On the convergence of certain sums of independent random elements

On the convergence of certain sums of independent random elements

J.C. Ferrando

Abstract. In this note we investigate the relationship between the convergence of the sequence {Sn} of sums of independent random elements of the formSn=

Pn i=1εixi

(whereεitakes the values±1 with the same probability andxibelongs to a real Banach spaceX for each i ∈ N) and the existence of certain weakly unconditionally Cauchy subseries of^P∞_n=1xn.

Keywords: independent random elements, copy ofc0, Pettis integrable function, perfect measure space

Classification: 46B15, 46B09

1. Preliminaries

Our notation is standard ([1], [3], [4], [9]). Throughout this note ∆ will denote the Cantor space {−1,1}^N, Σ the σ-algebra of subsets of ∆ generated by the n-cylinders of ∆ for eachn∈N, andν the Borel probability⊗^∞_i=1ν_i on Σ, where ν_i : 2^{−1,1} → [0,1] is defined by ν_i(∅) = 0, ν_i({−1}) = ν_i({1}) = 1/2 and ν_i({−1,1}) = 1 for eachi ∈ N. In what follows X will be a real Banach space andL₀(ν, X) will stand for the (F)-space overRof all [classes of] ν-measurable X-valued functions equipped with the (F)-norm

kfk₀= Z

∆

kf(ε)k

1 +kf(ε)kdν(ε)

of the convergence in probability. We shall represent by P₁(ν, X) the (real) normed space consisting of all those [classes of] ν-measurable X-valued Pettis integrable functionsf defined on ∆ provided with the semivariation norm

kfk_P1(ν,X)= sup Z

∆

|x^∗f(ω)| dν(ω) :x^∗∈X^∗,kx^∗k ≤1

.

As it is well known, P1(ν, X) is not a Banach space whenever X is infinite- dimensional. In the sequel we shall shorten by wuC the sentence ‘weakly uncon- ditionally Cauchy’.

Supported by DGESIC PB97-0342 and Presidencia de la Generalitat Valenciana.

In [5] we have shown that if a series of independent random elements of the formP∞

n=1f_n, withf_n(ω) =ω_nx_nforω∈∆ and{x_n} ⊆X, convergesν-almost surely inX, thenP∞

n=1xnhas a subseries which is unconditionally convergent in norm. In this note we continue the investigation on the relationship among the convergence of the functional series P_∞

n=1fn under different topologies and the existence of certain wuC subseries ofP_∞

n=1xn.

2. On certain weakly unconditionally Cauchy subseries

Lemma 2.1. If there are a closed setA in∆ withν(A)>1/2and a nonempty setS ⊆X^∗ such that P_∞

i=1x^∗f_i(ω)converges forω∈Aand x^∗ ∈S, then there exists a subsequence{xni}of{xn}such thatP_∞

i=1|x^∗xni|<∞for eachx^∗ ∈S.

Proof: The following fact is contained in the proof of [8, Proposition] (see also [5, Claim]). We shall denote byC_i1i2...ik or C_i1i2...ik(ε) any rectangle of ∆ with fixed coordinatesi₁, i₂,. . .,i_k, i.e.,C_i1i2...ik(ε) ={ω∈∆ :ω_ij =ε_j,1≤j≤k}

for someε∈∆. On the other hand, given a strictly increasing sequenceQ={n_i: i∈ N} of positive integers, for each ω ∈∆ we shall design byω^′ (as in [8]) the element of ∆ defined byω_i^′=ω_i ifi∈Qandω^′_i=−ω_i ifi /∈Q.

Fact. Let A∈Σ. Ifν(A)>1/2, there is a strictly increasing sequence{n_i} of positive integers such thatA∩A^′∩Cn1n2...nk 6=∅ for each Cn1n2...nk and each k∈N.

By hypothesis there is a closed setAin ∆ withν(A)>1/2 such thatP∞

n=1ω_nx^∗x_n converges for ω∈ Aand x^∗ ∈S. According to the preceding fact there exists a strictly increasing sequenceQ={n_i}of positive integers such that, givenε∈∆, then A∩A^′∩C_n1n²...nk(ε)6=∅ for each k ∈N. Since{A∩A^′∩C_n1n²...nk(ε) : k ∈ N} is a decreasing sequence of nonempty closed sets in the compact space

∆, there is a pointζ (which depends ofε) in ∆ which belongs to the intersection T_∞

k=1A∩A^′∩Cn1n2...nk(ε). Hence, for eachx^∗∈Sand each pair (r, s) of positive integers, withs > r, one has

s

X

i=r+1

ε_ix^∗x_ni

=

s

X

i=r+1

ζ_nix^∗x_ni

≤ 1 2

ns

X

i=nr+1

x^∗f_i(ζ)

+

ns

X

i=nr+1

x^∗f_i ζ^′

! .

Since ζ, ζ^′ ∈ A and x^∗ ∈ S, both series P_∞

i=1x^∗f_i(ζ) and P_∞

i=1x^∗f_i(ζ^′) are convergent. So, for a givenǫ >0 there is ak∈Nsuch that

Pns

i=nr+1x^∗f_i(ζ) < ǫ and

Pns

i=nr+1x^∗f_i(ζ^′)

< ǫfors > r≥k, which implies that Ps

i=r+1ε_ix^∗x_ni

≤ǫ for s > r ≥ k. Hence the numerical seriesP∞

i=1ε_ix^∗x_ni converges. Given that this is true for eachε∈∆, it follows thatP_∞

i=1|x^∗x_ni|<∞for eachx^∗∈S

and we are done.

Theorem 2.2. Assume thatkx_nk= 1for eachn∈NandX has a dual unit ball with countably many extreme points. If

sup

n∈N

Z

∆

|x^∗Sn(ω)|dν(ω)<∞ for eachx^∗∈ExtB_X^∗, thenX contains a copy ofc0.

Proof: By hypothesis, for eachx^∗ ∈ExtB_X^∗ there existsC_x^∗>0 such that

(2.1) sup

n∈N

Z

∆

n

X

i=1

x^∗f_i(ω)

dν(ω)< C_x^∗.

Hence, givenx^∗∈ExtB_X^∗, as a consequence of (2.1) and of Khinchine’s inequal- ities there exists aK >0 such that

(2.2)

( _n X

i=1

σ²(x^∗f_i) )1/2

= ( _n

X

i=1

(x^∗x_i)² )1/2

≤K Z

∆

n

X

i=1

x^∗f_i(ω)

dν(ω)< KC_x^∗ for each n ∈ N. Considering that the sequence {x^∗f_i} consists of independent random variables such that

E(x^∗f_i) = Z

∆

x^∗f_i(ω)dν(ω) = 0

for eachi∈N, according to [7, Section 46, Theorem B] equation (2.2) ensures that P∞

i=1x^∗f_i(ω) converges almost surely forω∈∆. Since ExtB_X^∗ is countable, it follows that there exists aν-null setN such thatP∞

i=1x^∗fi(ω) converges for each ω∈∆−N and eachx^∗ ∈ExtB_X^∗. So, using inner regularity we may choose a closed setAwithA⊆∆−N andν(A)>1/2 such thatP∞

i=1x^∗f_i(ω) converges for eachω ∈Aand eachx^∗∈ExtB_X^∗. On the basis of Lemma 2.1, this implies that there exists a subsequence{x_ni} of {x_n} such that P∞

i=1|x^∗x_ni| <∞ for eachx^∗∈ExtB_X^∗. SinceP∞

n=1x_ndiverges, Elton’s theorem guarantees thatX

contains a copy ofc₀.

Proposition 2.3. If the sums {S_n} are bounded inside of a complete linear subspaceLof P₁(ν, X), thenP∞

n=1x_nhas a wuC subseries.

Proof: Since{S_n}is bounded inside of a complete linear subspaceLofP₁(ν, X) and given that the canonical inclusion map fromP₁(ν, X) intoL₀(ν, X) has closed graph ([6, Lemma 4]), then Banach-Schauder’s theorem guarantees that{Sn} is stochastically bounded. So, according to [9, Section 5.2.3, Theorem 2.2] the sums {S_n} are bounded almost surely, i.e. ν({ω ∈ ∆ : sup_n∈N

P_n

i=1f_i(ω)

= ∞})

= 0. Hence Kwapie´n’s theorem [8, Proposition] assures the existence of a wuC subseries ofP∞

n=1xn.

Corollary 2.4. Assume that{fn} is a basic sequence inP₁\(ν, X)equivalent to the unit vector basis of c₀. If [fn] is contained inP₁(ν, X), then there exists a subsequence{fni} such that[fni] is isomorphic to a complemented copy of c₀. Proof: Since the seriesP∞

i=1f_i is wuC inP₁(ν, X), there isK >0 such that sup

n∈N

n

X

i=1

ξ_if_{i P}

1(ν,X)

< Kkξk_∞

for each ξ ∈ ℓ∞. Hence the sums {Sn} are bounded in the complete linear subspace [f_i] of P₁(ν, X) and Proposition 2.3 guarantees that P_∞

n=1xn has a wuC subseries. Sincekxnk =kfnk_P1(ν,X) for eachn∈N, then inf_n∈Nkxnk>0 and the classic Bessaga-Pe lczy´nski allows us to conclude that {x_n} contains a subsequence {x_ni} equivalent to the unit vector basis of c₀. Therefore, there exists a bounded sequence{y_i^∗} in X^∗ such that y^∗_ix_nj =δ_ij for each i, j ∈ N. Assuming without loss of generality thaty_i∈B_X^∗, setg_i(ε) =ε_iy^∗_i for eachi∈N and define

hg_i, fi= Z

∆

ε_iy^∗_if(ε)dν(ε)

for eachf ∈P₁(ν, X). So we havehg_i, fnji=δ_ij for eachi, j∈N. On the other hand, denoting byCn the rectangle of ∆ formed by all thoseε∈∆ withεn= 1 and noting thatν(E∩Cn)→ν(E)/2 for allE∈Σ, it follows that

E_Cn(ϕ) = 1 ν(C_n)

Z

Cn

ϕ dν→ Z

∆

ϕ dν=E(ϕ)

for each ν-simple function ϕ : ∆ → R. This implies that E_Cn(ϕ) → E(ϕ) for each ϕ ∈ L1(ν), which leads to R

∆ε_iϕ(ε)dν → 0 for each ϕ ∈ L1(ν). Since, in addition, (∆,Σ, ν) is a perfect measure space, it can be shown as in [2] that hg_i, fi →0 for eachf ∈P₁(ν, X). Consequently the mapP :P₁(ν, X)→P₁(ν, X) defined by

P f =

∞

X

i=1

hg_i, fifni

is a bounded linear projection operator from the barreled space P₁(ν, X)

onto [fni].

Proposition 2.5. If there exists a complete linear subspaceLin P1(ν, X)such that{f_i} ⊆LandP∞

i=1f_iconverges inP₁(ν, X)to some separably-valuedf ∈L, then there exists a subseries of P∞

i=1x_iwhich is unconditionally convergent inX.

Proof: Given thatP∞

i=1f_i=finP₁(ν, X) andLis complete, thenP∞ i=1f_i=f inL. Then, using the fact that the inclusion map fromP₁(ν, X) intoL₀(ν, X) has closed graph together with the Banach-Schauder theorem, we get thatP∞

i=1f_i = f in probability. Since the range off is separable in norm, then [9, Section 5.2.3, Theorem 2.1] guarantees that the seriesP_∞

i=1f_i(ω) converges inX tof(ω) almost surely forω∈∆. Hence [5, Theorem 2.1] establishes the existence of a subseries ofP_∞

n=1xn which is unconditionally convergent inX.

Question. We do not know whether the statement of Theorem2.2is true without the assumption thatB_X^∗ has countable many extreme points.

References

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[3] Diestel J.,Sequences and series in Banach spaces, GTM92, Springer-Verlag, New York- Berlin-Heidelberg-Tokyo, 1984.

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[5] Ferrando J.C.,On a theorem of Kwapie´n, Quaestiones Math.24(2001), 51–54.

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[9] Vakhania N.N., Tarieladze V.I., Chobanian S.A., Probability Distributions on Banach Spaces, D. Reidel Publishing Company, Dordrecht, 1987.

Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez, E-03202 Elche (Alicante), Spain

E-mail: [email protected]

(Received June 27, 2001)