We consider the matrix $H_t$, the stationary probability $\pi$, and $\lambda$ the smallest nonzero eigenvalue of $u : X \mapsto (I_N - K)X$. We have established that $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$ and that $$H_t[i,j] - \pi[j] = \sum_{k=1}^{N} \left(H_{t/2}[i,k] - \pi[k]\right)\left(H_{t/2}[k,j] - \pi[j]\right)$$ Deduce that for all $(i,j) \in \llbracket 1;N \rrbracket^2$ and all $t \in \mathbf{R}_+$, $$\left|H_t[i,j] - \pi[j]\right| \leq e^{-\lambda t} \sqrt{\frac{\pi[j]}{\pi[i]}}$$ Determine $\lim_{t \rightarrow +\infty} H_t[i,j]$.
We consider the matrix $H_t$, the stationary probability $\pi$, and $\lambda$ the smallest nonzero eigenvalue of $u : X \mapsto (I_N - K)X$. We have established that $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$ and that
$$H_t[i,j] - \pi[j] = \sum_{k=1}^{N} \left(H_{t/2}[i,k] - \pi[k]\right)\left(H_{t/2}[k,j] - \pi[j]\right)$$
Deduce that for all $(i,j) \in \llbracket 1;N \rrbracket^2$ and all $t \in \mathbf{R}_+$,
$$\left|H_t[i,j] - \pi[j]\right| \leq e^{-\lambda t} \sqrt{\frac{\pi[j]}{\pi[i]}}$$
Determine $\lim_{t \rightarrow +\infty} H_t[i,j]$.