A C T A U N I V E R S I T A T I S L O D Z I E N S I S
FO LIA O EC O N O M IC A 141, 1997
Grażyna Trzpiot*
L IM IT LAW S FO R M U L T IV A L U E D R A N D O M VA RIABLES
Abstract. In the probability theory, the strong law o f large num bers and the central lim it theorem are the most im portant convergence theorems.
Given a probability measure space (fi, A, P), random variable in classical definition is a m apping from Í2 to R. M ultivalued random variable is a m apping from О to all subset of X . F o r a real separable Banach space X with dual space X * , let L p(Cl, A), for 1 < p < oo, denote the X -valued //-sp a c e . In this paper we introduce the multivalued Ľ space, next the integral for m ultifunction and some property o f the sequences in X with respect to the HausdorfT distance convergence.
Probabilistic law for m ultifunctions are available, when multifunctions are viewed as point-valued m apping into a p p ro p ria te space in which the sets are embedded. In this paper, we will discuss limit laws for multivalued random variables whose values are com pact o r weakly com pact in Banach space.
Key words: multivalued random variable, multivalued function, multivalued L p space, B anach space.
1. IN T R O D U C T IO N
In the probability theory, the strong law of large num bers and the central limit theorem are the m ost im portant convergence theorems.
Probabilistic law for m ultifunctions are available, when m ultifunctions are viewed as point-valued m apping into appriopriate space in which the sets are embedded. In this paper, we will discuss a limit law for random variables whose values are weakly com pact in Banach space.
The paper is organized as follows. In section 2 we display the multivalued random variable; some properties o f which are discuss in section 3. The limits law in Banach space is presented in section 4.
2. M U L TIV A L U ED R A N D O M VARIA BLE
Given a probability m easure space (ÍÍ, A, P), a random variable in classical definition is a m apping from Í2 to R. M ultivalued random variable is a m apping from Q to all closed subset of X .
We have a real separable Banach space X with dual space X*. F o r any nonem pty and closed sets A, B a X we define the excess e(A, B) of A over B, the H ausdorff distance h(A, B) o f A and B, the norm \\A\\ of A, and the support function я(Л, •) o f A.
Definition 1. The excess for two nonem pty and closed sets is defined by e(A, B) = sup d(x, В), where d(x, В ) = inf II x - у II
х бА у е В
the H ausdorff distance of A and В is given by h(A, В) = тах {(еЛ , В), e(B, А)},
the norm ||А || of set A we get as IIA II = h(A, {0}) = sup II x II
x e A
and the support function:
s(4 , x*) = sup < x, x* > , x * e X * .
x e A
The set o f all nonem pty and closed subsets o f X is a m etric space with the H ausdorff distance. The set o f all nonem pty and com pact subsets of X is a complete, separable m etric space with the m etric h.
F o r sequence o f nonem pty and closed subsets o f X besides the H ausdorff distance, we use some notion for convergence sequence o f sets ( H a u s d o r f 1957; S a l i n e t t i , W e t s 1979).
Given a sequence {An} of nonem pty subsets o f X let:
- s-lim inf A„ be a set o f all x e X such th at || x„ - x || —* 0 for some x ne A n, n ^ l
- w-lim sup A„ be a set o f all x e X such that x t —> x (weakly) for some x keA„t (k > 1) and some subsequence { Ą J of {A„}.
Clearly
s-lim inf A n cz w-lim sup A n.
F o r a sequence o f nonem pty and closed sets {An}, s-lim inf A„ is also nonem pty and closed.
Definition 2. The sequence {/!„} converges to A, denoted by lim A n = A, if
n~* 00
s-lim inf A„ = A — w-lim sup A n.
Definition 3. A m ultivalued function (p: Q —>2X with nonem pty and
closed values, is said to be (weakly) m easurable if cp satisfies the following equivalent conditions:
a) cp~l (C) = {toeQ: ę ( w ) n C Ф 0 } e A for every open subset С of X , b) d(x, (fl(cj)) is m easurable in со for every x e X ,
c) there exists a sequence {/„} o f m easurable functions f„: Q — > X such th at (p(co) = cl{/„((y)} for all w e i l.
Definition 4. A measurable multivalued function q : Q —*-2x with nonempty
and closed values is called a m ultivalued random variable.
A m ultivalued function is called strongly m easurable if there exists a sequence {ę>„} o f simple functions (measurable functions having a finite num ber o f values in 2X), such that h(qn(co), (p(a>))— > - 0 a .s.
Since the set o f all nonem pty and com pact (or convex and com pact) subsets of X is a complete separable m etric space with the m etric h, so multifunction (p: f i —»-2х is measurable if and only if it is strongly measurable.
This is equivalent to the Borel m easurability o f (p.
Let K ( X ) denote all nonem pty and closed subsets o f X . As the tr-field on K ( X ) , we get the er-field G generated by cp~i (C) — {coeQ: ср(ш)n С Ф 0 ) , for every open subset С o f X.
Definition 5. We say th at a sequence o f m ultivalued random variables
<p„: Ü —>2K(X) is independent if so is {rp„} considered as m easurable functions from (Q, A, P) to ( K ( X) , G).
Definition 6. Two multivalued random variables ę , ф: П —► 2K{X) are
identically distributed if <p(eo) = ф(ы) a .s.
Particularly for q n with com pact values independence (identical distribu-tedness) o f {</>„} coincides with th at considered as Borel m easurable functions to all nonem pty, com pact subsets of X .
In this case, {<?„} is independent if and only if
p ( / W c Gii) = П р ( { ш: с G,})
3. M E A N O F M U L TIV A LU E D R A N D O M V ARIA BLE
Let Lp(Cl, A), for l < p < o o , denote the X - valued Lp - space. We introduce the m ultivalued L p space.
Definition 7. The m ultivalued space Lp[fJ, -K(AT)], for 1 oo denotes the space of all m easurable m ultivalued functions <p: Cl—*-2Km, such that
lipII = Il<p(-)ll is in L p.
Then L P[Q, K(X)] becomes a complete m etric space with the m etric H p given by I
H p( ę, ф) = { $nh(q>(w), 'K v ) pdP}i,p, for 1 со
H p(q>, Ф) = ess sup /i(<p(w), ф(а>),
I a>eil
where cp and ф are considered to be identical if ę(co) = ф(а>) a.s.
We can define similarly other L p space for the set o f different subsets o f X (convex and closed, weakly com pact or compact). We denote by L P[Q, K(A")] the space o f all strongly m easurable functions in L P[Q, K(X)]. Then all this space is complete m etric space with the m etric H p.
Definition 8. The m ean E ( ę) , for a m ultivalued random variables cp: Q — > 2K(X) is given as the integral Jaq>dP o f cp defined by
io cpdP = { \ J d P - . f e S ( ( p ) } where
S(cp) = { / e L1 [fl, X]: f(co)e<p(co) a.s.}
The m ean E(cp) exists if S((p) is nonempty. If cp have an integral the E((p) is compact.
This m ultivalued integral was introduced by A u m a n n (1965). F o r detailed argum ents concerning the m easurability and integratin o f m ultifun-ction we refer to ( C a s t a i n g , V a l a d i e r 1977; D e b r e n 1967; H i a i , U m e g a k i 1978). Now we present some properties o f m ean of m ultivalued random variables.
Let cp, ф: Q —*- 2K(X) be two multivalued random variables with nonem pty S(cp) and S(tI/), then:
1. cl Е((риф) — cl(jE(ę>) + Е(ф)), where (ф и ^ )(ш ) = cl(<p(cü) + ф(а>)). 2. cl E(côcp) = čoE(cp), where (c0ę)(a>) = coę(io), the closed convex hull. 3. s(cl E(cp), x*) =E(s(<p(-), x*)) for every x * e X * .
4 . Let Wc( X) denote all nonem pty and weakly com pact subsets o f X . I f ę> = L1[£2, Wc(X)], then E(q>)eWc(X).
4. M U L TIV A LU E D STRO N G LAW O F L A R G E N U M B E R S
A m ultivalued strong law of large num bers was proved by A r s t e i n and V i t a l e (1975) for a sequence o f independent identically distributed random variables having values in compact subset o f R n.
Given a probability m easure space (П, A, P) and a Banach space X , we have a theorem ( S e r f l i n g 1991, p. 134):
I f {/„} is a sequence of independent identically distributed random variables in L ^ Q , X], then
П
lim |j n _ 1 — m H = 0 a.s. (*)
n - x i=1 where m = E ( f n).
We now establish a strong law for multivalued random variables, which are generalization of this theorem , for the case o f independent identically distributed random variables with values are weakly com pact subset of Banach space. W e start by presenting two lemmas.
Lemma 1. If ( p e L ^ Q , K(X)] and £(</>) = {*}, then there exists an / e L ł [Q, X] such th at (p(o)) = {/(to)} a.s.
Proof. We can choose the sequence {x*} in X * which separates points o f X
By point 3) in section 3 we get
E(s(cp( ), x * ) ) = < X , x * > = £ ( inf < x , x * > ) ,
xe p ( )
and hence s((p(a>), x*) = inf < x, x* > a.s. for j ^ i. This implies rp(co) is *«?>(•)
a single point for a.s. coeQ. T hus the lemma is proved.
F o r each A e W c ( X ) and x * e l * , we define Ф(Х, х * ).е Ж ;(Х ) by Ф(А, x*) = { х б Х : < х , x* > = s(A, x*)}
Lemma 2. F o r each x* e X*, the m apping Ф( -, x*): Wc( X) —► Wc(X) is m easurable with respect to (G \ Wc (X ), G\Wc(X)).
Proof. Let G = G\Wc(X). Since each open subset o f X is a countable union o f closed balls, it suffices to show th a t { A e W c ( X ) \ Ф(А, х * ) . п У Ф 0 } e G for any closed ball V
{ A e K c ( X ) : s(A, x*) > а} = {А е К с ( Х ) : Л п { х : < х , x * > > a ] ^ 0 } e G for every a e R, so the m apping Ah-+s(A, x*) from K c ( X ) to R is G m e-asurable.
Let Кл = {х: H x — y H ^ r + n " 1} for n > l . If A e W c ( X ) and A n V „ ^ 0 for n ^ l , then {Л n c l V„} is a decreasing sequence of nonem pty weakly
П
com pact subsets of X , so that A n V = ( ~ ] { A n V n} Ф 0 . П= 1
Hence we get
{ А е Щ Х ) : A n V * 0 } = f ] { A e W c ( X ): A n V „ * 0 } e G . rt= 1
M oreover, for any closed ball V = {x: ||x - у || s^r'}, we similarly get { A e W c ( X ) : A n V + V ' * 0 } = ( \ { A e W c ( X ) : A n VnnV'„ Ф & ) e G ,
11= 1
where V'n = {x: ||x - y ' | | < r ' + и- 1 }.
H ence the m apping A\-+ A n V itova. Wc(X) to Wc(X) is m easurable with respect to (G, G). Thus we have
{ A e W c ( X y . Ф(A, x * ) . n ^ 0 } = { A e W c ( X ) \ A n V * 0 , s ( A n V , x*) = = s(A, x * )} eG .
The lemma is proved.
Theorem I . If { ę nj is a sequence o f independent identically distributed random variables in L l [Wc(X)], then
П
lim n- 1 Y, = M a.s.
П-* CO
where M = E(cpn).
П
Proof. F o r {<?„}, let Т„(ш) = и“ 1 £ <р,(ю) for и ^ 1. W e get E{q>„) e Wc(X)
i=l
by point 4) in section 3. As noted in the p ro o f o f Lem ma 2 the m apping A\ -*s(A, x*) from K c ( X ) to R is G m easurable, it follows th at {s(q>„(■)> x*)} is identically distributed for every x * e X * . Hence, by point 3) in section 2, we get that M = E(cpn) is independent o f n.
W e first show that M e :s-lim inf ^ „ (o j) a.s.
Since M is a closed convex hull of its strongly exposed points and s—lim inf T„(co) is closed and convex, it suffices to show that any exposed point o f M is contained in s-lim inf ^ ( ш ) for a.s. w e i l
Let x be any exposed point o f M , then there is an x * e X * with Ф(M , x*) = {x}. Lem ma 2 implies that {Ф(<ря( •), x*)} is a sequence o f i.i.d.
random variables in L ^Q , Wc{X)]. If f e S(<b((p„( ■), x *)), then we have E ( f ) e M and < E ( f ) , x* > = s ( M , x*), so that E(f) = Ф(М, x*) = {x}. Hence £(Ф(<р„(-), **)) = {x}. By Lemma 1, there exists an f ne L l [£l, X ] such th at Ф(<?„(•)> x*) = { f n(o))} a.s. We thus obtain {/„} of independent identically distributed random variables in L x[ii, X] with E(f„) = x. It
П П
follows from (*) that lim ||n_ 1 Yj fi (w) ~ x II — 0 a s - Since n- 1 £ / ;( с u) G'í#n(ctí)
" - ® i = l i = l
a.s., we get x e s-lim inf ^„(co) a.s
Now, we show that vv-lim inf '{'„(coJeM a.s.
Since X is separable we can choose a sequence {x*} in X * such that if < x , x * > < s ( M , x*) for all j > i then x e M . The sequence {s((p(-), x*)) is a sequence o f i.i.d. random variables in L1 with the m ean s(M ,
x*). Hence there exist B e A with P(B) = 0 such that lim яСРя(а>), Xj) = lim n ~ i £s(</>((co), x j ) = s(M, x*)
rt~* 00 n~* 00 1 - 1
for every _/ >i and coeil/B. If x e w -lim inf '{'„(со) with со е й / В then х л—»-x (weakly) for some x ke s - 4 ,„i(co) for /с > 1. Since
< x , x * > = lim < х А, x* > ) — limsOF^ci)), x*) = s(M, x*), for j > i,
Л - * 0 0 n ~ * CO
we get x e M . T hus w-lim inf x¥ n( o ) ) e M a.s.
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Grażyna Trzpiot
T W IE R D Z E N IA G R A N IC Z N E D LA W IELOW A RTO ŚCIOW Y CH Z M IE N N Y C H LOSOW YCH
W teorii praw dopodobieństw a mocne praw o wielkich liczb oraz twierdzenie graniczne stanow ią podstawowe teorie konwergencji.
Przy danej mierze przestrzeni probabilistycznej (fi, A, P) zm ienna losowa definiow ana jest klasycznie jak o odwzorowanie f i n a R. W ielowartościowa zmienna losowa jest odw zorowaniem f i na wszystkie podzbiory X . W pracy w prowadzono pojęcie przestrzeni L p, całki dla funkcji wielow artościowej oraz niektóre własności ciągu w przestrzeni B anacha ze względu na zbieżność według m etryki H ausdorfia.
Rozważane są również moce praw a wielkich liczb dla wielowartościowych zmiennych losowych i twierdzenia graniczne w przestrzeni Banacha.
