Independent products in real L1: asymptotic midpoint convexity without AUC renormings

1 Introduction

A norm may force the average distance to two opposite points to grow in every sufficiently large direction, yet have no equivalent norm with the corresponding one-sided growth. We construct two explicit real L^1 spaces that exhibit this distinction. In both spaces, multiplying along a tree has two effects: fresh variables force midpoint growth, while path products of L^1 norm one prevent asymptotically uniformly convex renormings.

For a norm N on an infinite-dimensional real Banach space X, let \cof(X) denote the closed linear subspaces of finite codimension. For t>0, define the averaged asymptotic midpoint modulus and the usual one-sided asymptotic modulus [6, Section 2.1] by

The norm is asymptotically uniformly convex (AUC) if \overline\delta_N(t)>0 for every t>0. The subspace F may depend on the center x and on t; it must work for all the indicated directions. The standard asymptotic midpoint modulus uses unit directions N(y)=1 and replaces the displayed average by the maximum of the two norms. A norm is asymptotically midpoint uniformly convex (AMUC) when this maximum modulus is positive for every t>0 [5, Definition 2.2]. All numerical midpoint estimates below concern the displayed average. They imply the same lower bounds for the maximum, and hence AMUC, because the maximum dominates the average.

Let \mathcal T=\N^{<\omega} be the tree of finite sequences of positive integers, with root \varnothing. For a nonroot vertex v, write v^- for its parent. Fix a positive nonconstant real random variable W such that

On a countable product probability space take independent copies W_v, indexed by the nonroot vertices, and define

We use the inherited real L^1 norm, denoted by \norm{\cdot}. Write X_{\mathrm{exp}} for the law with density e^{-w} on (0,\infty), and X_{\mathrm{sq}} for the law of G^2, where G is a standard real Gaussian. For K>0 set

The three estimates come from different scalar arguments, whose parameter-dependent forms are proved below. None of the displayed constants is asserted to be optimal. The general multiplier hypothesis is used for the renorming obstruction; the positive midpoint estimates are proved for the two specified laws.

Context and method

Dilworth, Kutzarova, Randrianarivony, Revalski, and Zhivkov introduced AMUC and distinguished it from AUC for a given norm: they constructed an equivalent norm on \ell_2 that is AMUC but is not AUC [5, Theorem 2.4]. The renorming question is different. They proved that an AMUC space with an unconditional Schauder basis admits an equivalent AUC norm, and asked whether this remains true for every AMUC space [5, Section 5 and Proposition 5.1].

Baudier answered that question negatively. He proved that a closed subspace of L^1 whose unit ball is relatively compact in measure is AMUC, and applied this result to the Kadets–Werner space, which admits no equivalent AUC norm [3, Theorem 1 and Corollary 1]. His example also satisfies a linear lower bound for the maximum midpoint modulus [3, final paragraph]. Theorem 1.1 instead gives explicit spaces obtained from one repeated multiplier law, with direct proofs of quantitative bounds for the averaged modulus and of the renorming obstruction.

The construction has a useful relation to the multiplication step in the Bourgain–Rosenthal construction [4] as developed by Kadets and Werner [7, Lemmas 2.2 and 2.4]. Their enlargement multiplies selected old vectors by fresh nonnegative variables, chosen anew at successive enlargement stages. Here one fixed positive law with finite second moment is repeated at every vertex. Our arguments concern this closed product span directly; they make no assertion of reflexivity, measure precompactness of its unit ball, or the Daugavet property.

The main difficulty in the midpoint estimate is uniformity over all directions in a subspace of finite codimension. Conditioning on the variables in a finite predecessor-closed subtree T produces a finite-rank projection on X_W. In a finite linear combination of products in its kernel, the terms beyond T can be grouped by their first vertices outside T. Each group factors into the multiplier at its first exit and a coefficient depending only on old coordinates and coordinates strictly below that exit. Fixing these coefficients leaves the exit multipliers independent. An independent-copy and random-sign argument controls the vector’s L^1 norm by the expected Euclidean size of its coefficient list. The scalar probability estimates then force a definite midpoint gain for every such list, even in the presence of an arbitrary shift.

The three scalar mechanisms are elementary but distinct. For exponential multipliers, a difference of independent copies is a Gaussian scale mixture, which gives positive expected excess past every fixed threshold. For Gaussian squares, an orthogonal change of variables turns the difference into a conditionally Gaussian variable. One proof transfers its tail probability back to the original shifted variable by a union bound. The other uses even convexity of the midpoint increment directly on the difference.

Section 2 establishes the finite-rank projections, the two coefficient estimates, and the common passage to completed tails and arbitrary centers. Section 3 treats the exponential law. Section 4 gives the two Gaussian routes. Section 5 proves the renorming obstruction locally: path products have norm one, but their child increments are weakly null with fixed positive norm. An AUC renorming would force geometric growth along a selected branch, contradicting boundedness.

2 Finite conditional kernels and fresh multipliers

We first construct the subspaces of finite codimension needed for the midpoint estimates. The point is that conditioning on finitely many coordinates has finite rank on the actual closed span X_W.

Independence and positivity give

As a closed subspace of L^1, X_W is a Banach space. The first-generation differences P_{(n)}-1=W_{(n)}-1 are nonzero and pairwise orthogonal in L^2, so every finite subset is linearly independent, also as elements of L^1. Thus X_W is infinite dimensional. The first-generation products themselves are also linearly independent: taking expectations in a relation \sum_n a_nW_{(n)}=0 gives \sum_n a_n=0, reducing it to the corresponding relation among the orthogonal differences. Rational finite combinations of the countable family of products are dense, so X_W is separable.

Let T\subset\mathcal T be finite, contain the root, and contain every predecessor of each of its vertices. Write \mathcal G_T for the \sigma-field generated by the coordinates W_v with v\in T\setminus\{\varnothing\}. For v\in\mathcal T, let v_T be its last ancestor in T. Conditional expectation satisfies

Consequently Q_T restricts to a contractive projection of X_W onto the finite-dimensional space

This assertion on the completion follows from the continuity of conditional expectation and the closedness of E_T. Applying I-Q_T to finite-product approximations shows that finite-product combinations in F_T are dense in F_T.

The same space and tails can also be described using increments. Put u_v=P_v-P_{v^-} for v\ne\varnothing. Telescoping along paths shows that \{1,u_v:v\ne\varnothing\} and \{P_v:v\in\mathcal T\} have the same algebraic span. Formula (5) gives Q_Tu_v=u_v for v\in T and Q_Tu_v=0 for v\notin T. Applying I-Q_T to the algebraic span and then passing to its closure proves

For t>0 and x,y\in X_W, define the midpoint increment

The real scalar identity (|a+b|+|a-b|)/2=\max\{|a|,|b|\} gives

For u\in E_T and y\in F_T, we seek a lower bound on this excess in terms of \norm{y} and \norm{u}, uniform over T. Since u is \mathcal G_T-measurable, we first separate the fresh multipliers in y from the coordinates determining u.

Now write a finite-product combination in the kernel as y=\sum_{v\in S}\alpha_vP_v\in F_T, where S\subset\mathcal T is finite. Write w\preceq v when w is an ancestor of v, including v itself, and use w\prec v for strict ancestry. Let I be the finite set of first vertices outside T on paths to vertices of S\setminus T. Factoring out the multiplier at each such exit gives

where an empty product is 1. Thus d is \mathcal G_T-measurable, and B_i depends only on old coordinates and on strict descendants of i, not on W_i. Put m_i=\E(B_i\mid\mathcal G_T). The factorization and the condition Q_Ty=0 yield

Distinct first-exit vertices have disjoint descendant subtrees. Hence, conditionally on \mathcal G_T, the B_i are independent, and their entire vector is independent of the first-exit variables W_i. In particular, conditioning further on all B_i leaves the W_i independent with their original law. Write

We will condition on \mathcal G_T when copying entire coefficients, and on \mathcal H_T when resampling only first-exit multipliers. Figure 1 illustrates this distinction. All quantities used here are integrable; in fact the finite products belong to L^2.

The next two estimates measure the size of a tail vector by the expected Euclidean size of its coefficient list. Their upper bounds on \norm{y} give lower bounds on \E\sigma, which the scalar estimates in Sections 3 and 4 will convert into midpoint gains. They use independent-copy symmetrization followed by averaging over independent signs; this is the standard symmetrization method, here applied conditionally on the old coordinates [2, proof of Theorem 8, p. 468]. Both constants will be useful: they come from different ways of centering the sum.

The preceding construction applies to finite-product tails. To pass midpoint estimates to their closure, the triangle inequality gives

Thus all tail estimates below pass to the completed tail subspace before we impose the condition \norm{y}\ge1.

The lemma is applied to norm inequalities, not to an infinite first-exit expansion. Thus no representation or extra integrability of a general completed tail is needed. It also fixes the order of choices: t and x determine T before the direction y is allowed to vary.

3 Exponential products: a uniform excess estimate

The exponential law supplies positive expected excess beyond every multiple of the coefficient size. A Gaussian scale mixture makes that excess uniform over all real coefficient lists and all shifts. Take W=Z to have the exponential distribution with density e^{-z} on (0,\infty), and write X_{\mathrm{exp}}=X_Z. Here m_2=2, so Lemma 2.1 gives \norm{y}\le3\E\sigma. Let G denote a standard real Gaussian and put, for K>0,

The Laplace distribution as a Gaussian scale mixture belongs to the theory of normal scale mixtures [1]; the characteristic-function calculation below proves the exact identity needed here. We prove the uniform conditional bound

After conditioning as indicated, all coefficients, d, and \sigma are fixed. Resample only the exponential multipliers to obtain y' with the same conditional law. The triangle inequality and the positive-part inequality give

The difference of two independent standard exponentials has characteristic function (1+q^2)^{-1} at q\in\R. This also equals \E\exp(-q^2U) for a standard exponential U, so that difference has the law of \sqrt{2U}\,G, with U and G independent. Using independent such representations for the different multipliers, if \sigma>0 the law of (y-y')/\sigma is a centered Gaussian mixture with variance

Since \E[V\mathbf1_{\{V\ge1\}}]\ge1, Cauchy–Schwarz gives \PP(V\ge1)\ge1/8. On that event the Gaussian excess above 2K is at least \E(|G|-2K)_+. Hence

which proves (13). For \sigma=0 the asserted lower bound is zero, so the conclusion holds without normalization.

4 Gaussian squares: two scalar mechanisms

Now take W=G^2, where G is standard Gaussian, and put X_{\mathrm{sq}}=X_{G^2}. Here m_2=3. In particular, the product and jump descriptions above give the same closed subspace with the same norm. We first establish the Gaussian difference calculation that both estimates will use, while keeping their subsequent probability and convexity steps separate.

For fixed real coefficients a_1,\ldots,a_k and a fixed b\in\R, put

where the G_i are independent standard Gaussians. Let H' be an independent copy with the same coefficients and shift. The Gaussian polarization identity used in decoupling [8, proof of Lemma 3.2] is especially simple here. The orthogonal change of variables A_i=(G_i+G_i')/\sqrt2, C_i=(G_i-G_i')/\sqrt2 shows that

where all A_i,C_i are independent standard Gaussians. Conditionally on the A_i, this is Gaussian with variance 4V, where

This is the elementary second-moment step usually called the Paley–Zygmund inequality [9, 10]. We include the argument: for s>0, the inequality \E[V\mathbf1_{\{V\ge s^2/2\}}]\ge s^2/2 and Cauchy–Schwarz yield \PP(V\ge s^2/2)\ge1/12. On this event the conditional standard deviation of H-H' is at least \sqrt2s. Consequently, for every K>0,

The event in (15) has probability one when s=0, so the same statement remains valid in that case.

The two proofs control different scalar quantities. Proposition 4.1 first estimates a shifted original variable and pays a factor of two in the union bound. Proposition 4.2 estimates a difference and uses even convexity directly in the midpoint increment, preserving the denominator 12. Their first-moment constants are respectively 2+\sqrt2 and 2\sqrt3. Neither proof requires the original shifted variable to be symmetric.

5 Bounded paths exclude AUC renormings

The midpoint arguments depend on the multiplier distribution. Here we use only positivity, a nonconstant mean-one law, and a finite second moment to rule out AUC renormings. At every vertex the child increments are weakly null and have the same positive norm. An AUC norm would therefore grow by a fixed factor at each step of a suitably chosen branch, whereas all path products have inherited norm one.

For each fixed vertex v and its children v^\frown n, put

Independence and (4) give

For fixed v the sequence (D_{v,n})_n is pairwise orthogonal in L^2 and has constant finite L^2 norm. Bessel’s inequality therefore makes it weakly null in L^2. The inclusion L^2\longrightarrow L^1 is continuous on this probability space. Moreover, each continuous linear functional on X_W extends to L^1 by Hahn–Banach and hence restricts to a continuous functional on L^2. Thus (D_{v,n})_n is weakly null in X_W itself.

Suppose N is equivalent to the inherited norm, with a\norm{z}\le N(z)\le b\norm{z} for 0<a\le b<\infty. We will show the precise obstruction

The usual one-sided modulus is nonnegative: for each unit center x, a norming functional gives a finite-codimensional kernel on which N(x+ty)\ge1. It remains to rule out positivity at the displayed scale.

Assume, to the contrary, that

At a vertex v, we have a\le N(P_v)\le b. The modulus at the unit center P_v/N(P_v) supplies a closed finite-codimensional subspace F such that

Indeed, its definition first gives the estimate at equality of the radius. For N(x)=N(w)=1, with w\in F and r\ge t_0, convexity yields

so the estimate extends to all larger radii.

Equivalent norms induce the same weak topology. The quotient images of the weakly null sequence D_{v,n} in the finite-dimensional space X_W/F therefore tend to zero in quotient norm. We may choose z_n\in F with N(D_{v,n}-z_n)\to0. Eventually

Choose such an n with approximation error at most \gamma N(P_v)/2. Then

The same \gamma works at every vertex, although F and n may change. Repeatedly selecting a child gives a depth-k product with norm at least a(1+\delta/4)^k, contrary to the uniform upper bound b. This proves (23) and the renorming assertion of Theorem 1.1.

Finally 0<\rho<2: nonconstancy gives positivity, and \E|W-1|=2-2\E\min\{W,1\}<2 because W>0 almost surely. Thus 0<t_0<1, so positivity of the one-sided modulus on (0,1) alone would already contradict the tree. For exponential multipliers \rho=2/e; indeed \E|Z-1|=2\E(Z-1)_+=2\int_1^\infty e^{-s}\,ds. The weak-null calculation requires only a finite L^2 bound at each fixed parent, not a bound uniform over the depth.