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I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but the difficulty arises due to the infinite-dimensional functional(operator) setting. Any tiny or partial comment and insight would be enormously appreciated.

I consider a functional time series $(f_t)_{t=1}^{T}$ taking values in an infinite-dimensional separable Hilbert space $H$, specified by $f_t=Af_{t-1}+\varepsilon_t$, where $A$ is a Hilbert-Schmidt operator whose operator norm $\Vert A\Vert_{ \infty}$ is less than 1 so that $(f_t)_{t=1}^{T}$ is stationary.

Then, I think of the variance operator $\Sigma_f = \mathbb{E}(f_t\otimes f_t)$ of the time series, where $\otimes$ denotes the tensor product in $H$. Then, it is well-known that $\Sigma_f$ is a self-adjoint and trace-class operator, and it admits the spectral representation: $\Sigma_f = \sum_{i=1}^{\infty}\lambda_i(v_i\otimes v_i)$, where $(\lambda_i,v_i)_{i=1}^{\infty}$ is the eigenvalue and (orthonormalized) eigenfunction pairs of $\Sigma_f$, and $\lambda_i \downarrow 0$ as $i \rightarrow \infty$.

Thus, $\Sigma_f^{-1}$ is an unbounded operator, but it seems to be usually assumed in the literature that $\Vert \Sigma_f^{-\frac{1}{2}}A \Vert_{\infty}<\infty$.(If you should be interested, see this paper.)

Then, I am given consistent estimators $\hat{A}$ and $\hat\Sigma_f$ respectively for $A$ and $\Sigma_f$. Here, $\hat A$ is of finite rank for each $T$, and $\Vert \hat A -A\Vert_{\infty} \rightarrow 0$ in probability as $T \rightarrow \infty$. Also, $\hat \Sigma_f$ is self-adjoint and of finite rank for each $T$, and $\Vert\hat \Sigma_f - \Sigma_f \Vert_{\infty} \rightarrow 0$ in probability as $T \rightarrow \infty$.

Since both $\hat A$ and $\hat \Sigma_f$ are of finite rank for each $T$, it always holds true that $\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A \Vert_{\infty} < \infty$ for each T.

Then, my three questions are

Q1: Is it guaranteed that $\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A - \Sigma_f^{-\frac{1}{2}}A \Vert_{\infty} \rightarrow 0$ in probability as $T\rightarrow \infty$ ? If not, could we have at least that $\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A \Vert_{\infty} = O_p(1)$ as $T \rightarrow \infty$ ?

Q2: If another estimator $\tilde \Sigma_f$ for $\Sigma_f$ is such that $\Vert \Sigma_f - \tilde \Sigma_f + A\Sigma_f A^{*} - \hat A\tilde \Sigma_f \hat A^{*} \Vert_{\infty} \rightarrow 0$ in probability as $T \rightarrow \infty$, where $A^{*}$ and $\hat A^{*}$ are respectively the adjoints of $A$ and $\hat A$, then does it imply that $\Vert \Sigma_f - \tilde \Sigma_f \Vert_{\infty} \rightarrow 0$ in probability as $T \rightarrow \infty$ ?

Q3: Could you please let me know, if any, book, monograph, or paper dealing with such materials in a relatively easily accessible way ?

Any tiny or partial comment and insight would be enormously appreciated!

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