# Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?

Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow, $$P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0,\forall t\geq0,$$ where $P_0$ is a positive definitive matrix.

Is it suffices to show that $\phi_t^TP_t^{-1}\phi_t\to 0$?

I have tried in this way: Since $\phi_t^TP_t^{-1}\phi_t\leq \lambda_{max}(P_t^{-1})\|\phi_t\|_2^2$, if $\lambda_{max}(P_t^{-1})\to 0$, we can obtain the conclusion. However, a counterexample is $\phi_t=[1\;0]^T$, $\forall t$, and $P_0=I_{2\times2}$, then $$P_t=\begin{bmatrix} t+2 & 0 \\ 0 & 1 \\ \end{bmatrix},$$ and $\phi_t^TP_t^{-1}\phi_t\to 0$ but $\lambda_{max}(P_t^{-1})=1$, $\forall t$.

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