Covariance matrix and the Cramer-Wold device

brownian motioncentral limit theoremprobability theoryprobability-limit-theorems

In Appendix B of Financial Statistics and Mathematical Finance, after the Cramer-Wold device (Theorem B.1.1) it is said:

In particular, the Cramer–Wold technique tells us that $$X_n \xrightarrow{D} N(\mu, \Sigma)$$ as $n \to \infty$, for some $\mu \in \mathbb{R}^d$ and $\Sigma \in \mathbb{R}^{d \times d}$, if and only if the univariate sequence $Y_n = Y_n(\lambda) = \sum_{j=1}^d \lambda_j X_{nj}$ satisfies a univariate central limit theorem such as Theorem B.7.3 or Theorem B.7.2, for each fixed vector $\lambda = (\lambda_1, \dots, \lambda_d)' \in \mathbb{R}^d$.

The theorems are written below for reference. My question is: what is this CLT that each sequence $Y_n(\lambda)$ must satisfy, and what role does $\Sigma$ play in it? Some versions of the CLT deal with the asymptotic covariance matrix explicitely, but the problem I have at hand involves a vector $J_n = (J_n^{(1)}, \dots, J_n^{(m)})$ where each component converges in distribution to a standard normal r.v. as $n \to \infty$ and isn't itself an average of $n$ statistics (also, the components are not necessarily independent). I have an interest in the asymptotic behaviour of the vector $J_n$ and the covariance matrix of the limit normal distribution.

Theorem B.7.2 (Lindeberg–Feller FCLT for Martingale Difference Sequences)
Suppose $\{ \xi_t \}$ is a square-integrable $\mathcal{F}_t$-martingale difference sequence. Let $V_t = \sum_{k=1}^t E[\xi_k^2 \mid \mathcal{F}_{k-1}]$. If

  1. $V_T/T \xrightarrow{P} \sigma^2 > 0$, as $T \to \infty$, and
  2. the Lindeberg condition is satisfied, i.e. $$\frac{1}{T}\sum_{t=1}^T E[\xi_t^2 I(\lvert \xi_t \rvert > \varepsilon \sqrt{T})] \to 0,$$ as $T \to \infty$, for any $\varepsilon > 0$, then $$\frac{1}{\sqrt{T}} \sum_{t=1}^{\lfloor T – \rfloor} \xi_t \implies \sigma B(-)$$ as $T \to \infty$.

Theorem B.7.3 (Lindeberg–Feller FCLT for Martingale Difference Arrays)
Suppose $\{ \xi_{Tt}: 1 \leq t \leq T, T \geq 1 \}$ is a $\mathcal{F}_{Tt}$-martingale difference array such that $E[\xi_{Tt}^2 \mid \mathcal{F}_{T, t-1}] < \infty$ for all $1 \leq t \leq T, T \geq 1$. Put $$V_{tk} = \sum_{i=1}^k E[\xi_{ti}^2 \mid \mathcal{F}_{t,i-1}], 1 \leq t \leq T, T \geq 1.$$ Suppose the following conditions are satisfied.

  1. $V_{T,\lfloor Tu \rfloor} \xrightarrow{P} u$ for all $u \in [0,1]$.
  2. The conditional Lindeberg condition holds true, that is $$\sum_{i=1}^T E[\xi_{ti}^2 I(\lvert \xi_{Ti} \rvert > \varepsilon ) \mid \mathcal{F}_{T,i-1}] \xrightarrow{P} 0,$$ as $T \to \infty$, for all $\varepsilon > 0$. Then $$\sum_{t=1}^{\lfloor Tu – \rfloor} \xi_{Tt} \implies \sigma B(u)$$ as $T \to \infty$.

Best Answer

It should be read as follows: $$ X_n\xrightarrow{d}N(\mu,\Sigma) $$ if and only if for any $\lambda\in\mathbb{R}^{d}\setminus\{0\}$, $$ \lambda^{\top}X_n\xrightarrow{d}N(\lambda^{\top}\mu,\lambda^{\top}\Sigma\lambda). $$

Related Question