matrix, we denote by ${\@ifstar{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  s
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  }
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{A}_\infty$ the $\ell_\infty$-operator norm of $A$: $${\@ifstar{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  s
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{  \mathopen{|\mkern-1.5mu|\mkern-1.5mu|}
  }
  \mathclose{|\mkern-1.5mu|\mkern-1.5mu|}
}{A}_\infty=\max_{1\leq i\leq m}\sum_{j=1}^d|A^{ij}|.$$ If $B$ is another $m\times d$ matrix, we denote by $A\cdot B$ the Frobenius inner product of $A$ and $B$. That is, $$A\cdot B:=\sum_{i=1}^m\sum_{j=1}^dA^{ij}B^{ij}.$$ For a $d\times d$ matrix $A$, we denote by $\operatorname{diag}(A)$ the $d$-dimensional vector consisting of the diagonal entries of $A$, i.e. $\operatorname{diag}(A)=(A^{11},\dots,A^{dd})^\top$.

For a random variable $\xi$ and a number $p>0$, we write $\|\xi\|_p=(E[|\xi|^p])^{1/p}$. We also use the notation $\|\xi\|_\infty$ to denote the essential supremum of $\xi$. We will denote by $L^{\infty-}$ the space of all random variables $\xi$ such that $\|\xi\|_p<\infty$ for every $p\in[1,\infty)$. The notation $\to^p$ stands for convergence in probability.

If $V$ is a real Hilbert space, we denote by $\langle\cdot,\cdot\rangle_V$ and $\|\cdot\|_V$ the inner product and norm of $V$, respectively. Also, we denote by $L^p(\Omega;V)$ the set of all $V$-valued random variables $\xi$ such that $E[\|\xi\|_V^2]<\infty$.

Given real Hilbert spaces $V_1,\dots,V_k$, we write their Hilbert space tensor product as $V_1\otimes\cdots\otimes V_k$. For a real Hilbert space $V$, we write the $k$th tensor power of $V$ as $V^{\otimes k}$, i.e. $$V^{\otimes k}:=\underbrace{V\otimes\cdots\otimes V}_{k}.$$ For 