  We consider a general class of total cost Markov decision processes (MDP) in
which the one-stage costs can have arbitrary signs, but the sum of the negative
parts of the one-stage costs is finite for all policies and all initial states.
We refer to this class as the General Convergence (GC for short) total cost
model, and we study the convergence of value iteration for the GC model, in the
Borel MDP framework with universally measurable policies. Our main results
include: (i) convergence of value iteration when starting from certain
functions above the optimal cost function; (ii) convergence of transfinite
value iteration starting from zero, in the special case where the optimal cost
function is nonnegative; and (iii) partial convergence of value iteration
starting from zero, for a subset of initial states.
  These results extend several previously known results about the convergence
of value iteration for either positive costs problems or GC total cost
problems. In particular, the first result on convergence of value iteration
from above extends a theorem of van der Wal for the GC model. The second result
relates to Maitra and Sudderth's analysis of transfinite value iteration for
the positive costs model. It suggests connections between the two total cost
models when the optimal cost function is nonnegative, and it leads to
additional results on the convergence of ordinary non-transfinite value
iteration, with a suitably defined dynamic programming operator, for finite
state or finite control GC problems. The third result on partial convergence of
value iteration is motivated by Whittle's bridging condition for the positive
costs model, and provides a novel extension of the bridging condition to the GC
model, where there are no sign constraints on the costs.
