  We propose and investigate a hidden Markov model (HMM) for the analysis of
aggregated, super-imposed two-state signal recordings. A major motivation for
this work is that often these recordings cannot be observed individually but
only their superposition. Among others, such models are in high demand for the
understanding of cross-talk between ion channels, where each single channel
might take two different states which cannot be measured separately. As an
essential building block we introduce a parametrized vector norm dependent
Markov chain model and characterize it in terms of permutation invariance as
well as conditional independence. This leads to a hidden Markov chain "sum"
process which can be used for analyzing aggregated two-state signal
observations within a HMM. Additionally, we show that the model parameters of
the vector norm dependent Markov chain are uniquely determined by the
parameters of the "sum" process and are therefore identifiable. Finally, we
provide algorithms to estimate the parameters and apply our methodology to
real-world ion channel data measurements, where we show competitive gating.
