Let $X^i_1,X^i_2,\ldots,X^i_n$ be independent stationary Markov chains on a same space $\Omega = \{s_1,\ldots,s_m\}$ with a same transition matrix $P$ and $X^i_1\sim \pi$ for $i=1,2,\ldots,N$ ($N>m$).
I would like to find a lower bound of the following random variable:
$$f(\{X^i_j\}_{i=1}^N):= E_\pi[\min_{s\in\Omega}\sum_{i=1}^N\mathbb{1}(X^i_j=s)],$$ where $\pi$ is the stationary distribution of $\{X^i_j\}$ and $\mathbb{1}(\cdot)$ is the indicator function.
Is there any useful inequality?