# Covariance matrix and the Cramer-Wold device

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 $$T \to \infty$$.

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