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Hilbert-valued CLT for L2-m-approximable Bernoulli shifts

Statement

Assume the Axiom of Choice. Let $H = L^{2} ([0, 1], d t; R)$ and let $(X_{t})_{t \in Z}$ be a centered, $L^{2}$ - $m$ -approximable causal Bernoulli shift as defined in L2-m-approximable causal Bernoulli shifts, with Bochner mean $E_{B} X_{0} = 0$ . Let $Γ$ be its long-run covariance operator from Long-run covariance of an L2-m-approximable shift is trace class, and let $G$ be a centered $H$ -valued Gaussian random element with covariance $Γ$ , whose existence is given by Positive trace-class covariance defines a Hilbert Gaussian. Then, for $S_{n} = \sum_{t = 1}^{n} X_{t}$ , $n^{- 1 / 2} S_{n} \Rightarrow G$ in the weak topology of Borel probability measures on $H$ . The covariance kernel is the $L^{2} ([0, 1]^{2})$ equivalence class $c$ in the long-run covariance lemma, and its integral operator is $T_{c} = Γ$ . This is a CLT for the $H$ -valued sample sums; it is not a functional CLT for the time-indexed partial-sum process.

Facts & Assumptions

Given: Assume AC and take the centered shift and notation in the Statement.

[A1]

AC implies Countable Choice and Dependent Choice, which supply the countable product constructions used in the definition and Gaussian supplier (The Axiom of Choice, AC supplies countable selections and prescribed serial paths, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F1]

The definition gives strict stationarity of $X_{t}$ and, for each fixed $m \geq 1$ , of $X_{t, m}$ . The joint process $(X_{t}, X_{t, m})$ is stationary because the original iid coordinates and the mutually independent, independently indexed replacement arrays are invariant in law under a common time shift. It also gives centered replacement differences, $δ_{m} = ∥ X_{0} - X_{0, m} ∥_{L^{2} (H)}$ , and $∥ X_{t} - X_{t, m} ∥_{L^{2} (H)} = δ_{m}$ (L2-m-approximable causal Bernoulli shifts). For $h \geq 1$ , $X_{0}$ is independent of $X_{h, h - 1}$ because their original innovation indices are disjoint and all replacement arrays are independent of the original family; the measurable maps defining the random elements preserve this independence (Measurable coordinatewise functions preserve independence).

[F2]

The long-run covariance lemma gives $Γ_{n} = n^{- 1} E_{B} [S_{n} \otimes S_{n}] \to Γ$ in trace norm, where $Γ$ is positive, self-adjoint and trace class (Long-run covariance of an L2-m-approximable shift is trace class). For trace-class operators $∥ T ∥_{o p} \leq ∥ T ∥_{1}$ (Trace class is a two sided Banach operator ideal).

[F3]

AC supplies a complete finite or countable orthonormal eigenbasis $(e_{k})$ for the positive compact self-adjoint operator $Γ$ ; its eigenvalues $λ_{k} \geq 0$ sum to $tr Γ < \infty$ (Trace class operator, Spectral theorem for compact self adjoint operators, Orthonormal eigenbasis for a compact self adjoint operator, Trace of a positive operator is the sum of its eigenvalues). The local basis-projection lemma gives norm convergence of $P_{K} x$ to $x$ and Parseval's identity for this basis (Complete orthonormal bases give norm-convergent projections and Parseval).

[F3a] Tonelli permits summing the nonnegative squared Gaussian coordinates before taking expectation (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F9]

Cauchy--Schwarz holds in the real Hilbert space $H$ and for square- integrable real random variables (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, Cauchy-Schwarz for random variables).

[F10]

Independent centered square-integrable random elements in $H$ have zero expected inner product; the local proof uses the complete countable orthonormal basis in [F3] (Independent centered Hilbert-valued elements have zero expected inner product).

[F4]

A centered stationary finite $m$ -dependent sequence of square-integrable vectors in $R^{d}$ has the CLT with covariance equal to the finite sum of its lag covariances, including singular covariance (Finite-dependent vector central limit theorem).

[F5]

Pointwise characteristic-function convergence to a specified real law implies weak convergence, and convergence of every real linear projection implies vector weak convergence (Characteristic function criterion for weak convergence, Cramer wold device).

[F11]

A centered real normal with variance $v \geq 0$ has characteristic function $t \mapsto e^{- t^{2} v / 2}$ , including $v = 0$ (Multivariate normal law, including singular covariance, Characteristic function of a normal law).

[F6]

Dominated convergence applies to the squared projection tails when they are bounded by $∥ X_{0} ∥_{H}^{2}$ (Dominated convergence).

[F7]

Markov's inequality bounds the probability that a nonnegative random variable exceeds a positive threshold by its expectation divided by that threshold (Markov's inequality for random variables).

[F8]

For Borel probability laws on a metric space, weak convergence is defined as convergence against bounded continuous tests and is equivalent to convergence of expectations against every bounded uniformly continuous real function (Convergence in distribution of random elements, Portmanteau theorem).

Proof

Given: The assumptions and supplier facts in the Statement and Facts & Assumptions.

1.1F1F9F10givenalgebra

Put $σ = ∥ X_{0} ∥_{L^{2} (H)}$ and $D_{t, m} = X_{t} - X_{t, m}$ for $m \geq 1$ . Both terms have the same centered law, so $D_{t, m}$ is centered. For $h \geq 1$ , $X_{h, h}$ is independent of $D_{0, m}$ : its original inputs have indices $1, \dots, h$ , while $D_{0, m}$ uses original indices at most $0$ and its own replacement array; $X_{h, h}$ uses a distinct independent replacement array. It has the centered law of $X_{0}$ , so [F10] gives $E ⟨ D_{0, m}, X_{h, h} ⟩_{H} = 0$ . Also $∥ X_{h} - X_{h, h} ∥_{L^{2} (H)} = δ_{h}$ . Hence Cauchy--Schwarz gives $∣ E ⟨ D_{0, m}, X_{h} ⟩_{H} ∣ = ∣ E ⟨ D_{0, m}, X_{h} - X_{h, h} ⟩_{H} ∣ \leq δ_{m} δ_{h} .$ If $h \geq m$ , the original inputs of $X_{h, m}$ have indices $h - m + 1, \dots, h$ , all positive, and its replacement array is independent of the coordinates in $D_{0, m}$ ; hence it is independent of $D_{0, m}$ and centered. [F10] gives $E ⟨ D_{0, m}, X_{h, m} ⟩_{H} = 0$ . If $1 \leq h < m$ , its marginal second moment is $σ^{2}$ , giving $∣ E ⟨ D_{0, m}, X_{h, m} ⟩_{H} ∣ \leq δ_{m} σ$ . Therefore $∣ E ⟨ D_{0, m}, D_{h, m} ⟩_{H} ∣ \leq {\begin{cases} δ_{m} (δ_{h} + σ), & 1 \leq h < m, \\ δ_{m} δ_{h}, & h \geq m . \end{cases}$ Stationarity makes the negative-lag inner products equal to the corresponding positive-lag ones. Expanding the squared norm of the normalized sum and bounding each finite-lag weight by one yields, for $U_{n} = n^{- 1 / 2} S_{n}$ and $U_{n}^{(m)} = n^{- 1 / 2} \sum_{t = 1}^{n} X_{t, m}$ , $\sup_{n} E ∥ U_{n} - U_{n}^{(m)} ∥_{H}^{2} \leq δ_{m}^{2} + 2 δ_{m} \sum_{h \geq 1} δ_{h} + 2 σ (m - 1) δ_{m} .$ The $m = 1$ empty middle-lag sum is zero.

2.1A1F2F9step 1.1choosealgebra

Since $\sum_{m \geq 1} δ_{m} < \infty$ , ${lim inf}_{m \to \infty} m δ_{m} = 0$ : otherwise $δ_{m} \geq c / m$ eventually for some $c > 0$ , contradicting summability. Recursively choose strictly increasing integers $m_{r}$ with $m_{r} δ_{m_{r}} < 1 / r$ ; the least eligible integer at each stage exists by the liminf statement. Thus the right side of step 1.1 tends to zero along $m = m_{r}$ . Write $ε_{m}$ for its square root. For each fixed $K$ , the covariance matrices of $P_{K} U_{n}$ are bounded because $Cov (U_{n}) = Γ_{n}$ and [F2] bounds $tr (P_{K} Γ_{n} P_{K})$ . For any $v \in P_{K} H$ , the scalar variance difference satisfies $∣ E ⟨ U_{n}^{(m)} - E U_{n}^{(m)}, v ⟩_{H}^{2} - E ⟨ U_{n}, v ⟩_{H}^{2} ∣ \leq ∥ v ∥_{H}^{2} ε_{m} (2 M_{K} + ε_{m}),$ where $M_{K} = \sup_{n} (E ∥ P_{K} U_{n} ∥_{H}^{2})^{1 / 2} < \infty$ ; this follows from $∣ a^{2} - b^{2} ∣ \leq ∣ a - b ∣ (∣ a ∣ + ∣ b ∣)$ and [F9].

3.1F2F3F4F5F11step 2.1algebra

For fixed $m$ and $K$ , the coordinate vectors $(⟨ X_{t, m}, e_{1} ⟩_{H}, \dots, ⟨ X_{t, m}, e_{K} ⟩_{H})_{t}$ are centered, stationary, $m$ -dependent and square-integrable in $R^{K}$ ; they identify $P_{K} X_{t, m}$ isometrically with $P_{K} H$ . By [F4], their normalized sums converge to a centered Gaussian with covariance matrix $Σ_{m, K} = \sum_{∣ j ∣ \leq m} Cov (P_{K} X_{0, m}, P_{K} X_{j, m})$ . The finite covariance expansion shows $Cov (P_{K} U_{n}^{(m)}) = \sum_{∣ j ∣ \leq m} (1 - ∣ j ∣ / n)_{+} Cov (P_{K} X_{0, m}, P_{K} X_{j, m}) \to Σ_{m, K}$ ; terms with $∣ j ∣ > m$ vanish by $m$ -dependence, and each of the finitely many coefficients tends to one. Letting $n \to \infty$ in step 2.1 and using [F2] gives $∥ Σ_{m, K} - P_{K} Γ P_{K} ∥_{o p} \leq ε_{m} (2 M_{K} + ε_{m}) .$ In particular $Σ_{m_{r}, K} \to P_{K} Γ P_{K}$ . Because these operators are symmetric and finite-dimensional, their operator norm is the supremum of the absolute quadratic form over unit vectors. For each real $v \in P_{K} H$ , [F11] therefore gives convergence of the characteristic functions of the scalar projections of $N (0, Σ_{m_{r}, K})$ to that of $N (0, P_{K} Γ P_{K})$ ; [F5] and Cramér--Wold give convergence of these finite-dimensional Gaussian laws.

4.1F4F5F11step 1.1step 3.1algebra

For $a \in P_{K} H$ , compare the characteristic functions of $⟨ U_{n}, a ⟩_{H}$ and $⟨ U_{n}^{(m)}, a ⟩_{H}$ . The inequality $∣ e^{i x} - e^{i y} ∣ \leq ∣ x - y ∣$ and step 1.1 bound their difference by $∥ a ∥_{H} ε_{m}$ . For each fixed $m$ , [F4] gives the limit law of the second projection; along $m = m_{r}$ , step 3.1 and [F11] make those Gaussian limit laws converge to $N (0, ⟨ Γ a, a ⟩_{H})$ . Taking first $n \to \infty$ and then $r \to \infty$ proves convergence of the characteristic functions of $⟨ U_{n}, a ⟩_{H}$ to that normal law. Explicitly, their limsup difference from the target characteristic function is at most $∥ a ∥_{H} ε_{m} + ∣ e^{- ⟨ Σ_{m, K} a, a ⟩_{H} / 2} - e^{- ⟨ Γ a, a ⟩_{H} / 2} ∣,$ which tends to zero along $m = m_{r}$ by steps 1.1 and 3.1. Since $a$ ranges over all vectors, for any fixed projection direction $b$ this also gives its characteristic-function convergence at every frequency $t$ by taking $a = t b$ . Thus [F5] and Cramér--Wold give $P_{K} U_{n} \Rightarrow P_{K} G$ for every fixed $K$ , since $G$ has covariance $Γ$ .

5.1F1F3F6F9F10step 4.1algebra

Let $R_{K} = I - P_{K}$ and $α_{K} = (E ∥ R_{K} X_{0} ∥_{H}^{2})^{1 / 2}$ . Completeness of the eigenbasis and Parseval give $R_{K} X_{0} \to 0$ almost surely in $H$ ; the bound $∥ R_{K} X_{0} ∥_{H}^{2} \leq ∥ X_{0} ∥_{H}^{2}$ and [F6] imply $α_{K} \to 0$ . For $h \geq 1$ , $X_{0}$ is independent of $X_{h, h - 1}$ by [F1]; bounded linear projection preserves independence and centering, so [F10] gives $E ⟨ R_{K} X_{0}, R_{K} X_{h, h - 1} ⟩_{H} = 0$ . Hence $∣ E ⟨ R_{K} X_{0}, R_{K} X_{h} ⟩_{H} ∣ = ∣ E ⟨ R_{K} X_{0}, R_{K} (X_{h} - X_{h, h - 1}) ⟩_{H} ∣ \leq α_{K} δ_{h - 1} .$ Expanding the variance of $R_{K} U_{n}$ and using [F9] now gives $\sup_{n} E ∥ R_{K} U_{n} ∥_{H}^{2} \leq α_{K}^{2} + 2 α_{K} (δ_{0} + \sum_{m \geq 1} δ_{m}) ⟶ 0.$ Also [F3], [F3a] and the covariance of $G$ give $E ∥ R_{K} G ∥_{H}^{2} = \sum_{k > K} λ_{k} \to 0$ ; if the basis is finite, take $P_{K} = I$ once all basis vectors are included.

6.1F7F8step 4.1step 5.1∎

Let $ϕ : H \to R$ be bounded and uniformly continuous, with $M = \sup_{x} ∣ ϕ (x) ∣$ . For $η > 0$ put $ω_{ϕ} (η) = \sup_{∥ x - y ∥ \leq η} ∣ ϕ (x) - ϕ (y) ∣$ , which tends to zero as $η ↓ 0$ . Markov's inequality [F7] and step 5.1 imply $∣ E ϕ (U_{n}) - E ϕ (P_{K} U_{n}) ∣ \leq ω_{ϕ} (η) + 2 M η^{- 2} \sup_{n} E ∥ R_{K} U_{n} ∥_{H}^{2},$ and the same bound for $G$ and $P_{K} G$ with $E ∥ R_{K} G ∥_{H}^{2}$ . For fixed $K$ , step 4.1 gives $E ϕ (P_{K} U_{n}) \to E ϕ (P_{K} G)$ . First let $K \to \infty$ in these inequalities and then let $η ↓ 0$ . It follows that $E ϕ (U_{n}) \to E ϕ (G)$ for every bounded uniformly continuous $ϕ$ . By [F8], $U_{n} \Rightarrow G$ in the Borel weak topology of $H$ . The kernel assertion is the conclusion of the long-run covariance lemma; the theorem asserts no time-indexed process convergence.

Depends on

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