THE DISTRIBUTION OF

VECTOR-VALUED RADEMACHER SERIES

S.J. Dilworth and S.J. Montgomery-Smith

Abstract. Let X = ∑ ε_{n}x_{n} be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable ||X|| in terms of its mean and a certain quantity derived from the K-functional of interpolation theory. Several applications of the formula are given.

1. Results

In [6] the second-named author calculated the distribution of a scalar Rademacher series ∑ ε_{n}a_{n}. The principal result of the present paper extends the results of [6] to the case of a Rademacher series ∑ ε_{n}x_{n} with coefficients (x_{n}) belonging to an arbitrary Banach space E. Its proof relies on a deviation inequality for Rademacher series obtained by Talagrand [9]. A somewhat curious feature of the proof is that it appears to exploit in a non-trivial way (see Lemma 2) the platitude that every separable Banach space is isometric to a closed subspace of ℓ_{∞}. The principal result is applied to yield a precise form of the Kahane-Khintchine inequalities and to compute certain Orlicz norms for Rademacher series.

First we recall some notation and terminology from interpolation theory (see e.g. [1]). Let (E_{1}, ||.||_{1}) and (E_{2}, ||.||_{2}) be two Banach spaces which are continuously embedded into some larger topological vector space. For t > 0, the K-functional K(x, t; E_{1}, E_{2}) is the norm on E_{1} + E_{2} defined by

K(x, t; E_{1}, E_{2}) = inf{||x_{1}||_{1} + t||x_{2}||_{2} : x = x_{1} + x_{2}, x_{i} ∈ E_{i}}.

For a sequence (a_{n}) ∈ ℓ_{2}, we shall denote the K-functional K((a_{n}), t; ℓ_{1}, ℓ_{2}) by K_{1,2}((a_{n}), t) for short. For 1 ≤ p < ∞, a sequence (x_{n}) in a Banach space (E, ||.||) is said to be weakly-ℓ_{p} if the scalar sequence (x^{∗}(x_{n})) belongs to ℓ_{p} for every x^{∗} ∈ E^{∗}. The collection of all weakly-ℓ_{p} sequences is a Banach space, denoted ℓ_{p}^{w}(E), with the norm given by l_{p}^{w}((x_{n})) = sup_{||x^{∗}||≤1} ||(x^{∗}(x_{n}))||_{p} (where ||(a_{n})||_{p} = (∑ |a_{n}|^{p})^{1/p}). If (x_{n}) ∈ ℓ_{2}^{w}(E), we make the following definition:

K_{1,2}^{w}((x_{n}), t) = sup_{||x^{∗}||≤1} K_{1,2}((x^{∗}(x_{n})), t).

Observe that K_{1,2}^{w}((x_{n}), t) is a continuous increasing function of t. In fact, it is a Lipschitz function with Lipschitz constant at most ℓ_{2}^{w}((x_{n})).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 46B20; Secondary 60B11, 60G50.

Key words and phrases. Rademacher series, K-functional, Banach space.

The second author was supported in part by NSF DMS-9001796

Next we set up some function space notation. Let (Ω, Σ, P) be a probability space. A Rademacher (or Bernoulli) sequence (ε_{n}) is a sequence of independent identically distributed random variables such that P (ε_{n} = 1) = P (ε_{n} = −1) = \frac{1}{2}. For a random variable Y defined on Ω, its decreasing rearrangement, Y^{∗}, is the function on [0, 1] defined by Y^{∗}(t) = inf{s > 0 : P (|Y | > s) ≤ t}. For 0 < p < ∞, the weak-L_{p} norm of Y, denoted ||Y ||_{p,∞}, is given by ||Y ||_{p,∞} = sup_{0<t<1} t^{\frac{1}{p}} Y^{∗}(t). As usual, ||Y ||_{p} denotes (𝔼|Y |^{p})^{1/p}. Let Ψ be an Orlicz function on [0, ∞). The Orlicz norm, ||Y ||_{Ψ} , is given by ||Y ||_{Ψ} = inf{c > 0 : 𝔼Ψ(|Y |/c) ≤ 1}. We shall be particularly interested in the Orlicz functions Ψ_{q}(t) = e^{t^{q}} − 1 for 2 < q < ∞. The weak-ℓ_{p} norm of the scalar sequence (a_{n}) is defined by ||(a_{n})||_{p,∞} = sup n^{\frac{1}{p}} a_{n}^{∗}, where (a_{n}^{∗}) is the decreasing rearrangement of (|a_{n}|).

Finally, we shall write A ≈ B to mean that there is a constant C > 0 such that \frac{1}{C} A ≤ B ≤ CA. We shall try to indicate in each case whether the implied constant is absolute or whether it depends on some parameter, typically p ∈ [1, ∞), entering into the expressions for A and B.

Now we can state the principal result of the paper.

Main Theorem. Let X = ∑ ε_{n}x_{n} be an almost surely convergent Rademacher series in a Banach space E. Then, for t > 0, we have

(1)  P (||X|| > 2𝔼||X|| + 6K_{1,2}^{w}((x_{n}), t)) ≤ 4e^{−t^{2}/8},

and, for some absolute constant c, we have

(2)  P (||X|| > \frac{1}{2}𝔼||X|| + cK_{1,2}^{w}((x_{n}), t)) ≥ ce^{−t^{2}/c}.

The proof of the Main Theorem will be deferred until the end of the paper in order to proceed at once with the applications.

Corollary 1. Let X = ∑ ε_{n}x_{n} be an almost surely convergent Rademacher series in a Banach space. Then, for 0 < t ≤ \frac{1}{10}, we have

(3)  S^{∗}(t) ≈ 𝔼||X|| + K_{1,2}^{w}((x_{n}), \sqrt{log(1/t)}),

where S denotes the real random variable ||X||. The implied constant is absolute.

Proof. (1) and (2) give rise to the inequalities S^{∗}(4e^{−t^{2}/8}) ≤ 2𝔼||X||+6K_{1,2}^{w}((x_{n}), t) and S^{∗}(ce^{−t^{2}/c}) ≥ \frac{1}{2}𝔼||X|| + cK_{1,2}^{w}((x_{n}), t), respectively, whence (3) follows for all sufficiently small t by an appropriate change of variable. To see that the lower estimate implicit in (3) is valid in the whole range 0 < t < \frac{1}{10}, we recall from [2] that 𝔼||X||^{2} ≤ 9𝔼^{2}||X||. Hence, by the Paley-Zygmund inequality (see e.g. [4,p.8]), for 0 < λ < 1, we have

P (||X|| > λ𝔼||X||) ≥ (1 − λ)^{2} \frac{𝔼^{2}X}{𝔼X^{2}}

≥ \frac{1}{9}(1 − λ)^{2},

whence P (||X|| > (1 − \frac{3}{\sqrt{10}})𝔼||X||) ≥ \frac{1}{10}, which easily implies (3). □

In [4] Kahane proved that if P (||X|| > t) = α, where X is a Rademacher series in a Banach space, then P (||X|| > 2t) ≤ 4α^{2}. By iteration this implies P (||X|| > st) ≤ \frac{1}{4}(4α)^{s} for s = 2^{n}. According to our next corollary the exponent s in the latter result may be improved to be a certain multiple of s^{2}.

Corollary 2. Let X = ∑ ε_{n}x_{n} be an almost surely convergent Rademacher series in a Banach space. Then, for t > 0 and s ≥ 1, we have

P (||X|| > st) ≤ (\frac{1}{c_{1}} P (||X|| > t))^{c_{1}s^{2}}

for some absolute constant c_{1}.

Proof. By choosing c_{1} < c, where c is the constant which appears in (2), the result becomes trivial whenever P (||X|| > t) ≥ c. Hence we may assume that P (||X|| > t) < c. Choose α > 0 such that P (||X|| > t) = ce^{−α^{2}/c}. Then (2) gives t ≥ \frac{1}{2}𝔼||X|| + cK_{1,2}^{w}((x_{n}), α). Thus,

st ≥ \frac{s}{2}𝔼||X|| + scK_{1,2}^{w}((x_{n}), α)
≥ 2𝔼||X|| + K_{1,2}^{w}((x_{n}), csα)

provided s ≥ max(4, 1/c). Now (1) gives

P (||X|| > st) ≤ 4e^{−(csα)^{2}/8}

= 4 (\frac{1}{c}(ce^{−α^{2}/c}))^{c^{3}s^{2}/8}

= 4 (\frac{1}{c}(P (||X|| > t))^{c^{3}s^{2}/8} ,

which gives the result. □

Our next corollary, which is the vector-valued version of a recent result of Hitczenko [3], is a rather precise form of the Kahane-Khintchine inequalities.

Corollary 3. Let X = ∑ ε_{n}x_{n} be a Rademacher series in a Banach space. Then, for 1 ≤ p < ∞, we have

(𝔼||X||^{p})^{1/p} ≈ 𝔼||X|| + K_{1,2}^{w}((x_{n}), \sqrt{p}).

The implied constant is absolute.

Proof. We may assume that p ≥ 2. It follows from a result of Borell [2] that (𝔼||X||^{2p})^{1/2p} ≤ \sqrt{3}(𝔼||X||^{p})^{1/p}. Since \frac{1}{2}||Y ||_{p} ≤ ||Y ||_{2p,∞} ≤ ||Y ||_{2p} for every random variable Y (as is easily verified), it follows (letting S denote the random variable ||X||) that \frac{1}{2}||S||_{p} ≤ ||S||_{2p,∞} ≤ \sqrt{3}||S||_{p}. So it suffices to prove that ||S||_{p,∞} ≈ 𝔼S + K_{1,2}^{w}((x_{n}), \sqrt{p}) to obtain the desired conclusion. By Corollary 1, we have

||S||_{p,∞} ≈ 𝔼S + sup_{0<t<1} t^{1/p}K_{1,2}^{w}((x_{n}), \sqrt{log(1/t)})

= 𝔼S + sup_{0<t<1} \{t^{1/p} sup_{||x^{∗}||≤1} K_{1,2}((x^{∗}(x_{n})), \sqrt{log(1/t)})\}

= 𝔼S + sup_{||x^{∗}||≤1} \{ sup_{0<t<1} t^{1/p}K_{1,2}((x^{∗}(x_{n})), \sqrt{log(1/t)})\} .

To evaluate the expression in brackets we shall make use once more (see Corollary 2) of the elementary inequality K_{1,2}((a_{n}), s) ≤ max(1, s/t)K_{1,2}((a_{n}), t). Thus,

sup_{0<t≤e^{−p}} t^{1/p}K_{1,2}((x^{∗}(x_{n})), \sqrt{log(1/t)}) ≤ ( sup_{0<t≤e^{−p}} t^{1/p}\sqrt{\frac{log(1/t)}{p}}) K_{1,2}((x^{∗}(x_{n})), \sqrt{p})
= e^{−1}K_{1,2}((x^{∗}(x_{n})), \sqrt{p}).

Moreover,

sup_{e^{−p}<t<1} t^{1/p}K_{1,2}((x^{∗}(x_{n})), \sqrt{log(1/t)}) ≤ K_{1,2}((x^{∗}(x_{n})), \sqrt{p}).

Finally, we obtain

\frac{1}{e}K_{1,2}^{w}((x_{n}), \sqrt{p}) ≤ sup_{||x^{∗}||≤1} \{ sup_{0<t<1} K_{1,2}((x^{∗}(x_{n})), \sqrt{log(1/t)})\} ≤ K_{1,2}^{w}((x_{n}), \sqrt{p}),

which gives the desired result. □

Our final application is to the calculation of the Orlicz norms ||S||_{ψ_{q}} for 2 < q < ∞. The proof will use the scalar version of the result, which was obtained by Rodin and Semyonov [8] (see also [7]). (Recall that by a result of Kwapien, [5], ||S||_{ψ_{q}} ≈ 𝔼||X|| in the range 0 < q ≤ 2.)

Corollary 4. Let X = ∑ ε_{n}x_{n} be an almost surely convergent Rademacher series in a Banach space. Then, for 2 < q < ∞, we have

||S||_{ψ_{q}} ≈ 𝔼||X|| + sup_{||x^{∗}||≤1} ||(x^{∗}(x_{n}))||_{p,∞},

where \frac{1}{p} + \frac{1}{q} = 1 and S denotes ||X||. The implied constant depends only on q.

Proof. It is easily verified that ||f||_{ψ_{q}} ≈ sup_{0<t<1}(log(1/t))^{−1/q}f^{∗}(t). Hence, by Corollary 1, we have

||S||_{ψ_{q}} ≈ 𝔼||X|| + sup_{0<t<1}(log(1/t))^{−1/q}K_{1,2}^{w}((x_{n}), t)

≈ 𝔼||X|| + sup_{0<t<1} {(log(1/t))^{−1/q} sup_{||x^{∗}||≤1} K_{1,2}((x^{∗}(x_{n})), t)}

≈ 𝔼||X|| + sup_{||x^{∗}||≤1} { sup_{0<t<1}(log(1/t))^{−1/q}K_{1,2}((x^{∗}(x_{n})), t)}

≈ 𝔼||X|| + sup_{||x^{∗}||≤1} || ∑ ε_{n}x^{∗}(x_{n})||_{ψ_{q}}

≈ 𝔼||X|| + sup_{||x^{∗}||≤1} ||(x^{∗}(x_{n}))||_{p,∞},

where the last line follows from the result of Rodin and Semyonov. □

2. Proof of main result

The principal ingredient in the proof of the Main Theorem is the following deviation inequality of Talagrand [9].

Theorem A. Let X = ∑_{n=1}^{N} ε_{n}x_{n} be a finite Rademacher series in a Banach space and let M be a median of ||X||. Then, for t > 0, we have

P (||| ∑_{n=1}^{N} ε_{n}x_{n}|| − M| > t) ≤ 4e^{−t^{2}/8σ^{2}},

where σ = ℓ_{2}^{w}((x_{n})_{n=1}^{N}).

Lemma 1. Let X = ∑_{n=1}^{N} ε_{n}x_{n} be a finite Rademacher series in a Banach space E. Then, for t > 0, we have

P (||X|| > 2𝔼||X|| + 3K((x_{n})_{n=1}^{N}, t; ℓ_{1}^{w}(E), ℓ_{2}^{w}(E))) ≤ 4e^{−t^{2}/8}.

Proof. It follows from Theorem A that for all y_{1}, …, y_{N} in E, we have

(4)  P (|| ∑ ε_{n}y_{n}|| > 2𝔼|| ∑ ε_{n}y_{n}|| + tℓ_{2}^{w}((y_{n}))) ≤ 4e^{−t^{2}/8}.

On the other hand, since max || ∑ ε_{n}y_{n}|| = ℓ_{1}^{w}((y_{n})), we have the trivial estimate

(5)  P (|| ∑ ε_{n}y_{n}|| > ℓ_{1}^{w}((y_{n}))) = 0.

Let x_{n} = x_{n}^{(1)} + x_{n}^{(2)} for 1 ≤ n ≤ N, let X^{(1)} = ∑ ε_{n}x_{n}^{(1)}, and let X^{(2)} = ∑ ε_{n}x_{n}^{(2)}. Then

ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)})) + 2𝔼||X^{(2)}|| ≤ ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)})) + 2𝔼||X^{(1)}|| + 2𝔼||X||
≤ 3ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)})) + 2𝔼||X||
≤ 2𝔼||X|| + 3(ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)}))).

Let Q denote 2𝔼||X|| + 3(ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)}))). Then, by (4) and (5) and by the above inequality, we have

P (||X|| > Q) ≤ P (||X^{(1)}|| + ||X^{(2)}|| > ℓ_{1}^{w}((x_{n}^{(1)})) + tℓ_{2}^{w}((x_{n}^{(2)})) + 2𝔼||X^{(2)}||)

≤ P (||X^{(1)}|| > ℓ_{1}^{w}((x_{n}^{(1)}))) + P (||X^{(2)}|| > 2𝔼||X^{(2)}|| + tℓ_{2}^{w}((x_{n}^{(2)})))

< 0 + 4e^{−t^{2}/8}

The desired conclusion now follows from the definition of the K-functional. □

Lemma 2. Let x_{1}, …, x_{N} be elements of the Banach space ℓ_{∞}. Then

K((x_{n})_{n=1}^{N}, t; ℓ_{1}^{w}(ℓ_{∞}), ℓ_{2}^{w}(ℓ_{∞})) ≤ 2K_{1,2}^{w}((x_{n})_{n=1}^{N}, t).

Proof. For 1 ≤ n ≤ N, let x_{n} = (x_{n,j})_{j=1}^{∞} ∈ ℓ_{∞}. A simple convexity argument gives

||(x_{n})||_{ℓ_{p}^{w}(ℓ_{∞})} = sup_{1≤j≤∞} (∑_{n=1}^{N} |x_{n,j}|^{p})^{(1/p)} .

It follows that the mapping ϕ which associates an element (y_{n})_{n=1}^{∞} ∈ ℓ_{p}^{w}(ℓ_{∞}) with the element in ℓ_{∞}(ℓ_{p}) whose jth coordinate equals (y_{n,j})_{n=1}^{∞} is an isometry. Hence K((x_{n}), t; ℓ_{1}^{w}, ℓ_{2}^{w}) = K(ϕ((x_{n})), t; ℓ_{∞}(ℓ_{1}), ℓ_{∞}(ℓ_{2})). Let (y_{n})_{n=1}^{∞} ∈ ℓ_{∞}(ℓ_{2}) and let ε > 0. For each n there exists a splitting y_{n} = z_{n}^{(1)} + z_{n}^{(2)} such that

||(z_{n,j}^{(1)})_{j=1}^{∞}||_{1} + t||(z_{n,j}^{(2)})_{j=1}^{∞}||_{2} ≤ K_{1,2}((y_{n,j})_{j=1}^{∞}, t) + ε.

It follows that

||(z_{n}^{(1)})||_{ℓ_{∞}(ℓ_{1})} + t||(z_{n}^{(2)})||_{ℓ_{∞}(ℓ_{2})} = sup_{1≤n<∞} ||(z_{n,j}^{(1)})_{j=1}^{∞}||_{1} + t sup_{1≤n<∞} ||(z_{n,j}^{(2)})_{j=1}^{∞}||_{2}

≤ 2 sup_{1≤n<∞} K_{1,2}((y_{n,j})_{j=1}^{∞}, t) + 2ε

≤ 2K_{1,2}^{w}((y_{n}), t) + 2ε.

Since ε is arbitrary, the result now follows from the definition of the K-functional. □

Proof of Main Theorem. First we prove (1) for a finite Rademacher series ∑_{n=1}^{N} ε_{n}x_{n}. Since every separable Banach space embeds isometrically into ℓ_{∞}, we may assume that E is a closed subspace of ℓ_{∞}. Recall that K_{1,2}^{w}((x_{n}), t) was defined as sup_{||x^{∗}||≤1} K_{1,2}((x^{∗}(x_{n})), t). By the Hahn-Banach Theorem, the supremum is the same whether it is taken over elements of E^{∗} or over elements of ℓ_{∞}^{∗}. Hence (1) follows by combining Lemmas 1 and 2. The result for an infinite series follows from the result for ∑_{n=1}^{N} ε_{n}x_{n} by taking the limit as N → ∞. To prove (2), we use the result from [6] that there exists an absolute constant d such that P (∑ ε_{n}a_{n} > dK_{1,2}((a_{n}), t)) ≥ de^{−t^{2}/d} for every sequence (a_{n}) ∈ ℓ_{2}. Hence

P (|| ∑ ε_{n}x_{n}|| > \frac{d}{2}K_{1,2}^{w}((x_{n}), t)) ≥ inf_{||x^{∗}||≤1} P (|| ∑ ε_{n}x_{n}|| > dK_{1,2}((x^{∗}(x_{n})), t))

≥ inf_{||x^{∗}||≤1} P (∑ ε_{n}x^{∗}(x_{n}) > dK_{1,2}((x^{∗}(x_{n})), t))

≥ de^{−t^{2}/d}.

The Paley-Zygmund inequality now gives

P (||X|| > \frac{1}{2}𝔼||X|| + \frac{d}{6}K_{1,2}^{w}((x_{n}), t))

≥ min (P (||X|| > \frac{3}{4}𝔼||X||) , P (||X|| > \frac{d}{2}K_{1,2}^{w}((x_{n}), t)))

≥ min (\frac{1}{144}, de^{−t^{2}/d}.) . □

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Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A.

Department of Mathematics, University of Missouri, Columbia, Missouri 65211, U.S.A.