Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n_k\}_{k=1}^\infty$ such that $$ \lim_{k\to\infty}\frac{1}{n_k}\sum_{i=1}^{n_k}\sum_{j=1}^{n_k}a_{ij} >0?$$
In a previous post A question on the partial sum of infinite doubly stochastic matrix, Iosif Pinelis constructed a counterexample for the whole sequence.
No. Enumerate all positive integers which are not powers of $2$: $3=n_1<n_2<n_3<\dots$ and partition positive integers into two-element sets $\{n_k,2^{k-1}\}$. Let $a_{i,j}=1$ if the set $\{i,j\}$ is such a two-element set, and let $a_{i,j}=0$ otherwise. We get a symmetric bistochastic matrix, and $1$'s are only in the rows or columns indexed by a power of $2$. We have $O(\log n)$ such entries in a square $\{1,2,\dots,n\}^2$.
