Theory for distribution matchers (DMs) is extended to distributions with memory. The divergence-minimizing DM is shown to select the sequences with the highest target probability, as in the memoryless case. A scaling law for divergence is extended to distributions driven by innovation processes. The theory is applied to parallel additive white Gaussian noise channels. A modified enumerative sphere-shaping (ESS) method with a weighted energy constraint is used in implementations. An illustrative example with three channels shows that joint ESS across channels reduces the rate loss by a large factor compared to product DMs at short blocklengths. The gains are confirmed by simulations with probabilistic amplitude shaping and a 5G-NR low-density parity-check code.

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