A Proofs 😱

A.1 Proof of Theorem 3.1

We let $$X_t = W_t + \mu$$, where $$\mu < \infty$$ and $$(W_t)$$ is a strong white noise process with variance $$\sigma^2$$ and finite fourth moment (i.e. $$\mathbb{E} [W_t^4] < \infty$$).

Next, we consider the sample autocovariance function computed on $$(X_t)$$, i.e.

$\hat \gamma \left( h \right) = \frac{1}{n}\sum\limits_{t = 1}^{n - h} {\left( {{X_t} - \bar X} \right)\left( {{X_{t + h}} - \bar X} \right)}.$

For this equation, it is clear that $$\hat \gamma \left( 0 \right)$$ and $$\hat \gamma \left( h \right)$$ (with $$h > 0$$) are two statistics involving sums of different lengths. As we will see, this prevents us from using directly the multivariate central limit theorem on the vector $$[ \hat \gamma \left( h \right) \;\;\; \hat \gamma \left( h \right) ]^T$$. However, the lag $$h$$ is fixed and therefore the difference in the number of elements of both sums is asymptotically negligible. Therefore, we define a new statistic

$\tilde{\gamma} \left( h \right) = \frac{1}{n}\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)},$

which, as we will see, is easier to used and show that $$\hat \gamma \left( h \right)$$ and $$\tilde{\gamma} \left( h \right)$$ are asymptotically equivalent in the sense that:

$n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)] = o_p(1).$

Therefore, assuming this results to be true, $$\tilde{\gamma} \left( h \right)$$ and $$\hat \gamma \left( h \right)$$ would have the same asymptotic distribution, it is sufficient to show the asymptotic distribution of $$\tilde{\gamma} \left( h \right)$$. So that before continuing the proof the Theorem 1 we first state and prove the following lemma:

Lemma A1: Let

$X_t = \mu + \sum\limits_{j = -\infty}^{\infty} \psi_j W_{t-j},$ where $$(W_t)$$ is a strong white process with variance $$\sigma^2$$, and the coefficients satisfying $$\sum \, |\psi_j| < \infty$$. Then, we have

$n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)] = o_p(1).$

Proof: By Markov inequality, we have

$\mathbb{P}\left( |n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]| \geq \epsilon \right) \leq \frac{\mathbb{E}|n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]|}{\epsilon},$ for any $$\epsilon > 0$$. Thus, it is enough to show that

$\mathop {\lim }\limits_{n \to \infty } \; \mathbb{E} \left[|n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]|\right] = 0$

to prove Lemma A1.By the definitions of $$\tilde{\gamma} \left( h \right)$$ and $$\hat \gamma \left( h \right)$$, we have

\begin{aligned} n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)] &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n}(X_t - \mu)(X_{t+h} - \mu) \\ &+ \frac{1}{\sqrt{n}} \sum_{t = 1}^{n-h}\left[(X_t - \mu)(X_{t+h} - \mu) - (X_t - \bar{X})(X_{t+h} - \bar{X})\right]\\ &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n}(X_t - \mu)(X_{t+h} - \mu) + \frac{1}{\sqrt{n}} \sum_{t = 1}^{n-h}\left[(\bar{X} - \mu)(X_t + X_{t+h} - \mu - \bar{X})\right]\\ &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n} (X_t - \mu)(X_{t+h} - \mu) + \frac{1}{\sqrt{n}} (\bar{X} - \mu)\sum_{t = 1}^{n-h}(X_t + X_{t+h} - \mu - \bar{X})\\ &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n} (X_t - \mu)(X_{t+h} - \mu) + \frac{1}{\sqrt{n}} (\bar{X} - \mu)\left[\sum_{t = 1+h}^{n-h}X_t - (n-h)\mu + h\bar{X}\right]\\ &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n} (X_t - \mu)(X_{t+h} - \mu) + \frac{1}{\sqrt{n}} (\bar{X} - \mu)\left[\sum_{t = 1+h}^{n-h}(X_t - \mu) - h(\mu - \bar{X})\right]\\ &= \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n} (X_t - \mu)(X_{t+h} - \mu) + \frac{1}{\sqrt{n}} (\bar{X} - \mu)\sum_{t = 1+h}^{n-h}(X_t - \mu) + \frac{h}{\sqrt{n}} (\bar{X} - \mu)^2, \end{aligned} where $$\bar{X} = \frac{1}{n}\sum_{t=1}^n X_t = \mu + \frac{1}{n}\sum_{t=1}^n\sum_{j=-\infty}^{\infty} \psi_j W_{t-j} = \mu + \frac{1}{n} \sum_{j = -\infty}^{\infty} \sum_{t=1}^n \psi_j W_{t-j}$$.

Then, we have \begin{aligned} \mathbb{E}\left[\left|n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]\right|\right] &\leq \frac{1}{\sqrt{n}} \sum_{t = n-h+1}^{n} \mathbb{E}\left[\left|(X_t - \mu) \, (X_{t+h} - \mu)\right|\right]\\ &+ \frac{1}{\sqrt{n}} \mathbb{E} \left[\left|(\bar{X} - \mu) \, \sum_{t = 1+h}^{n-h}(X_t - \mu)\right|\right] + \frac{h}{\sqrt{n}}\mathbb{E} \left[ (\bar{X} - \mu)^2 \right]. \end{aligned}

Next, we consider each term of the above equation. For the first term, since $$(X_t - \mu)^2 = \left(\sum_{j = -\infty}^{\infty} \psi_j W_{t-j}\right)^2$$, and $$\mathbb{E}[W_iW_j] \neq 0$$ only if $$i = j$$. By Cauchy–Schwarz inequality we have

$\mathbb{E}\left[|(X_t - \mu)(X_{t+h} - \mu)|\right] \leq \sqrt{\mathbb{E}\left[|(X_t - \mu)|^2\right] \mathbb{E}\left[|(X_{t+h} - \mu)|^2\right]} = \sigma^2 \sum_{i = -\infty}^{\infty}\psi_i^2.$

Then, we consider the third term, since it will be used in the second term

$\mathbb{E}[(\bar{X} - \mu)^2] = \frac{1}{n^2} \sum_{t = 1}^{n} \sum_{i = -\infty}^{\infty} \psi_i^2 \mathbb{E}\left[ W_{t-i}^2 \right] = \frac{\sigma^2}{n} \sum_{i = -\infty}^{\infty}\psi_i^2.$

Similarly, for the second term we have

\begin{aligned} \mathbb{E}\left[\left|(\bar{X} - \mu) \sum_{t = 1+h}^{n-h}(X_t - \mu)\right|\right] &\leq \sqrt{\mathbb{E}\left[|(\bar{X} - \mu)|^2\right] \mathbb{E}\left[|\sum_{t = 1+h}^{n-h}(X_t - \mu)|^2\right]}\\ &= \sqrt{\mathbb{E}\left[(\bar{X} - \mu)^2\right] \mathbb{E}\left[\sum_{t = 1+h}^{n-h}\left(X_t - \mu \right)^2 + \sum_{t_1 \neq t_2}(X_{t_1} - \mu)(X_{t_2} - \mu) \right]}\\ &\leq \sqrt{\frac{\sigma^2}{n} \sum_{i = -\infty}^{\infty}\psi_i^2 \cdot (n-2h)\sigma^2 \left( \sum_{j = -\infty}^{\infty} |\psi_j| \right)^2}\\ &\leq \sqrt{\frac{n-2h}{n}}\sigma^2 \left(\sum_{i = -\infty}^{\infty}|\psi_i| \right)^2. \end{aligned}

Combining the above results we obtain

\begin{aligned} \mathbb{E}|n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]| &\leq \frac{1}{\sqrt{n}} h \sigma^2 \sum_{i = -\infty}^{\infty}\psi_i^2 + \sqrt{\frac{n-2h}{n^2}}\sigma^2 \left(\sum_{i = -\infty}^{\infty}|\psi_i| \right)^2 + \frac{h}{n\sqrt{n}}\sigma^2 \sum_{i = -\infty}^{\infty}\psi_i^2\\ &\leq \frac{1}{n\sqrt{n}} (nh + \sqrt{n - 2h} + h) \sigma^2 \left(\sum_{i = -\infty}^{\infty}|\psi_i|\right)^2, \end{aligned}

By the taking the limit in $$n$$ we have

$\mathop {\lim }\limits_{n \to \infty } \; \mathbb{E} \left[|n^{\frac{1}{2}}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)]|\right] \leq \sigma^2 \left(\sum_{i = -\infty}^{\infty}|\psi_i|\right)^2 \mathop {\lim }\limits_{n \to \infty } \; \frac{nh + \sqrt{n - 2h} + h}{n\sqrt{n}} = 0.$

We can therefore conclude that

$\sqrt{n}[\tilde{\gamma} \left( h \right) - \hat \gamma \left( h \right)] = o_p(1),$

which concludes the proof of Lemma A1. $$\;\;\;\;\;\;\;\; \blacksquare$$

$$\\$$

$$\\$$

Returning to the proof of Theorem 1, since the process $$(Y_t)$$, where $$Y_t = \left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)$$, is iid, we can apply multivariate central limit theorem to the vector $$[ \tilde \gamma \left( h \right) \;\;\; \tilde \gamma \left( h \right) ]^T$$, and we obtain

\begin{aligned} \sqrt{n}\left\{ \begin{bmatrix} \tilde{\gamma} \left( 0 \right) \\ \tilde{\gamma} \left( h \right) \end{bmatrix} - \mathbb{E}\begin{bmatrix} \tilde{\gamma} \left( 0 \right) \\ \tilde{\gamma} \left( h \right) \end{bmatrix} \right\} &= \frac{1}{\sqrt{n}}\begin{bmatrix} \sum\limits_{t = 1}^{n}(X_t - \mu)^2 - n\mathbb{E}\left[ \tilde{\gamma} \left( 0 \right) \right]\\ \sum\limits_{t = 1}^{n}\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right) - n\mathbb{E}\left[ \tilde{\gamma} \left( h \right) \right] \end{bmatrix} \\ & \overset{\mathcal{D}}{\to} \mathcal{N}\left(0, n \, \text{var} \left(\begin{bmatrix} \tilde{\gamma} \left( 0 \right) \\ \tilde{\gamma} \left( h \right) \end{bmatrix} \right)\right) \end{aligned}

Moreover, by Cauchy–Schwarz inequality and since $$\text{var}(X_t) = \sigma^2$$, we have

$\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)} \leq \sqrt{\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)^2} \sum\limits_{t = 1}^{n} {\left( {{X_{t + h}} - \mu} \right)^2}} < \infty.$

Therefore, by bounded convergence theorem and $$(W_t)$$ is iid, we have

\begin{aligned} \mathbb{E}[\tilde{\gamma} \left( h \right)] &= \frac{1}{n}\mathbb{E}\left[\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)}\right]\\ &= \frac{1}{n}\left[\sum\limits_{t = 1}^{n} { \mathbb{E}\left( {{X_t} - \mu} \right)\mathbb{E}\left( {{X_{t + h}} - \mu} \right)}\right] = \begin{cases} \sigma^2, & \text{for } h = 0\\ 0, & \text{for } h \neq 0 \end{cases}. \end{aligned}

Next, we consider the variance of $$\tilde{\gamma} \left( h \right)$$ when $$h \neq 0$$,

\begin{aligned} var[\tilde{\gamma} \left( h \right)] &= \frac{1}{n^2}\mathbb{E}\left\{\left[\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)}\right]^2\right\}\\ &= \frac{1}{n^2}\mathbb{E}\left\{\left[\sum\limits_{i = 1}^{n} {\left( {{X_i} - \mu} \right)\left( {{X_{i + h}} - \mu} \right)}\right] \left[\sum\limits_{j = 1}^{n} {\left( {{X_j} - \mu} \right)\left( {{X_{j + h}} - \mu} \right)}\right]\right\}\\ &= \frac{1}{n^2}\mathbb{E}\left[\sum\limits_{i = 1}^{n}\sum\limits_{j = 1}^{n} {\left( {{X_i} - \mu} \right)\left( {{X_{i + h}} - \mu} \right)}{\left( {{X_j} - \mu} \right)\left( {{X_{j + h}} - \mu} \right)}\right]. \end{aligned}

Also by Cauchy–Schwarz inequality and the finite fourth moment assumption, we can use the bounded convergence theorem. Once again since $$(W_t)$$ is white noise process, we have

$\mathbb{E}\left[{\left( {{X_i} - \mu} \right)\left( {{X_{i + h}} - \mu} \right)}{\left( {{X_j} - \mu} \right)\left( {{X_{j + h}} - \mu} \right)}\right] \neq 0$ only when $$i = j$$.

Therefore, we obtain

\begin{aligned} var[\tilde{\gamma} \left( h \right)] &= \frac{1}{n^2}\sum\limits_{i = 1}^{n} \mathbb{E}\left[ {\left( {{X_i} - \mu} \right)^2\left( {{X_{i + h}} - \mu} \right)^2}\right]\\ &= \frac{1}{n^2}\sum\limits_{i = 1}^{n} \mathbb{E}{\left( {{X_i} - \mu} \right)^2\mathbb{E}\left( {{X_{i + h}} - \mu} \right)^2} = \frac{1}{n}\sigma^4. \end{aligned}

Similarly, for $$h = 0$$, we have

\begin{aligned} var[\tilde{\gamma} \left( 0 \right)] &= \frac{1}{n^2}\mathbb{E}\left\{\left[\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)^2}\right]^2\right\} - \frac{1}{n^2}\left[\mathbb{E}\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)^2}\right]^2 = \frac{2}{n}\sigma^4. \end{aligned}

Next, we consider the covariance between $$\tilde{\gamma} \left( 0 \right)$$ and $$\tilde{\gamma} \left( h \right)$$, for $$h \neq 0$$, and we obtain

\begin{aligned} cov[\tilde{\gamma} \left( 0 \right), \tilde{\gamma} \left( h \right)] &= \mathbb{E}[\tilde{\gamma} \left( 0 \right) \tilde{\gamma} \left( h \right)] - \mathbb{E}[\tilde{\gamma} \left( 0 \right)] \mathbb{E}[\tilde{\gamma} \left( h \right)] = \mathbb{E}[\tilde{\gamma} \left( 0 \right) \tilde{\gamma} \left( h \right)]\\ &= \mathbb{E}\left[\left[\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)^2}\right]\left[\sum\limits_{t = 1}^{n} {\left( {{X_t} - \mu} \right)\left( {{X_{t + h}} - \mu} \right)}\right]\right] = 0. \end{aligned}

Therefore by Slutsky’s Theorem we have,

\begin{aligned} \sqrt{n}\left\{ \begin{bmatrix} \hat{\gamma} \left( 0 \right) \\ \hat{\gamma} \left( h \right) \end{bmatrix} - \begin{bmatrix} \sigma^2 \\ 0 \end{bmatrix} \right\} &= \sqrt{n}\left\{ \begin{bmatrix} \tilde{\gamma} \left( 0 \right) \\ \tilde{\gamma} \left( h \right) \end{bmatrix} - \begin{bmatrix} \sigma^2 \\ 0 \end{bmatrix} \right\} + \underbrace{\sqrt{n}\left\{ \begin{bmatrix} \hat{\gamma} \left( 0 \right) \\ \hat{\gamma} \left( h \right) \end{bmatrix} - \begin{bmatrix} \tilde{\gamma} \left( 0 \right) \\ \tilde{\gamma} \left( h \right) \end{bmatrix} \right\}}_{\overset{p}{\to} 0}\\ &\overset{\mathcal{D}}{\to} \mathcal{N}\left(0, \begin{bmatrix} 2\sigma^4 & 0\\ 0 & \sigma^4 \end{bmatrix} \right). \end{aligned}

Next, we define the function $$g\left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = b/a$$, where $$a \neq 0$$. For this function it is clear that

$\nabla g\left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \begin{bmatrix} -\frac{b}{a^2} \\ \frac{1}{a} \end{bmatrix}^{T} ,$

and thus using the Delta method, we have for $$h \neq 0$$

\begin{aligned} \sqrt{n}\hat{\rho}(h) = \sqrt{n}\left\{g\left( \begin{bmatrix} \hat{\gamma} \left( 0 \right) \\ \hat{\gamma} \left( h \right) \end{bmatrix} \right) - {\mu} \right\} &\overset{\mathcal{D}}{\to} \mathcal{N}\left(0, \sigma_r^2 \right), \end{aligned}

where

\begin{aligned} {\mu} &= g\left(\begin{bmatrix} \sigma^2 & 0 \end{bmatrix} \right) = 0,\\ \sigma_r^2 &= \nabla g\left(\begin{bmatrix} \sigma^2 \\ 0 \end{bmatrix} \right) \begin{bmatrix} 2\sigma^4 & 0\\ 0 & \sigma^4 \end{bmatrix} \nabla g\left(\begin{bmatrix} \sigma^2 \\ 0 \end{bmatrix} \right)^{T} = \begin{bmatrix} 0 & \sigma^{-2} \end{bmatrix} \begin{bmatrix} 2\sigma^4 & 0\\ 0 & \sigma^4 \end{bmatrix} \begin{bmatrix} 0 \\ \sigma^{-2} \end{bmatrix} = 1. \end{aligned}

Thus, we have

$\sqrt{n}\hat{\rho}(h) \overset{\mathcal{D}}{\to} \mathcal{N}\left(0, 1 \right),$

which concludes the proof the Theorem 1. $$\;\;\;\;\;\;\;\; \blacksquare$$

A.2 Proof of Theorem 4.1

Consider:

${\left( {{X_t} - m} \right)^2} = {\left[ {\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right) + \left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right]^2}$

where $$m = m\left( {{X_{1}}, \cdots ,{X_{t}}} \right)$$ and $${\Omega _t} = \left( {{X_{1}}, \cdots ,{X_{t}}} \right)$$.

Therefore we can write

${\left( {{X_t} - m} \right)^2} = {\left( {{X_t} - E\left[ {{X_t}|{\Omega _T}} \right]} \right)^2} + {\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right)^2} + 2\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right)\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right)$

Focusing on only the last term (and dropping the constant 2), we have that

$\underbrace {\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right)}_{ = {\varepsilon _t}}\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right).$

At this point, let us study the value of $$E\left[ {{\varepsilon _t}|{\Omega _t}} \right]$$ (the reason for this will become apparent in the next steps of the proof):

$E\left[ {{\varepsilon _t}|{\Omega _t}} \right] = E\left[ {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]|{\Omega _t}} \right] = E\left[ {{X_t}|{\Omega _t}} \right] - E\left[ {{X_t}|{\Omega _t}} \right] = 0$

Given this, we now consider the law of total expectation (i.e. $$E[X] = E[E[X|Y]]$$ which allows us to rewrite the expectation of the last term as follows

$E\left[ {{\varepsilon _t}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right] = E\left[ {E\left[ {{\varepsilon _t}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)|{\Omega _t}} \right]} \right] = E\left[ {\underbrace {E\left[ {{\varepsilon _t}|{\Omega _t}} \right]}_{ = 0}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right] = 0$

Since we have shown that the expectation of the last term is zero, we have that

${\left( {{X_t} - m} \right)^2} = {\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right)^2} + {\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)^2}.$

Now the first term is positive and doesn’t depend on $$m$$, so we focus on the second term which is minimized for $$m = E\left[ {{X_t}|{\Omega _t}} \right]$$ thereby minimizing the entire expression in terms of $$m$$.

A.3 Proof of Theorem 4.2

We begin by examining the norm and notice that

$\begin{equation*} \left\| {X_0 - \hat{X}} \right\|_2^2 = {X_0^T}X_0 + \hat{X}^T \hat{X} - 2X_0^T \hat{X} \end{equation*}$

Therefore, we obtain

\begin{align*} \mathbb{E}\left[ \mathbb{E}_0 \left[ \left\| {X_0 - \hat{X}} \right\|_2^2\right] \right] &= \mathbb{E}\left[ \mathbb{E}_0 \left[{X_0^T}X_0\right] \right] + \mathbb{E}\left[ \mathbb{E}_0 \left[\hat{X}^T \hat{X}\right] \right] - 2 \mathbb{E}\left[ \mathbb{E}_0 \left[X_0^T \hat{X}\right] \right]\\ &= \mathbb{E}_0 \left[{X_0^T}X_0\right] + \mathbb{E}\left[ \hat{X}^T \hat{X}\right] - 2 \mathbb{E}\left[ \mathbb{E}_0^T \left[X_0\right] \hat{X} \right]. \end{align*}

Next, we let $${C^*} \equiv \mathbb{E}\left[ {\left\| {X - \hat X} \right\|_2^2} \right]$$ which can be expressed as follows:

$\begin{equation*} \mathbb{E}\left[ \left\| {X - \hat{X}} \right\|_2^2\right] = \mathbb{E} \left[{X^T}X\right] + \mathbb{E}\left[ \hat{X}^T \hat{X}\right] - 2 \mathbb{E}\left[ X^T \hat{X} \right]. \end{equation*}$

Since $$X$$ and $$X_0$$ have the same distribution we have that $$\mathbb{E}[X] = \mathbb{E}_0 [X_0]$$ and $$\mathbb{E}[X^T X] = \mathbb{E}_0 [X_0^T X_0]$$. By taking the difference between $$C$$ and $$C^*$$ we obtain

\begin{align*} C - {C^*} &= 2 \mathbb{E}\left[ X^T \hat{X} \right] - 2 \mathbb{E}\left[ \mathbb{E}_0^T \left[X_0\right] \hat{X} \right] = 2 \mathbb{E}\left[ \left(X - \mathbb{E}_0 \left[X_0\right]\right)^T \hat{X} \right]\\ &=2 \mathbb{E}\left[ \left(X - \mathbb{E} \left[X\right]\right)^T \hat{X} \right] = 2 \mathbb{E}\left[ tr \left(\left(X - \mathbb{E} \left[X\right]\right)^T \hat{X}\right) \right]\\ &= 2 \mathbb{E}\left[ tr \left(\hat{X} \left(X - \mathbb{E} \left[X\right]\right)^T \right) \right] = 2 tr \left( \mathbb{E}\left[ \hat{X} \left(X - \mathbb{E} \left[X\right]\right)^T \right] \right)\\ &= 2 tr \left( \text{cov}\left( X - \mathbb{E} \left[X\right], \hat{X} \right)\right) + 2 tr \left( \mathbb{E}\left[ X - \mathbb{E} \left[X\right] \right] \mathbb{E}^T[\hat{X}]\right)\\ &= 2 tr \left( \text{cov}\left( X, \hat{X} \right)\right) + 2 tr \left(\left( \mathbb{E}\left[ X \right] - \mathbb{E} \left[X\right] \right) \mathbb{E}^T[\hat{X}]\right)\\ &= 2 tr \left( \text{cov}\left( X, \hat{X} \right)\right). \end{align*}

Thus, we have

$\begin{equation*} C = \mathbb{E}\left[ {\left\| {X - \hat X} \right\|_2^2} \right] + 2 tr \left( \text{cov}\left( X, \hat{X} \right)\right), \end{equation*}$

which concludes the proof.