... , are defined by the
branches of the theory of stochastic processes is the relations
theory of linear extrapolation and filtration, which F2(1, 2) = F 1(l)F1(2)+G2(1, 2),
yields optimal linear algorithms for the prediction and F3(1, 2, 3) = Ft(l)Ft(2)Ft(3)+Ft(1)G2(2, 3)+
approximation of stochastic processes; this theory is
+Ft(2)G2(1, 3)+Ft(3)G2(1, 2)+63(1, 2, 3), ... ,
based on a knowledge of the correlation function.
A characteristic property of the correlation function where the symbols .1, 2, ... , in the arguments of the
is the fact that it is positive definite: functions denote the sets of coordinates r and momenta
n p of the 1-st, 2-nd, ... , particles, respectively, and
~ C;SB(t;, tj) ;;;;, 0, Fs(l, ... ,s) are the reduced distribution functions
i,j=l
for any n, any complex ci, ... , en and any F,(1, ... ,s) = V [ 1- ~] · ..
t I , ... , tn E T. In the most important case of a station-
ary process in the broad sense, B(t, s) depends (only)
.. . [ 1 - s; 1 ) j D1 d(s + 1) .. · dN,
on the difference between the arguments:
B(t, s)=R(t-s). The condition that it be positive where V is the volume of the system, N is the number
definite then becomes of particles and the D1 = D1(1, ... , N) are the distribu-
n tion functions in the phase space at time t, normalized
~ C;SR(t;-tj);;;;, 0.
i,j=l so that
jD,(l, ... ,N)d1 · · · dN = 1.
If R(t) is also continuous at t =0 (in other words, the
process X(t) is mean-square continuous), then The variation of D1 in time is characterized by the
Liouville equation aD, I at =AD,, where A represents
R(t) = j eit>. F(d'A),
the Liouville operator, which is not explicitly dependent
where F(dA.) is a positive finite measure; here A. runs on time. One usually considers the case in which A is
over the entire real line if T = ( - oo, oo) (the case of the sum of an additive part and a binary part charac-
'continuous time'), or over the interval [-'IT, 'IT] if terizing the interactions of the particles:
T= { ... , -1, 0, 1, ... } (the case of 'discrete time'). A= ~ A(j)+ ~ A(jt>h).
The measure F(dA.) is known as the spectral measure of l.;.j.;.N I.;.j,<}2.;.N
the stochastic process. Thus, the correlation and spec- According to the principle of correlation damping, the
tral properties of a stationary stochastic process prove correlation functions satisfy the boundary conditions
to be closely related; for example, the rate of decrease Gs(l, . .. ,s)~O as
in correlations as t~oo corresponds to the degree of max{ I ri-r21• ... , I ri-rs I, ... , I fs-1-rs I }~oo.
smoothness of the spectral density j(A.)=F(dA.) IdA., The correlation functions
etc. G1(l)=F1(l),G2(1,2), ... ,Gs(l, ... ,s) are the func-
In statistical mechanics, the term is also used for the tional derivatives,
joint probability density p(x I> .•• , Xm) of m distinct
c5'A1(u) l
particles of the system under consideration placed at G,(l, ... ,s) = [ c5u(1)6u(2) ... 6u(s) u=O·
points xI, ... , Xm; the totality of these functions
uniquely determines the corresponding discrete stochas- of a functional A1(u) which is related to the so-called
tic field. generating functional
References
[1] DooB, J.L.: Stochastic processes, Chapman and Hall, 1953. ft,(u) = f{ I.;I.Tj<. N [I+ ;uv>]}D,d1···dN
[2) Lo~VE, M.: Probability theory, Princeton Univ. Press, 1963.
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