randomization of random variables and their distributions

RANDOMIZATION OF RANDOM VARIABLES AND THEIR DISTRIBUTIONS

L. N. Gurvits and A. M. Zakharin UDC 519.21:519.7

Finite or denumerable randomization substantially simplifies the investigation of compound semi-Markov processes and flows: the corresponding integral relationships are replaced by algebraic ones, and piecewise-linear components of the Markov model are replaced by discrete components [1-4].

The main applied value of randomization is that it actually allows us to pass from dependence on piecewise-continuous components to dependence on a single discrete component. Randomization decomposes the distribution characterizing the stochastic system into simpler distributions. This of course involves certain loss of information, but it is a reasonable Uprice" to pay for the simpler methods of analysis and simpler models.

We should stress that the value of randomization is not restricted to transformation to simpler Markov models with lower computational requirements. Randomization also simplifies the analysis of the original Marker models which contain additional piecewise-continuous components [1-4].

In what follows we introduce the notation of exact (finite, almost finite, denumerable) randomization of a random variable (r.v.), of its distribution function (d.f.), and of a family of d.f.'s, and also the important applied notion of approximate randomization of a family of d.f.'s. The main goal of this study is to elucidate the conditions for exact and approximate randomization.

i. Exact Randomization of a Random Variable and of Its Distribution

Definition I. Let ~ = r be a r.v. on the probability space (~, ~, P);{fla;ada} tion of the set a, i.e.,

a parti-

~6a

Let P(Q~)=~0 (sEa), so that the index set a is at most denumerable. Let ~a denote the restriction of ~ to ~ "with normalization," i.e., ga is a r.v. on (~, o~, P~), where

P(B~) (B~ C ~)

The representation of $ in the form {$= = %IQ~;=da} will be called randomization. By the com- plete probability formula, randomization of a r.v. induces randomization of its d.f. P{~ < x} =

F~ (x) = E P(~) F~ (x). s e a

A r a n d o m i z a t i o n i s c a l l e d t r i v i a l i f Ft (x) = Fi~ (x)(~ E a).

Thus, if the randomization is nontrivial, the corresponding d.f. F is not an extreme point in the set ~ of alld.f.'s, i.e., there exist F,--/=F 26~ such that

F = cF~ + ( 1 - - c ) F z ( O < c < [ ) , (i)

LEMMA i. For any decomposition (i) there are a probability space (~, o, P) and a r.v. on it which admits randomization intor.v.'s {~i, ~2} such that F = F$; F i = F$i (i = I, 2).

Proof. Let F i be the d.f. of gi on (Qi, ~ Pi) (i = I, 2). We may assume that ~iN~

= ~ . Let ~(m)=~(~)(~EQ~); ~ = Q ~ U ~ ; o={A~UA~:A~oi ; i = l , 2 } ; P(A~UA~)=cPt(A~)+(1-- c) P~(A~)(A~E%; i = 1,2). It is easy to see that these relationships specify the required representation which defines the sought randomization.

Translated from Kibernetika, No. i, pp. 108-115, January-February, 1987. Original ar- ticle submitted November II, 1984.

136 0011-4235/87/2301-0136512.50 �9 1987 Plenum Publishing Corporation

Remark i. Let $ be a r.v. on (~, o, P) whose d.f. can be randomized as Fg = cF l + (i - c)F 2. It is not always possible to randomize g into $i, ~2 with the d.f.s F~, F2~ The following examples prove this point.

! 1. Le t a = { O , 1}; P ( { O I ) = P ( { I } ) = T = c; ~ ( 0 ) = 0 ; ~ ( | ) = 1;

F(x) =

I 0 ,

F~(x)= { 2i_ --4 [

1,

O, x<~ O; 1 y , O < x G 1 ;

1, ~> I;

x~<0;

, 0 < x ~ l ;

x > 1;

i = 1,2.

The representation of $ to any proper subset of the space ~ is a deterministic quantity. Therefore, neither F l nor F 2 may be d.f.'s of these restrictions "with normalization~

2. Let Q=[0,1]; ~(~)=m; P(B)= m(B)(BEo[0.~I), where m is the Lebesgue measure on the corresponding o-algebra o[0,1 ]. For any B6~[0,11 of nonzero measure m(B) > 0, the restriction ("with normalization") of ~ to B has the d.f. PB(dx) such that PB(a \ B) = 0. Consider the following randomization of the d.f. of $:

1 2 1 1 Y l ( x ) + l F~ (x) = x = -~- x + ~- ( 2 x - - x ~) = -~ F2(x).

Since the d. f.'s F~,Fa are not concentrated on any proper subset of [0, i], there is no randomization of the r.v. ~ corresponding to the above randomization of its d.f. However, there do exist r.v.'ssuchthat to each randomization of their d. f, ts corresponds a randomization of the r.v.'s.

3. Let

Q = [ 0 , I]; Im~ = {1, 2, ..., n};

P (B) = m (B) (B 6 e[0,~l)-

Clearly, for any randomization of the d.f. F~ = cF I + (i - c)F 2, the d.f.s F~, F 2 are also concentrated on {i, 2,...,n}, so that the measures of these atoms are related by the equality

;j + (I --c) p< 9 (/=: 1, 2, .. . , n).

Let ~ - l ( j ) = Bj and p a r t i t i o n each Bj i n t o two s u b s e t s ( u s i n g c o n t i n u i t y o f t h e Lebes-. gue m e a s u r e ) :

~(lh -/n(') U B} ~ = &; I"11 (tS i ) = cp}');

( B } = ( I - - c) ._(/)

Setting Qi---- B/ (i = i, 2), we obtain the sought randomization of the ~(~o). /'=i

Note that it is interesting to explore the description of the set of all r.v. 's on the probability space ([0, i], o, P) such that randomization of their d.f.'salwayscorrespondsto randomization of the r.v. 's themselves.

2. On Extreme Points of Families of Distribution Functions

THEOREM i. Let M be a separable metric space, o M any o-algebra which includes all the open subsets of M, r M) the setfd.f.'softhe r.v.'s with values in M. Then the extreme points of the set ~(o M) are thed.f.'s of deterministic quantities and only they.

Proof. Note that the d.f. of a deterministic quantity x is the ~-measure 6x(dY), Clearly, ~-measures are extreme points of the set r so that "one direction" of the theorem is obvious. Let us prove that any extreme point is a 6-measure.

Assume that some d.f. F(A) is an extreme point with the corresponding r.v. ~:~ § M on (~, o, P). Then, by the definition of extreme point, for any X ~ ~, P (X)=/= 0, the d.f. of the

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restriction ~ ~ X on any Borel set A c M is defined by the equality F$~x(A) = F(A). Let F(A) ~ 0, i.e., P($-~(A)) ~ 0. Then 1 =F~_~c~(A): :F(A).

Thus, for any Borel set A c M either F(A) = 0 or F(A) = i.

Using separability of M, we choose in it a denumerable everywhere dense subset {y~}.

Then clearly M=UB(yl, l), where B(x, r) is a ball of radius r centered at x. By countable i

additivity of the measure and the above result, there exists a i I such that F(B(y~I, i)) = i. i

Let c I = B(Yiz, i). Since any subset of a separable space is separable, there exists a de-

numerable subset {y~} everywhere dense in c I. Clearly,

Similar argument shows that we can choose a subset c 2 Y~, N we define a nested system of sets

{c~}; c.+l = B Y~n+x' n + 1 flcn;

F (c~) = 1 (n ~> 1).

By upper continuity of o-additive measure and the last equality,

% F(Q)= 1. By induction,

F((-'le.] = l i m F ( c ~ ) = 1,

2 t h e r e f o r e ~ ] e ~ @ ~ , S i n c e by c o n s t r u c t i o n c ~ - ~ - n - - > O = d i a m ~ ] c n , t h e n ~]c~ c o n s i s t s o f

n~I n=l n=l

a s i n g l e p o i n t , i . e . , F i s a 8 - m e a s u r e .

T h i s t h e o r e m , in p a t i c u l a r , d e s c r i b e s c o l l e c t i o n s o f e x t r e m e p o i n t s o f t h e f o l l o w i n g s e t s : d . f . ' s o f r e a l o n e - d i m e n s i o n a l r . v . ' s , r a n d o m v e c t o r s , s t o c h a s t i c p r o c e s s e s w i t h o u t d i s - c o n t i n u i t i e s o f s econd k i n d .

Note that the theorem looked at extreme points of the set of all d.f.'s. Its subsets, obviously, may contain extreme points which are not 6-measures or no extreme points at all. In particular, the set of d.f.'s with continuous density contains no extreme points.

Let us prove another fact of this kind.

LEMMA 2. The set of all absolutely continuous d.f.'s contains no extreme points.

Proof. Consider an arbitrary d.f., i.e., an almost everywhere measurable nonnegative

T function f(x) such that j f(x)dx= I. From the last condition (normalization) it follows

that for some s > 0 the measurable set Ag = f-1([e, ~)) is of positive measure m(Ae) > 0. Using continuity of the Lebesgue measure, we partition the set A e into two measurable sets of equal measure, Ae = AIUAi; m(Ai)=m(Ai), AinA~= ;~.

e(x) =

[

Consider the function

8 x 6 A1;

2 '

8 x 6 Ai; 2 '

0, xEi &.

T Since At, A 2 are measurable, the function g(x) is measurable and by construction ~g(x)dx

=0; [~g ; therefore the functions f(x) + g(x), f(x) - g(x) are densities, and the lemma ! I

follows from the representation f(~=-~If(x)+g(x)]@-~[/(x)--g(x)].

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It is easy to show that the set of densities of probability measures is a convex bounded closed subset of the space LI(-~ , ~). We have therefore constructed an example of a closed bounded convex set in a Banach space which has no extreme points.

3. Exact Randomization of Families of Distribution Functions

This section explores the question of randomization of families of d.f.~s which is of considerable importance in both theory and applications~ The mathematical propositions will be stated for d.f.'s of realr.v.'s, although they can be readily generalized to the case of d.f.'s on more genera] spaces.

Definition 2. Let F(x/y) be the d.f. of x given y. We say that the family {F(x/y)} is finitely randomizable if there exists a finite collection of d.f.'s[F~(x); ~ = I-~} and a family of randomization d.f.'s

%

such that for any y

(x/y) = E c~ (u) F~ (~). (3)

Definition 3. The family {F(x/y)} is called almost finitely randomizable if there exists a denumerable collectionofd.f.'s {F~(x); ~ = i, 2,...}, a set of natural numbers {n(y):sup x

Y

n(y) = ~}, and a family of randomizationofd.f.'s(2) such that (3) is satisfied for all y, with n = n(y). If n(y) = ~ for some y, we call this denumerable randomization~ Randomization of density functions is defined similarly~

Definition 4. A family of functions is called m-dimensional if its linear hull is a m- dimensional linear function space.

The geometrical meaning of finite randomizability is the following: the convex set of all d.f.'sfrom the hypersphere of functions of unit variation in the linear space V( -~, ~) of functions of bounded variation contains a convex* polyhedron which includes a randomizable (finite-dimensional) family.

Finite-dimensionality is obviously a necessary condition for finite randomizability, but, as we shall see below, it is not sufficient.

Since the intersection of a polyhedron with a space is again a polyhedron, the randomization elements {F~} may be chosen from the finite-dimensional set W=L({F(x/y)}N(D, where L(A) is the linear hull of the set A, r = %(-~, ~) is the set ofalld.f.'s. If W is a polyhedron, then obviously finite randomization is possible, if not, then, as is easily seen, W (a finite- dimensional set!) is not finitely randomizable. An example of this situation will be given below. If the family {F(x/y)} is two-dimensional, then W is a line segment in the space V • (-~, ~). For randomizable families of higher dimension, the set W is not always a polyhedron.

LEMMA 3. Let the finite family of densities {f(x/y)} have the following property: Yl ~ y~=>N(yJ=/=N(y~), where N(y) is the set of zeros of the density f(x/y). Then the family {f(x/y)} is not finitely randomizable, and if it is nondenumerable, then it is not almost finitely randomizable.

Proof is by contradiction. Assume that there exist m (m < ~) densities fl, f2 ..... fm such that {f(x/F)}~co(f1(x ) ..... fm(x)). Since m < ~, and the family {f(x/y)} is infinite, there exist l~<m densities f~1' f~2 ..... f~% such that for an infinite set Y,

l

f (x/v) = .,~ c~ (y) f~,k (x) (c~ k (y) > O; y E r) .

Since f=k(x)~O<c~k(y), then for all !IEY,

*In what follows, the adjective "convex" is sometimes omitted, but it is always implied.

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N (9) = ]1 ~v~6 N~ = {x :/~. (x) = 0}, k--I

a contradiction.

The second proposition of the lemma is proved similarly, since the set of all finite subsets of a denumerable set is also denumerable.

If we agree to distinguish between densities as elements of Ll, then the lemma clearly implies that the family of d.f.'s corresponding to the conditions of the lemma is not finitely randomizable in the set of all d.f.'s.

As an example of the application of the lemma, consider a three-dimensional family of quadratic densities on [0, i],

3 ( x - - Y)2 (x, y ~[0, 1]). [ (x/y) = I - - 3y + 3 f

Clearly, this family satisfies the conditions of the lemma. Therefore, it is neither finitely nor almost finitely randomizable, and the set W----K[0,1ln~[0. n is not a polyhedron, where q0[0.ll is the set of densities on [0, i], K[0,~l ={a+bx~, cx2}=L(x',x,l). Thus finite-dimensionality is not sufficient for finite or almost finite randomizability.

Similar argument obviously may be developed for any families of densities of the form

f ( x /v ) = ~ t v ) l x - - v l ~ (x, yC[0, ll; n = 1, 2, ...),

where ~n(Y) i s a n o r m a l i z a t i o n f a c t o r . I t s h o u l d be s t r e s s e d , however , t h a t in d i s t i n c t i o n from the families {(x - y)2n}, already the family { [x - y[ } is not finite-dimensional.

In view of the last remark, note the following point. When investigating randomization of concrete families of d.f. 's, we have to allow for the fact that the transformed family of d.f.'s {G(F(x/y))} (here G is some d.f. on [0, i]) may be infinite-dimensional even if the family {F(x/y)} is finite-dimensional. Consider the following example. Let

0, x~< O; F l ( x ) = x, O ~ x ~ l ;

[1, x ~ 1;

0, x<~ 1; P2(x )= 1 - - 6 - % x ~ > l .

Then c o ( F 1 , Fa) i s a t w o - d i m e n s i o n a l f a m i l y o f d . f . ' s , w h i l e I JI its transform sin nZ [cF 1 -}- (1 -- c) F=],

to [0, 1] i s s in-~cx; O < c < l , 0 ~ c ~ l / is infinite-dimensional. Indeed, its restriction

but for any n = 2, 3 .... the functions {sink~x} (~i~=~j; i, ] = l,n), are easily seen to be linearly independent.

THEOREM 2. Let B be a convex bounded closed subset of a locally convex space, D c B, and for any finite subset E c D there exists a convex polyhedron

W (E) = co (yl e, V2 e ..... Yf(e~) ~ B; E ~ W (E),

and supn(E)~ no< co. Then there exists a convex polyhedron E

W" ---- co (y~, y~, ..., y~) ~ B; D ~ W'.

Proof. From the assumptions of the theorem it follows that D is finite-dimensional, and therefore the linear hull L(D) is finite-dimensional and thus closed [5, 6], while the set B 0 = L(D) n B is a convex finite-dimensional compactum. Let WB0(E) = W(E) n B 0.

We know [7] that the number of vertices v(W) of any convex polyhedron W and the number of of its defining inequalities ~(W) are related by v(W)<~C~(wl; n=dimL(W) From obvious con- siderations we obtain the bound ?(W) ~p(v(W))= 2 ~(w>, and therefore*

v (WBo (E)) <~ C~cz--~J; ~P~e~= co (v (W (f)) = 2 ~(veCe~ <~ 2%

*Note that this is a very rough bound, but the proof of the theorem does not require anything better.

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Thus, ~'z~o ( E ) = co (t]f, ..., !/~,(m ), ~ " sup ~* (E) = n o < oc

Using the fact that D is finite dimensional and the space is locally convex, we choose in D a denumerable everywhere dense subset {xi; i~ I} [5, 6]. Then for any n ~ 1 there are y~n)..., ycn2EBo (not necessarily all different) such that x I ..... xn6co(y~ n) .... , u(n2) Since B 0 is com-

n o ~Yn 0 "

pact, there exists a subsequence {n~}; |imy~ ~> ----z~B 0 (i v---~ ~ = l, no). Let co ~ (z~, ..., z~) denote the e-

neighborhood of co(z~ .... ,z%). Since for any s > 0 there is k 0 such that for all ~

co (z~% ..., ~c~)) c co~ (z~, ..., z @ - - n 0

then {x~; i~ l}c co 8 (z~,..., zn~ ). And since I~ c~ (zl .... , zn~ ) = co(zl, ..., zn~ ) and co(zl, .o., z~) is a closed ~>0

set, we obtain DcOr-co(zl,...,Zn~ ).

COROLLARY i. Let {F(x/y); y6Y} be a family of d.f.'s and for any finite subset E c y we have the randomization

F (~y) = ~ c= (y) F~ ) (x) (y E E); sup n (E) < ~ .

Then t h e f a m i l y { F ( x / y ) } i s f i n i t e l y r a n d o m i z a b l e .

The f o l l o w i n g p r o p o s i t i o n p o l i s h e s t h e c o n d i t i o n s o f a l m o s t f i n i t e r a n d o m i z a b i l i t y o f f a m i l i e s o f d . f . ' s

THEOREM 3~ L e t t h e f a m i l y o f d . f . ' s S = { F ( x / y ) } be f i n i t e - d i m e n s i o n a l and t h e s e t S N Fr (co ( S ) ) a t most d e n u m e r a b l e , where Fr (co ( S ) ) i s t h e bounda ry o f t h e s e t co (S) in g (S ) , and ~(S} i s t h e h y p e r p l a n e in L(S) which c o n t a i n s S. Then t h e f a m i l y S i s a l m o s t f i n i t e l y r a n d o m i z a b l e . *

The proposition of the theorem easily follows from the next "geometrical '~ lemma.

LEMMA 4. Let K be a convex compact set in R TM. Then there exists a sequence of poly-

hedra {T~K:TncTn+~} such that K~ UT~, where K is the interior of the set K.

Proof. Because of separability of R TM, we can choose a denumerable subset {x l, x2,...} which is dense in the set of extreme points of K. Let T n = co (x I .... ,Xn). Clearly, T n c K,

T n c Tn+ I. Therefore UTn- is a convex subset of K and by the Krein-Mil'man theor~ UT n =

K [7]. ~

Let x06K; xo~UT~. Then by the finite-dimensional version of the Hahn-Banach theorem

[5] there exist a linear(continuous) functional~=O and number r such that ~(x0)~r; ~(x)~r

n

Take yER'n such that ~(y)>0. But then W(xo+~y)>r and (since :% is an interior point of K) ~@~y6K for sufficiently small s > 0, which by definition and continuity of ~ con-

tradicts the denseness of U Tn in K, so that any interior point of K is contained in ~Tn,

i.e., ~ bTr~. n ,~ tz

These propositions lead to some interesting corollaries.

Theorem 3 and Lenmm 3 lead to

LEMMA 5. Let s be a finite-dimensional nondenumerable family of densities which satisfies the conditions of Lemma 3. Then, with the exception of an at most denumerable set, all

*To date, we have managed to prove that for a wide class of families of functions S~the en- tire boundary Fr (co (S}) may be replaced by its part - the set of extreme points co (S)o

141

t

the elements of the family s belong to the boundary Fr (co (s)) of the closure of its convex hull co s in the hyperplane Z(s).

Clearly, the conclusions of this lemma hold for the quadratic densities considered in the example of Lemma 3. However, for these densities we have a still stronger result.

LEMMA 6. The extreme points of the set of quadratic densities on [0, i] are the densities

{6x(1--x); (y~--y-i-1)-~(x--g)~; x, y6[O, 1]}

and only they.

This lemma elucidates the geometrical structure of the set of quadratic densities. This is a convex subset of some two-dimensional linear space. Its boundary consists of a convex

curve formed by the p lanes y~--y-{-~- (x--y)";x, y610, and two po lygona l segments of equal I

l en g th wi th a v e r t e x a t the p o i n t 6x(1 - x ) ; the symmetry ax i s of the s e t of q u a d r a t i c dens i - t i e s passes through t h i s v e r t e x and the p o i n t 12(x - 1/2) 2 of the curve .

The following proposition establishes sufficient conditions of finite randomizability.

THEOREM 4. Let R n ~ K be a convex compactum, B(x, e) a ball of radius r centered at the point x, Ks={x6K:B(x,e)~K }. Then there exists a convex polyhedron in K which contains Ke.

Proof. Since K e c K c R n, then Ke is (completely) bounded. We can therefore choose a finite system of neighborhoods covering Ke, and (by definition of Kr such that its elements are contained in K. It is easy to see that hyperpyramids with n + 1 vertices may be used as the neighborhoods. The sought polyhedron is obviously spanned by the vertices of the hyperpyramids forming a finite cover of the set K e.

COROLLARY 2. Let (--oo, oo)~ X be a Lebesgue measurable set of finite measure, ~x the set of all densities on X, ~x ~ s a finite-dimensional family of densities on X uniformly separable from 0. Then s is finitely randomizable.

Proof. Finite-dimensionality implies closure of L(s), so that K=L(s)N~x is a convex finite-dimensional compactum. Note that by finite-dimensionality, all the norms in L(s) are equivalent and all linear functionals are continuous, therefore the set I d = I[E i(s):~[(x)dx=

~ ~ X

d} (d~(--oo, oo)) is a hyperplane in L(s), and Kis a convex body in I I. Let g~K. Then the setK--g={f--g:f6K} is a convex body in the linear space I 0. Using equivalence of norms and uniform separability from zero of the densities from s, we can easily show that s - g c (K - g)e for some e > 0. The last inclusion, by Theorem 4, indicates that s - g is finitely randomizable in K - g, and therefore s is finitely randomizable in K.

Since K----UKI , then Theorem 4 leads to Lemma 4 (without resorting to the Krein-Mil'- n

man and Hahn-Banach theorems).

4. Approximate Randomization

The above results indicate that exact finite randomization is a highly desirable, yet very rare property, and it is therefore appropriate to consider the development of approximate randomization methods.

The geometric approach to this problem seeks to establish conditions for approximation of convex sets by convex polyhedra.

Definition 5. We say that the family ofd.f.'s {F(x/y)} is approximately randomizable

if c-o{F(x/9) } == UTn, where {Tn} is a monotone increasing sequence of polyhedra in the set of r~

d.f.'s, and the closure of the convex hull and the union of polyhedra are taken in the same topology in which the approximation is sought.

By the Krein-Mil'man theorem [7], the representation specified in the definition is realizable if the original family is relatively compact, since a convex hull of a relatively compact set is relatively compact.

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Thus, the problem of approximate randomization of a family ofd.f.~s reduces to finding conditions of their relative compactness in the corresponding topology.

Let C = C(-~, ~) be the space of continuous functions bounded on (-~, ~) with the norm i[[(x)][c: sup [ / (x) l; V c : Vc(--oo, ~) the space of continuous functions of bounded variations,

i . e . , 0; I!/ll = V If(,))<

Using the standard methods of the theory of functions of real variable, we can easily prove the following proposition.

LEMMA 7. Let ~=d~(_oo, oo) be a set of d.f.'s. Then ~ N C=~ N V~ is closed in C and V C.

THEOREM 5. The subset A c~ N C is relatively compact in C if and only if all the functions from A are equicontinuous and uniformly in A

l i m F ( x ) = 1, lim FIx)=O(FEA).

Proof. Necessity is easily proved using Hausdorff~s criterion [5].

Sufficiency. Let {Fn(x) } be a sequence of d,f.'s fromA, By Helly's second theorem [5], we can extract from this sequence a subsequenceF~k(x)~> F(x) pointwise convergent on (-~,

~). We will show that it is uniformly convergent (in the norm of C).

We know [5] that equieontinuity and pointwise convergence on a compactum imply uniform convergence.

Take a fixed e > 0. By the assumptions of the theorem, there exists b > 0 such that for all k ~ i,

Y n k ~ ) - ~ < ~ ( x < - - b ) ; F,,k(x)~l--~Lx>b).

On [-b, b], the sequence {Fnk(x)} is uniformly convergent if there is k 0 such that

sup I &k (x) -- y (x) I ~< ~ (k ~> k0). ~E[--b,b]

By t h e c h o i c e o f b , we h a v e ~ sup [F(x)--Y,,k(x)[. T h u s , .r~(- -oo,--b) d (b, ~)

sup I Fn~ -- F (x)[ <~ ~ (k ~> ko). xE(--oo.:o)

THEOREM 6. The subset A~NF c is relatively compact in V C if all the d.f.~s from A are absolutely equicontinuous and uniformly in F E A

lim F(x) = 1, lim F(x) = O(FCA).

Proof. Absolute equicontinuity implies that density exists for any d.f. FCA and on each finite interval [a, b] the family of all these densities is relatively compact in L1[a, b] [ 5 , 8 ] .

Consider an arbitrary sequence {Fn(x) } c A. Let fn(X) be the density of Fn(X). Using relative compactness of the family of densities in L1[a, b] and Cantor's diagonal method, we extract a subsequence {fnk(x)} which is convergent on each [-n, n] in the sense of L1[-n, n]~

b b

Since V[F(x)]--~_ i'f(x)dx, then {Fnk(x) } is convergent in variation on each of the inter- a

vals I-n, n].

Using uniform convergence at the end points and reasoning as in the proof of the pre- ceeding theorem, we find that f,~k(x)1,~F(x) in the sense of convergence in V C, where, by Lem-

ma 7, F(x) is a d.f.

Remark 2. Relative compactness of the family S = {F(x/y)} makes it possible, fo___r any e > 0, to choose from the set of extreme points of the closure of its convex hull co S a finite e-grid (F l, F 2 .... ,Fn(e) which constitutes an aggregate [7] for the subsequent problem of

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convex finite-dimensional programming - the problem of choosing a family of randomization d.f.'s {c~(y);~ = i, n(e)} from a given class (e,g., the class of d.f.'s continuous in y).

I ,,~ I I As a c r i t e r i o n f o r t h i s c h o i c e , we n a t u r a l l y t a k e sup F ( x , ' t t ) - - ~ ca(!/)F~(x)

With this approach, approximate randomization of relatively compact families is a two- stage procedure. In the first stage, we find the vertices of the randomization convex polyhedron (the randomization elements), and in the second stage we construct the optimal randomization d.f.'s.

5. Application of the Results

l. The notion of randomization of a r.v. introduced in Sec. 1 is particularly useful in experimental design and modeling of compound stochastic systems, when we have to manipulate the r.v.'s themselves.

Note that randomization of r.v.'sisrelevant and topical also in the context of the von Mieses approach to the analysis of stochastic system. Of particular importance is the problem of description of the set of allr.v.'s on [0, i] such that finite randomizations ofd.f.'s always correspond to randomization of the r.v.'s themselves, since especially with the yon Mieses approach the choice of the probability space is restricted in many models to ([0, i], o , P).

2. As we have noted above, finite-dimensionality is a necessary condition for finitely randomizability of the family ofd.f.'s. In practice, it is often quite difficult to check the condition of finite-dimensionality. However, the analysis of compound stochastic systems may be based on an approximation of the defining families of d.f.'s by linear combinations of finitely many elements in some basis, which obviates the need to check for finite- dimensionality. This situation arises in cases when the randomizable d.f.'s are estimated from statistical analysis.

After establishing finite-dimensionality, we go through the following stages: find a convex polyhedron which includes the randomizable family and determine the family of random- izationd.f.'s - collections of coefficients of convex linear combinations. This opens certain prospects for applying the results of the first two sections.

If (despite finite-dimensionality) it is impossible to implement exact finite randomization, we may try to implement almost finite exact randomization, denumerable randomization, or possibly approximate (finite) randomization, which is particularly important because of its relatively wide scope of applications.

3. Let us describe one of the simplest approaches which applies approximate randomization to the analysis of compound stochastic systems.

Let the customer service time in a single-channel queueing system have the conditional distribution F(x/y), where y is the service time of the preceding customer. For many real systems, we may naturally assume that the service time is bounded with probability i, i.e., 9~y0<~, and the function F(x/y) is continuous in y uniformly in x, so that the family

{f(x'y); 9C[0, yo]} is a continuous curve Z(t) (t= 9----C[0,1]l in the set of all d.f.'s ~, i.e., \ Y0 ]

{Z(t) = F(x/y0) } is continuous (in the uniform topology) image of the interval [0, I].

Any continuous curve, being a continuous image of a compactum, is also compact, so that (in view of the above) the orginal family of d.f.'sof customer service time admits (finite) approximate randomization. Consider the polygon Zn(t) inscribed in the curve Z(t),

Z,~G):=(tn--k)Z\~]+(k@l--tn)Z ~ l < ~ - - ; n k = = O , n - - 1 .

We c l e a r l y have u n i f o r m c o n v e r g e n c e Z,~(t)~i Z(t) and f o r any n 9 1 t h e p o l y g o n Z n ( t ) ( r e g a r d e d

as a f a m i l y o f d . f . ' s ) i s f i n i t e l y r a n d o m i z a b l e w i t h r a n d o m i z a t i o n e l e m e n t s { Z ( k / n ) ; k = O, n} . A similar construction is feasible when there is dependence on several previous service times. In this case, the randomizable family is a surface in ~.

The proposed approach obviously leads to approximate randomization of compound input streams and semi-Markov processes, so that we can apply the results of [1-4] to investigate

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a fairly wide class of stochastic systems with partial memory: systems describable by compound semi-Markov processes and also systems serving compound recurrent streams.

Note that numerous studies have been published on questions of stability (in corresponding topologies) of the characteristics of various queueing processes and systems (see, cog., [9-12]).

By analogy with the discussion of Sec. 4, we can find other necessary and sufficient or only sufficient conditions of relative compactness of families of d.f.'s defining stochastic systems in the sense of various metrics in which stability of the relevant characteristics is guaranteed by appropriate known results. In this way, we will find the conditions of stable (for the relevant characteristics and in the sense of the given metrics) approximation of the corresponding stochastic systems by randomized systems.

If the conditions ensuring relative compactness of the defining families of d.f.~s are satisfied, the problem of determining the corresponding system characteristics with accuracy

may be divided into two stages. In the first stage, standard methods and results are applied to calculate from the given s the required accuracy ~ of the approximation of the de- finingd.f.'s(i.e., the initial system characteristics); in the second stage, the above-de- scribed scheme is applied to solve the problem of approximate randomization with the given accuracy 6 of the family of defining d.f.'s and to calculate the sought characteristics of the randomized model.

Although a direct check of the conditions of the theorems of Sec. 4 and their analogs is often quite complicated, the simplicity of randomized models and the stability theorems mentioned above are sufficiently attractive for application of approximate randomization.

LITERATURE CITED

i. I. I. Ezhov and A. M. Zakharin, "Sojourn time of a compound randomized semi-Markov pro- cess in a fixed subset of states," Dokl. Akad. NaukUkrSSR, Ser. A, Noo4, 345-348 (1978) o

2. A. M. Zakharin, "The ergodic theorem for singly compound randomized semi-Markov processes," Ukr. Mat. Zh., No. 6, 809-811 (1978).

3. A. M. Zakharin, "Queueing processes of compound randomized recurrent streams," Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, 126-131 (1979).

4. A. M. Zakharin, "Distribution of the busy period in the system (IRRG, G, E/I, i), '~ Kibernetika, No. 3, 145-146 (1980).

5. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972).

6. M. Reed and B. Simon, Methods of Modern Mathematical Physics: Functional Analysis [Rus- sian translation], Vol. i, Mir, Moscow (1977).

7. B. N. Pshenichnyi, Convex Analysis and Extremal Problems [in Russian], Nauka~ Moscow (1980).

8. P.A. Meyer, Probability and Potentials [Russian translation], Mir, Moscow (1973). 9. A. A. Borovkov, Asymptotic Methods in Queueing Theory [in Russian], Nauka, Moscow (1980).

i0. V. M. Zolotarev, "Quantitative estimates in continuity problems of queueing systems," Teor. Veroyatn. Ee Primen., 20, No. i, 215-218 (1975).

ii. V. M. Zolotarev, "Metric distances in spaces of random variables and their distributions," Mat. Sb., !01 , No. 3(11), 416-454 (1976).

12. V. V. Kalashnikov, Qualitative Analysis of the Behavior of Complex Systems by the Methods of Sampling Functions [in Russian], Nauka, Moscow (1978).

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randomization of random variables and their distributions

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