Interchange summation order in the limit of number of elements going to $\infty$

Question

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but some more hypothesis are required to allow the change of order (e.g., by Fubini Theorem, we are allowed if $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} |a_{ij}|<\infty$).

Consider now the sum $\sum_{i=0}^{N}\sum_{j=0}^{L} a_{ij}$ , $N=cL$ , $c \in \mathbb{N}$. Our sum doesn't respect Fubini hypothesis, i.e. $\sum_{i=0}^{N=\infty}\sum_{j=0}^{L=\infty} |a_{ij}|=\infty$.

If now we consider the limit sum $\lim_{N\rightarrow \infty} \sum_{i=0}^{N}\sum_{j=0}^{L} a_{ij}$, are we allowed then to interchange the summation order, since $\sum_{i=0}^{N}\sum_{j=0}^{L} |a_{ij}|< \infty$ $\forall N<\infty$?

In other words when can we say that

$\lim_{L\rightarrow \infty} \sum_{i=0}^{cL}\sum_{j=0}^{L} a_{ij} = \lim_{L\rightarrow \infty} \sum_{j=0}^{L}\sum_{i=0}^{cL} a_{ij}$ ?

Answer 1

Comment. I think it should work like this. Let $\alpha = \sum_{i=0}^\infty\sum_{j=0}^\infty a_{ij}, \beta = \sum_{j=0}^\infty\sum_{i=0}^\infty a_{ij}$ both exist. Assume $\alpha < \beta$. Let $\varepsilon>0$. Then there exist sequences $(N_k)_{k=1}^\infty$, $(M_k)_{k=1}^\infty$ so that $N_k \to \infty$, $M_k \to \infty$ and $$ \lim_{k\to\infty}\sum_{i=0}^{N_k}\sum_{j=0}^{k} a_{ij} < \alpha+\varepsilon \\ \lim_{k\to\infty}\sum_{i=0}^{M_k}\sum_{j=0}^{k} a_{ij} > \beta-\varepsilon $$

But we cannot assume that $N_k, M_k$ grow more slowly that $ck$ for some constant $c$, nor that they grow more rapidly than $ck$ for some $c>1$.