  For a flat $p$-adic formal family $S$ of log points over a complete discrete
valuation ring with perfect residue field of mixed characteristics $(0,p)$ and
for a simple normal crossing log scheme $X$ over an exact closed log subscheme
of $S$ defined by an element of the maximal ideal of the dvr, we construct two
fundamental filtered complexes in the convergent topos the underlying scheme of
$X$ over the underlying scheme of $S$. We prove that they are canonically
isomorphisc. Because one of the complex is shown to calculate the log
convergent cohomology sheaf of $X/S$, the filtered complex produces the weight
filtration on the log convergent cohomology sheaf if the underlying scheme of
$X$ is proper over the underlying scheme of $S$. We give a comparison theorem
between the projection of the filtered complex in the Zariski topos of the
underlying scheme of $X$ and the isozariskian weight-filtered complex
constructed in the author's previous work.
