Unit 4, continued: Conditional laws in the square array and a nonzero correlation

This part studies B §§6.4–6.5, pp.21–24, principally Lemmas 6.4–6.6 and Proposition 6.7. Retain the preceding construction: the five conditional expectations on $Y$ are stable, the nonzero correlation has the maximal number $r$ of auxiliary positions, $b\notin A$, and $Z=M_b(Y)$ has been regrouped into a three-group system.

The preceding selected-law, group-preservation, and relative structure results remain hypotheses. We show how, once they hold, the square-array argument produces one more auxiliary position and contradicts maximality.

1. Rows, columns, parameters, and the common corner

A state of $Z$ is a quadruple of $Y$-states with measure $m_Z=\lambda_Y$. Taking the selected law $\lambda_Z$ produces states represented by a $4\times4$ square array:

Define the old and new parameters by

Each $E_i$ contains all old parameters $(P_{ij})_j$ of its row, by B Lemma 3.3.

For column $b$, write

Column $b$ is obtained through the factor $Z\to Y$, $(y_j)_j\mapsto y_b$, and has law $\lambda_Y$. Row $b$ also has law $\lambda_Y$ because every single $Z$-state has marginal $m_Z=\lambda_Y$. This identifies two marginals; it does not assert invariance of the full array under transposition.

The functions from the preceding part are

and

We know that $g(P,y_b)$ is nonzero in $L^2(\lambda_Y)$. The crucial claim is

The first $q$ is evaluated along column $b$, the second along row $b$. Neither includes the separate corner function $h_b$.

2. Stability determines the full column law

Source: B Lemma 6.4, pp.21–22, the first paragraph of its proof.

Apply stability (6.3) to the extension $\theta:Z\to Y$, $\theta(z)=y_b$. On selected-law spaces, $\theta^4$ is exactly column $b$. The old parameter array pulls back to $p$, and the new type $\mathcal D_0(Z)$ is $\sigma(E)$.

Thus, for every bounded measurable $f:Y^4\to\mathbb R$,

On the standard spaces, apply this identity first to a countable generating algebra of tests, then identify the conditional probability. This gives

The law of the full four-state column does not follow directly from conditional independence for at most three states. The stable extension is genuinely used here.

Take marginal $b$ of this conditional law and apply the one-position case of B Proposition 3.2. Then

Since $w$ is a function of $E_b$, conditioning again on the smaller information $E_b$ preserves the conclusion. Taking the marginal of row $b$ shows that, in one $Z$-state, conditional on $e=\omega_Z$, its old state $b$ has law $m_{Y,\kappa(e)}$, where $\kappa(e)$ is that state's old parameter.

3. A row function conditional on the full column

Source: B p.22, equations (6.7), (6.9)–(6.10).

3.1 Temporarily omit one row

Fix $l\ne b$. Conditional on $E$, after omitting row $l$ the three rows $z_i$ are independent, with respective conditional measures $m_{Z,E_i}$. This is the proper-subset conclusion of Proposition 3.2 applied to $Z$.

Under this conditional product law, $q(z_b)$ uses only row $b$. Condition further on the column coordinates $y_i=\theta(z_i)$, $i\ne l$, in those rows. Observations in the other two rows do not change the conditional average in row $b$. Hence

This can be checked on product tests: integrate against $\prod_{i\ne l}H_i(\theta z_i)$, first work in row $b$ and use the definition of $u(e,y)$ to replace $q$, and multiply the integrals for the other rows. A monotone-class argument extends the identity.

3.2 Restoring the omitted column state to the conditioning

Write

Then $\mathcal B\subseteq\mathcal C$. Stability for type $l$ says that, for every bounded state function $F$,

where the right side is pulled back along column $b$ and is therefore $\mathcal B$-measurable.

It follows that, conditional on $\mathcal B$, the state $y_l$ is independent of the extra information in $\mathcal C$. Indeed, for bounded $G$ measurable with respect to $\mathcal C$,

Take $G=q(z_b)$, which belongs to $\mathcal C$ because $b\ne l$. Product tests and a monotone-class argument show that adding $y_l$ to the conditioning leaves its conditional expectation unchanged. Thus (ThreeRows) becomes

No joint independence of all four complete rows has been asserted. Proper-subset independence is used for three rows; stability handles the omitted column state.

4. The two-line correlation becomes a fiber inner product

Source: B (6.11), p.22.

First condition on the full column and $E$. By (Row),

By (Column), conditional on $E$ the full column has the old law $\lambda_Y$ conditioned on the parameter $p$. Within that old law, condition further on the corner state. The definition of $g$ and (Corner) yield

This uses an identity of conditional probability kernels. Choose a regular conditional probability version of $\lambda^p=\lambda_Y(\cdot\mid P=p)$. For a countable Borel generating algebra of $Y$ and each of its sets $B$, the definition of $g$ and the one-position conditional law give

simultaneously for almost every $p$. Both sides define bounded finite signed measures. Uniqueness of measures extends the identity to every bounded Borel test in $y$. In particular, use the test $y\mapsto u(E_b,y)$ without choosing a new exceptional set for each parameter. Equation (Column) is likewise a kernel identity, permitting the integrand to depend on the given $E$. The column parameter $p$ has the old parameter distribution as its marginal, so its exceptional set is still null in the square array.

This integral has no immediate sign. Equal row and column marginals alone do not make it a square. We still need the relation between $p$ and $E_b$ conditional on the common parameter $w$.

5. Parameter independence from a change of time origin

Source: B Lemma 6.5, p.23. The change of variables below takes place on the original flow $Z$, without assuming a factor model strictly equivariant at every real time.

5.1 The old corner parameter is a function of the new parameter

The factor $\theta:Z\to Y$ has speed one. Write its induced distal factor map as $\kappa:W_Z\to W_Y$. Modulo null sets,

Take bounded $F:W_Z\to\mathbb R$ and $H_i:W_Y\to\mathbb R$. Proposition 3.2(2) determines the full new parameter law through ordinary continuous averages. Therefore

equals the Cesàro limit of the following fixed-time integral:

5.2 Changing variables on the original state space

Fix $t$ and make the measure-preserving substitution $z'=S_Z^{bt}z$. The expression in (Time) becomes

Except for $F$, every factor is $\kappa\circ\omega_Z$-measurable: $\theta$ commutes with the original flow, and each fixed time preserves $W_Y$. These are fixed-time statements about function classes modulo null sets, sufficient for the integral.

Replace $F(\omega_Z(z'))$ by its conditional expectation onto $\kappa\circ\omega_Z$, written $\overline F(\kappa(\omega_Z(z')))$. Reverse the substitution. The spatial integrals before and after replacement already agree at each fixed $t$. Factor equivariance is needed only modulo null sets at that fixed time. The integrands built from the original flow and Borel functions are jointly measurable and bounded, so they may be integrated over $t\in[0,L]$. The averages are therefore exactly equal for every deterministic $L>0$, with no endpoint error.

Neither the time variable nor the integration interval has changed. This uses the deterministic-length parameter law of Proposition 3.2(2); the point-dependent selector $N_a(z)$ no longer occurs. For $i<b$, the given invertible flow defines the negative times. Taking the deterministic-length limit gives

The one-coordinate marginal of $E_b$ is $\nu_Z$. Since $w=\kappa(E_b)$, we have $\overline F(w)=\mathbb E[F(E_b)\mid w]$. Parameter rectangle tests on $p$ generate its $\sigma$-algebra, so

Since $w$ is a coordinate of $p$, this is equivalent to

The conclusion follows from the change of variables and deterministic-length parameter law, not from transpose symmetry.

6. Compatibility of two conditional averages on one state

Source: the beginning of the proof of B Proposition 6.7 and equation (6.12), pp.23–24.

6.1 Two parameter laws and one fiber Hilbert space

In one $Z$-state, let $R(e)$ be the complete old $W_Y$-parameter array. It is a function of $e=\omega_Z$. Conditional on its old parameter at position $b$, denoted $w$, write

Since one $Z$-state has measure $\lambda_Y$, we have $R_*\beta_w=\alpha_w$. The column parameter $p$ and row parameter $R(E_b)$ both, conditional on the common corner parameter $w$, have law $\alpha_w$. Equation (PI) gives

For almost every fixed $w$, in the real Hilbert space

regard $g(p,\cdot)$ and $u(e,\cdot)$ as random vectors.

6.2 Measurability as Hilbert-space-valued functions

The standard Borel space $Y$ has a countable generating algebra. Its rational simple functions have dense span in every $L^2(m_{Y,w})$, so each $\mathcal H_w$ is separable.

Fix bounded Borel versions of $g,u$. Choose increasing finite partitions from the generating algebra and condition $g(p,\cdot)$ and $u(e,\cdot)$ on them. This gives finite-dimensional approximations whose coefficients, obtained by integration against the fixed fiber measure, are measurable in $p$ or $e$. Set coefficients on null atoms to zero. The representatives and their conditional averages may all retain the bound $\|q\|_\infty$. As the partitions generate the Borel $\sigma$-algebra, these approximations converge in $L^2(m_{Y,w})$ to the original functions.

Both maps are therefore strongly measurable Hilbert-space-valued functions. Uniform boundedness of $q$ bounds their norms, so they are square-integrable. The scalar conditional identities may first be established for countably many generating tests; Fubini then gives one common $w$-full-measure set.

6.3 The corner state's conditional law is essential

The one-state consequence of (Corner) says that, given $e$, the corner state $y$ has conditional law exactly $m_{Y,w}$. Thus, conditional on $w$, the joint law of $(e,y)$ is

After conditioning further on $R(e)$, the law of $e$ has no additional dependence on $y$. On the other hand, the tower property gives

To obtain this on one $w$-full-measure set, let $s_j$ range over rational simple functions from a countable generating algebra of $Y$, and let $B$ range over a countable generating algebra of the old parameter space. The scalar tower property and the product law above imply

First intersect the full-measure sets for the countably many $(B,j)$, obtaining a common $w$-full-measure set. Fix $w$ there. Uniqueness of measures or a monotone-class argument extends $B$ to all Borel sets, and density of $s_j$ identifies the Hilbert-space vector. Since $R_*\beta_w=\alpha_w$, equation (T) gives precisely all the test identities for the Hilbert-space-valued conditional expectation

This is the paper's equation (6.12). Without first identifying the exact fiber law of $y$ given $e$, one could not pass directly from the scalar tower property to this identity.

7. A nonzero random vector cannot be almost surely orthogonal to an independent copy

Source: B Lemma 6.6, p.23.

Let $V$ be a square-integrable random vector in a separable real Hilbert space and $V'$ an independent copy with the same law. If $\langle V,V'\rangle=0$ almost surely, then $V=0$ almost surely.

Otherwise some $\varepsilon>0$ satisfies $\mathbb P(\|V\|\ge\varepsilon)>0$. Separability gives a countable cover by balls of radius $\varepsilon/4$, one of which meets this event with positive probability.

By independence, both copies lie in that intersection with positive probability. Their norms are then at least $\varepsilon$ and their distance at most $\varepsilon/2$, so

a contradiction.

8. The fiber inner product cannot vanish almost everywhere

Suppose the right side of (Pair) vanished almost everywhere. By (PI), for almost every $w$,

Conditional on $R(e)=p'$, average over $e$. Equation (Tower) replaces the second vector by $g(p',\cdot)$. Since $R_*\beta_w=\alpha_w$,

The random-vector lemma forces $g(p,\cdot)=0$ for $\alpha_w$-almost every point, for almost every $w$.

In the old selected law, conditional on the complete parameter $P$, the corner state has distribution $m_{Y,P_b}$. Therefore

This contradicts $g\ne0$, which the preceding part deduced from the original nonzero correlation. Thus (Target) holds.

9. One additional auxiliary position

Source: the end of the proof of B Proposition 6.7, p.24, equation (6.13).

Set

Take $\Psi(E)=\operatorname{sgn}K(E)$. Then

Check each of the four positions of this correlation in the new system $Z$.

• For $i\ne b$ and $i\in A$, the column function uses $d_Y(y_{ib})$. This is retained in $Z$ as the old auxiliary list at position $b$, so the position remains auxiliary.
• For $i\ne b$ and $i\notin A$, the function uses $x_Y(y_{ib})$, precisely the original spatial variable of $Z$. It remains an original-variable position.
• At position $b$, the function $q(z_b)$ uses the three newly added lists $Q_j$ and therefore supplies a new auxiliary position. It is centered for the product spatial law of all new auxiliary variables, because the three lists are independent and each factor is centered.

The old position functions retain mean zero, and $\Psi$ is a bounded function of the full new parameter array $E$. Thus (New) has exactly the form of (6.1), but with $r+1$ auxiliary positions, contradicting maximality of $r$.

10. Completion of the paper's fourfold deduction

Conditional on the previously stated results, pointwise failure produces an initial nonzero correlation. The maximal auxiliary-position count exists and survives a stable extension. Regrouping and the square-array calculation then produce one more auxiliary position, ruling out failure.

Finally, the known $L^2$ limit identifies the product of the four integrals. Pull back from the standard model for the fixed functions to the original probability space to obtain the conclusion of B Theorem 1.1.

This explains the deductions in B §§3–6 without certifying all their premises. Major inputs still requiring independent verification include the higher-order mixing assertion in R, the local analytic proofs in H/D and Appendix A, and the previously specified nonergodic structure results.

Guide to the original text

• B p.21: the square-array definition and the statement of Lemma 6.4.
• B p.22, equations (6.7)–(6.10): stability and conditional independence after omitting one row.
• B p.22, equation (6.11): the fiber integral for the row–column correlation.
• B p.23, Lemma 6.5: independence of parameters conditional on the common corner parameter.
• B p.23, Lemma 6.6: the independent-copy lemma in a separable Hilbert space.
• B pp.23–24, Proposition 6.7, equations (6.12)–(6.13): the Hilbert-space-valued tower property, nonvanishing, and the new auxiliary position.