Absolute summability, square-summability and $\sum_t(\log t)^2 x_t^2<\infty$, which is the strongest? I'm working through a few convergence conditions for time series. In particular, I have run into
$$
\sum_{t=1}^\infty|x_t|<\infty\tag{$*$},
$$
$$
\sum_{t=1}^\infty x_t^2<\infty,\tag{$**$}
$$
$$
\sum_{t=1}^\infty(\log t)^2x_t^2<\infty.\tag{$***$}
$$
In all series above, $\{x_t\}$ denotes a sequence of real numbers. I have already proved that absolute summability is a strictly stronger condition than square-summability. I have also proved that ($***$) implies ($**$). But how do I compare ($*$) and ($***$) please? Could you also please give a hand on the reverse implication ($**$) $\implies$ ($***$)? I suspect it doesn't hold but I don't have a counter example.
Thank you very much.

$(***)\nRightarrow(*)$: try $x_t=1/t$
$(**)\nRightarrow(***)$: $x_t=\frac{1}{\sqrt{t}\log(t)}$ does the trick
I'm still pondering on $(*)\nRightarrow(***)$.
