  We study the vanishing cycles of a one-parameter smoothing of a complex
analytic space and show that the weight filtration on its perverse cohomology
sheaf of the highest degree is quite close to the monodromy filtration so that
its graded pieces have a modified Lefschetz decomposition. We describe its
primitive part using the weight filtration on the perverse cohomology sheaves
of the constant sheaves. As a corollary we show in the local complete
intersection case that 1 is not an eigenvalue of the monodromy on the reduced
Milnor cohomology at any points if and only if the total space and the singular
fiber are both rational homology manifolds. Also we introduce quasi-semistable
degenerations and calculate the limit mixed Hodge structure by constructing the
weight spectral sequence. As a corollary we show non-triviality of the space of
vanishing cycles of the Lefschetz pencil associated with a tensor product of
any two very ample line bundles except for the case of even-dimensional
projective space where two has to be replaced by three.
