  We establish the pointwise convergence of the iterative Lloyd algorithm, also
known as $k$-means algorithm, when the quadratic quantization error of the
starting grid (with size $N\ge 2$) is lower than the minimal quantization error
with respect to the input distribution is lower at level $N-1$. Such a protocol
is known as the splitting method and allows for convergence even when the input
distribution has an unbounded support. We also show under very light assumption
that the resulting limiting grid still has full size $N$. These results are
obtained without continuity assumption on the input distribution. A variant of
the procedure taking advantage of the asymptotic of the optimal quantizer
radius is proposed which always guarantees the boundedness of the iterated
grids.

Probability