In this paper, we introduce a method of extending the domain of a random mapping admitting the series expansion.. Keywords and phrases : random operator, bounded random operator, domain
Trang 1VNU Journal of Science, Mathematics - Physics 25 (2009) 237-248
Series representation of random mappings and their extension
Dang Hung Thang*, Tran Manh Cuong Department of Mathematics, Vietnam National University, 334 NguyenTrai Str, Hanoi, Vietnam
Received 28 February 2009
Abstract In this paper, we introduce a method of extending the domain of a random mapping
admitting the series expansion This method is based on the convergence of certain random
series Some conditions under which a random mapping can be extended to apply to all X -
valued random variables will be presented
AMS Subject classification 2000: Primary 60H05; Secondary: 60B11, 60G57, 60437,
37155
Keywords and phrases : random operator, bounded random operator, domain of extension,
action on random inputs
1 Series representation of random mappings
Let X,Y be separable metric spaces By a random mapping from X into Y we mean a rule
® that assigns to each element x € X a unique Y - valued random variable a Equivalently, it is a mapping ®: 2 x X — Y such that for each fixed x € X, the map ®(.,2): 2 — Y is measurable
In this point of view, two mappings ®; : Qx X — Y, ®2: Ox X — Y define the same random mapping if for each x € X
®)(2,w) = Oa(xz,w) as
Noting that the exceptional set can depend on x In this case, we say that the random mapping ©®2 is
a modification of the random mapping ®,
Definition 1.1 A random mapping ® from X into Y is said to admit the series expansion if there exists a sequence (f,,) of deterministic measurable mappings from X into Y (rep from X into R) and a sequence (a,,) of real-valued random variables (rep Y-valued r.v.’s) such that
œœ›
Oz = ) Anfn&,
n=1 where the series converges in L)
In the case the sequence (a,,) are independent we say that ® admits an independent series expansion
* Corresponding authors Tel +844.38581135:
E-mail: hungthang.dang@gmail.com
237
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Proposition 1.2 Let đ be a random operator from X into Y and suppose that X is a Banach space
with the Shauder basis e = (€,)°°, and the conjugate basis e* = (e%)°°_, Then đ admits the series
expansion
Recall that, a random mapping đ is called a random operator if it is linear and stochastically
continuous, 1.e
đ(AiZi + Àa#2) = ÀiđZ1 + ÀzđZ2¿, đ.4 Vq,a2€C X,ÀI,Àa2€R,
and
plimz,„„ đứœ — đứn
Note that the exceptional set may depend on 1, Àas, #1, 8a
Proof For each x € X, we have
œ
4= So (a, Cen
n=1
Since â is linear and stochastically continuous, we get
œ
Ox = S (a, e* ) Pe,
n=1
where the series converges in 1)
Put ayn = Pen, fr(x) = (x, e%) (Qn) is a sequence of Y-valued and (f,,) of deterministic
measurable mappings from X into Y We have
œ
Ox = ằ ằn lạa
n=1
O
A random mapping đ from X into Y is called a symmetric Gaussian random mapping if for
each k € N and for each finite sequence (;, y7)*_, of X x Y* the R* - valued random variable {(z;, y*) }&_, is symmetric and Gaussian
Theorem 1.3 Let đ be a symmetric stochastically continuous Gaussian random mapping Then đ admits an independent series expansion
eo
Ox = › On frđ,
n=1
where (Qi) is a sequence of real-valued Gaussian i.i.d random variables and f, : X — Y is continuous (so is measurable)
Proof Let [đ] denote the closed subspace of L2((Q2) spanned by random variables {(2, y*), x € X,y* € Y*} Then [9] is a separable Hilbert space and every element of [â] is a symmetric Gaussian random variable Let (a,,) is an orthonormal basis of [â] Since the sequence (a) is orthogonal,
symmetric and Gaussian, it is a sequence of real-valued Gaussian 1.1.d random variables Now for each
n, we define a mapping f, : X — Y by
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Here the Bochner (1) exists because by Cauchy inequality
[ llan(w)ba(w)||dP(w) < {El|®z|f}' 7 (2)
Fix « € X For cach y* € Y*, (a, y*) € [®] is expanded in the basis (a,,) in the form
(ba, y*) SỨ, [ive yond Pw ) Op
So ( [ontear(o).u") on n=1
oo
— (an fan, 1)
n=1
where the series converges in L2(Q) so it is convergent in probability Since the sequence (a,,f;,2) is
a sequence of symmetric independent Y - valued r.v.’s, by the Ito - Nisio theorem, we conclude that
œ
Ox = » œnyfạa a.S
Finally, fixing ø„ we show that ƒ„ is continuous Let (z„) C X such that lim, 2, = 2 From (2) we
have
|./s+ — f„e|lŸ < E\| Oa, — ®z'|Ï
By the assumption p— lim ®z„ —= ®z and the faet that in [®] all the convergence in L,(Q), (p > 0)
are equivalent, we have lim Ea, — b2||? = 0 Therefore, lim, frxe = fr® L] Next, we shall be interested in possible extensions of Theorem 1.3 to the case of symmetric stable random mappings
Let ® be a random mapping from X into Y ® is said to be a symmetric p-stable random mapping
(SpS random mapping in short) if the real process {(®z, y*)} defined on X x Y* is symmetric p -
stable In this case, for each x € X, a is a Y-valued SpS random variable
Let [®] denote the closed subspace of Lo(Q) spanned by random variables {(®z,”),œ € X,y* © Y*} If € € [® then € is SpS so the chf of € is of the form exp{—c|t|?}, where c = c(€) is
a non-negative number depending on € The length of € denoted by ||é]|, is defined by
Elle = £e(€) 37”
It is known that (see [1])
Lemma
i) The correspondence & += ||E||, is an F-norm on |®| and in fact is a norm in the case p > 1
ii) [®] is a linear subspace of each L,(Q),0 <r < p and all topologies L,(Q), 0 <r <p coincide with the topology induced by ||&|| - norm on [®}
iti) The F - space |®] can be isometrically embedded into some L,(S, A, H)
Theorem 1.4 Let ® be SpS stochastically continuous random mapping and suppose that |®| is isometric
to l, Then ® admits an independent series expansion
œ
Pz = » ằn lạ,
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where (Q;,) is a sequence of real-valued SpS i.i.d random variables and f,,: X — Y is continuous (so it is measurable)
Proof Let I be an isometry from [®] onto J, and J = I~ Put
An = J(En),
I ((@z, y")) = B(x, y") € ly
At first, we shall show that (a,,) is a sequence of real-valued SpS i.i.d random variables Indeed, the
joint chf f(t), tz, .,t,) of the random variable (a1, a2, ., %,) is equal to
f (th, ta, «5 tn) = Eexp {Soto = Eexp {Some
J [done]
k=1
= Eexp { [doues) f = exp {-
op} S| ther } —en{-S iar |
as desired
For each (a, y*) € X x Y*, we have
J
hence
where b„(œ, *) is the n-th coordinate of B(x, y*) € l, and the series (3) converges in the norm || ||
so converges in probability
Fix n We show that there exists a mapping f, : X — Y such that for each x € X,y* € Y*
Fix « € X Since the mapping y* +> (®z, y*) is linear so the mapping y* +> B(x, y*) is linear which implies the mapping b, : y* +> b,(x,y*) from Y* into R is linear In addition, the ch.f of ®zx is
7(Y*, Y)- continuous on Y*, where r(Y*, Y) is the topology of uniform convergence on compact sets
of Y, and it is equal to
Ay(y") = exp{—||(®x, y) [EF = exp{-||B@, ”)ÏJƑ}
Consequently, b, : Y* — R is linear and 7(Y*, Y)- continuous on Y* Since the dual space of Y* under the topology 7(Y*, Y) is Y we conclude that there exists a unique element denoted by f,,x such that
bry") = (fn®,y") > On(x,y") = (fre, 9”).
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Now, the equality (3) becomes
(bz, y*) >>
œœ›
=Ss\ AnfnX, 1))
n=1
The rest of proof is carried out similarly as in the proof of Theorem 1.3
Finally, fixing n, we show that f, is continuous Let (v,) be a sequence of X such that lim xv, = x By the assumption p-lim a, = ®a, we have
œ
Ox, — Oa = So ay (fire — ;#)
j=l Since p < 2 by Corrolary 7.3.6 in [2], we get
œ
fn — fu#llP < Ð 2 | fies — Fpl? < CLE ||bay — dar",
jal
where r < pand the constant C > 0 depends only on, p From 2 of Lemma we obtain lim, { F'||®2,—
2 The extension of random mappings admitting series expansion
Let ® be a random mapping from X into Y admitting the series expansion
œ
n=1
where (/,,) is a sequence of deterministic measurable mappings from X into Y (rep from X into R) and (a,,) is a sequence of real - valued random variable (rep Y - valued r.v.) The series converges
in Li
Denote by D(®) the set of all X - valued rv u such that the series
n=1
converges in probability Here f,u(w) = fp (w(w)) is a random variable because f,, is measurable
Clearly, X C D(®) c Lự
Definition 2.1 D(®) is called the domain of extension of ® If u € D(®) then the sum (5) is denoted
by ®w and it is understood as the action of ® on the random variable w
Theorem 2.2 [fu is a countably - valued rv
œœ›
tụ — ) 1m;
¿=1
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where (E;,i = 1,2, ) is a countable partition of Q and x; € X, then u € D(®) and
œœ›
Ou = » 1p, Đã
i=l Proof Put Z, = S37, ai fiu and Z = SY, 1p, Pax; We have to show that
lim P(||Z, — Z|| > £) = 0
Since w € Hy > Z(w) = xp, Zp(w) = YO ai five so
¿=1
œ P(|Z¿ — Z|Í> #9 = 3 ` P(|Z2 — Z| >t Ex)
y=
k
N
<yP|
k =1
n
» œ4 q4 — ®Đãy
i=l
œœ›
> J + P(Ex)
k=N+1 For each k = 1,2, , N we have
lim P(| » ằœ;f;ag — ®a„|| > £) = 0
i=l Let n — co and then N — co, we get lim, P(||Z, — Z|| >t) = 0 L]
For each random mapping ® admitting the representation (4), let F(a) denote the o-algebra
generated by the family {a,,} A random variable u € Lx is said to be independent of © if F(u) and F(a) are independent
Theorem 2.3 Suppose that u is independent of ®, then u € D(®)
Proof Let t > 0 By the independence of u and the sequence (a,,) we have
(ee) Lo
where pz is the distribution of u Because for each « € _X
n
» ag fru
=m
n
» ai fix)
=m
By the dominated convergence theorem, we infer that
n
?—†n
Therefore, the series
œ
› ay fu
i=l
Theorem 2.4 Let ® be a random mapping from X into Y admitting the series expansion of the form (4) Suppose that Elaz|? < C for all k, where p > 1 and q is the conjugate number of p (i.e
Trang 7D.H Thang, TM Cuong / VNU Journal of Science, Mathematics - Physics 25 (2009) 237-248 243
1/p+1/q =) For u € Lx to belong to D(®), a sufficient condition is
k Proof Put
req) = {El fueull 9}
Applying the Hoélder inequality, we get
IS) ă/xull < 3” Eleslllfeall
< SP {Flow}? {Bl full} 2
k=m
<Œ ` ry(g) — 0 as M,N — Od
k=m ϡ
Hence, the series > ` œ ƒ„ converges in LY so converges In Ly LÌ
k=1 Corrolary 2.5 Suppose that ® is a symmetric stochastically continuous Gaussian random mapping and if
STEM (few) 4/4 < 00
k
for some q > 1 then u € D(®)
3 When a random mapping can be extended to the entire space Le
Let ® be a random operator from X into Y and suppose that X is a separable Banach space
+
with the Shauder basis e = (e,)°°., and the conjugate basis e* = (e%)°°, By Proposition 1.2, ®
admits the series expansion
œ
Ox = So (a, e*) Pen
n=1
Theorem 3.1
i) If ® is a bounded random operator then D(®) = L¢& and ®u does not depend on the basis (En)
ii) Conversely, if D(®) = Lé then © must be a bounded random operator
Recall that (see|3|) a random operator ® is said to be bounded if there exists a real-valued
random variable k(w) such that for each x © X
|Paw)|| <k&)Ilal] as
Noting that the exceptional set may depend on x
Proof i) Since ® is bounded, by Theorem 3.1 [3] there exists a mapping
T:O— L(X,Y)
such that for each vw €
®z(œ) = T()# as
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As a consequence, there is a set D with P(D) = 1 such that for each w € D and for all n we have
Be, (w) = T(w)en
Hence, for each w € D
Therefore, the series `” ¡(œ,c5)®e„ converges a.s so converges in probability Consequently,
u € D(®) and Su(w) = T(w)(u(w)) does not depend on the basis e = (en)
ii) Put
®,u = So(u, c)®Đeu
i=l
Then ©,, is a linear continuous mapping from Li into LY By the assumption lim, ®,u = ®u for all u € Lx Hence, by the Banach-Steinhaus theorem ® is again a linear continuous mapping from
Lé into L In addition, we have
®(u) = » 1p, Pa;
i=1 for u = >> 1lg,a; where (F;,7 = 1, ,) is a partition of Q and x; € X By Theorem 5.3 [3] we
Theorem 3.2 Let © be a random operator admitting the series expansion of the form (4), where (œ„)
is a sequence of real-valued random variables and (f,,) is a sequence of continuous linear mappings from X into Y Then
i) If ® is bounded then D(®) = Le
ii) Conversely, if D(®) = Lx then ® must be bounded
Proof:i) Since ® is bounded, by Theorem 3.1 [3] there exists a mapping
T:O— L(X,Y)
such that for each vw €
®z(œ) = T()# as
For this reason, there is a set D with P(D) = 1 such that for each w € D and for all & we have
Bez, (w) = » An(w) frer = T(w)er
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As a consequence, for each w € D
» An(w) fru(w) = » a„() (3 — < u(w), eZ > ex)
= » œ„(() » < u(w), CC > fr€k
= » < u(w), ej > So nw) fre
= » < (0), cÿ > T'(0)€k
k
= 7œ) (= < ulw), eZ > 2)
k
=T)(())
ii) Put
®„u — » œ; ;u
i=l Then ©,, is a linear continuous mapping from L¢ into L} By the assumption lim, 6,u = ®u for all u € Lx Hence, by the Banach - Steinhaus theorem © is again a linear continuous mapping from
Lx into L) In addition, for wu = 32 Le, where (£;,i = 1, ,n) is a partition of Q and x; € X,
we have
®(u) = » œ ƒyu
k=1
= » Oe » ln, frei = » lz; » œ_ fkữi
n
i=l
By Theorem 5.3 [3] we conclude that ® is bounded L]
Theorem 3.3 Let X be a compact metric space and ® be a random mapping from X into Y admiting the series expansion of the form (4), where (a) is a sequence of real-valued symmetric independent random variables and (fn) is a sequence of continuous mappings from X into Y
i) If ® has a continuous modification then every u € LẺ belongs to D(®) ie D(®) = Lự
ii) The converse is not true i.e there exists a random mapping ® from X into Y admiting the series expansion of the form (4), where (a) is a sequence of real-valued symmetric independent
random variables and (f,) is a sequence of continuous mappings from X into Y such that D(®) = Lé£
but ® has not a continuous modification
Proof Let V = C(X,Y) be the set of all continuous mappings from X into Y It is known that V is a separable Banach space under the supremum norm
I/lly = sup |ƒ()|:
zcxX
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For each pair (x, y*) € X x Y* the mapping x @ y* : V — R given by
(@ OV )f) = (Fe), 9")
is clearly an element of V* Let Tl = {(a@ @ y*), (w, y*) © X x Y*} It is easy to check that [ is a
separating subset of V* Let U(a, w) be a continuous modification of © Define a mapping T : Q — V
by
T(w) =a U(a,w)
We show that T is measurable i.e T is a V-valued random variable Indeed, for each (2 @ y*) € T
the mapping w > (T(w), x @y*) = (Tw)a,y*) = ((œ,@), 0Ÿ) = (aw), y*) as is measurable
Since V is separable and T° is a separating subset of V*, the claims follows from the theorem 1.1 in
({4))
Note that for each w the mapping 2 +> a,(w) f,x is an element of V Hence a, f;, isa V-valued
rv Now for each (x @ y*) € T we have
(T(w), 7 @ y") = (T(w)x, y") = (Paw), y")
œ
= You @)fam,W”) — >) (Anlw) fr, et Oy") as
Since (Qf) is a sequence of V-valued symmetric independent r.v.’s in view of Ito - Nisio theorem
ϡ
we conelude that the serles 5” œ„(œ2) Í„ converges a.s to T in the norm of V This implies that there
n=1
exists a set D of probability one such that for each w € D,x € X, we have
ja = Yale \fn&-
Consequently, for u € Lx we have
= 3 Oin(w) fal => On(w)fru(w) Ww € D
n=1
i.e the series $7 | an(w) fru(w) converges a.s
i1)The following example shows that the converse is not true
Example Let X = [0;1], Y = R Consider the sequence (€,,) of real-valued independent rv.’s given
by
1
P (Sn — 0) = =l—-=
1
Then (€,,) are real-valued symmetric independent r.v.’s and
1
Let (a) be sequence of positive numbers defined by
1
Jnlogsn
and put a, = G, Then (a,,) are real-valued symmetric independent rv.’s and
an =
E(an) = 0,E|o,| =, Ela,,|? = a2