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.