When I do my research, I found that there are two definitions of uniformly observable, I can't help thinking are they equivalent?
These two definitions are listed as follows.
For a linear stochastic system with the description $$x_{k+1} = F_{k}x_{k}+w_{k}$$ and $$z_k = H_k x_k+v_k$$ where $F_{k}$ and $H_k$ are system matrix and measurement matrix, respectively, $x_k$ is the system state, $z_k$ is the output, while $w_k$ and $v_k$ are Gaussian while noise sequences, $R_k$ is the covariance of $v_k$.
The first definition of uniformly observable is coming form
J. C. Spall and K. D. Wall, “Asymptotic distribution theory for the Kalman filter state estimator,” Communications in Statistics - Theory and Methods, vol. 13, no. 16, pp. 1981–2003, Jan. 1984.
it says
The system is said to be uniformly completely observable if there exists an integer $1 \leq m< \infty$ and constants $0<\beta_1 \leq \beta_2 <\infty$ such that \begin{align} \beta_1 I \leq \mathcal{O}(l,k) \triangleq \sum_{l=k-m}^{k} \phi^T(l,k) H_l^T R_l^{-1} H_l \phi(l,k) \leq \beta_2 I \end{align} for all $k\geq m$, where $\phi(k,k) =I$ and \begin{align} \phi(l,k) = (F_l)^{-1} (F_{l+1})^{-1} \cdots (F_{k-1})^{-1}. \end{align}
The second one is found in
K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 714–728, Apr. 1999.
it says
Consider time-varying matrices $F_k$, $H_k$, $k \geq 0$, and let the observability gramian be given by \begin{align} M_{k+m,k}=\sum_{l=k}^{k+m} \phi^T(l,k) H_l^T H_l \phi(l,k) \end{align} for some integer $m\geq 0$ with $\phi(k,k)=I$ for \begin{align} \phi(l,k) = (F_{l-1}) (F_{l-2}) \cdots (F_{k}), \end{align} for $i>n$. The matrices $F_k$, $H_k$, $k \geq 0$ are said to satisfy the uniform observability condition, if there are real numbers $\alpha_1$, $\alpha_2$ and an integer $m>0$, such that the following inequality holds: \begin{align} \alpha_1 I \leq M_{k+m,k} \leq \alpha_2 I. \end{align}
My question is that are those two definition equivalent?