Our main result is the proof of the existence of random stationary tessellations in d-dimensional Euclidean space
with the following stability property: their distribution is invariant with respect to the operation of iteration (or
nesting) of tessellations with an appropriate rescaling. This operation means that the cells of a given tessellation
are individually and independently subdivided by independent, identically distributed tessellations, resulting in a new
tessellation. It is also shown that, for any stationary tessellation, the sequence that is generated by repeated
rescaled iteration converges weakly to such a stable tessellation; thus, the class of all stable stationary
tessellations is fully characterized.
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