An inverse of the Borel–Cantelli lemma (without requiring the independence) There is a theorem, a converse to the Borel-Cantelli lemma (without requiring the independence), that I learnt from a talk of which I don't know the source:
Let $(f_n)$ be a sequence of nonnegative functions on the probability space $(X,\mathcal{B},\mu)$. Assume that there exists a constant $C>0$ such that for every $N\ge 1$,
$$\sum_{m,n=1}^{N}|\int_X f_m f_n d\mu-\int_X f_m d\mu \int_X f_n d\mu|\le C\sum_{n=1}^N \int_X f_n,$$
then
$$\sum_{n=1}^N \int_X f_n=\infty ~\text{implies}~ \sum_{n=1}^{\infty}f_n(x)=\infty$$
for almost every $x\in X$.
Can anyone show me any reference for this theorem (or a direct proof if not too long)?
 A: Let's

*

*Rewrite in terms of RVs and Expectations.

*Weaken the condition.

Suppose $X_1,X_2,\dots$ are nonnegative RVs. Let $S_N = \sum_{n=1}^N X_n$. Suppose that for some constant  $C>0$,
$$(*)\quad \mbox{Var} (S_N)\le C E[S_N]$$
for all $N$.
Note that
\begin{align*} \mbox{Var}(S_N) &= E[ (S_N- E[S_N])^2] \\
& = E[ \sum_{n=1}^N \sum_{m=1}^N (X_n-E[X_n])(X_m-E[X_m]) ] \\
& = \sum_{n=1}^N \sum_{m=1}^N E[X_n X_m] -E[X_n]E[X_m].
\end{align*}
Then $\lim_{N\to\infty} E[S_N]=\infty$ implies $\lim_{N\to\infty} S_N = \infty$, a.s.
The idea is this: use $(*)$ and Chebychev to show that $S_N$ is typically "close" to $E[S_N]$. Let's do it.
$$ P(|S_N  - E[S_N]| > \frac 12  E[S_N]) \overset{\mbox{Chebychev}}{\le} \frac{4\mbox{Var} (S_N)}{E[S_N]^2 }\overset{(*)}{\le} \frac{4C}{E[S_N]}.$$
Now take a subsequence $N_1< N_2 < \dots$ such that $\sum_{k=1}^\infty \frac{1}{E[S_{N_k}]}< \infty$. It follows form Borel-Cantelli that $|S_{N_k} -E[S_{N_k}]|> \frac 12 E[S_{N_k}]$ finitely often a.s. In particular, $S_{N_k} \ge \frac 12 E[S_{N_k}]$, eventually a.s. This implies $\lim_{N\to\infty} S_N =\infty$ a.s.
A comment: the argument shows that we can replace the RHS in $(*)$ by $C (E[S_N])^{\alpha}$ with $\alpha \in [0,2)$.
