2 research outputs found
Asymptotic and Non-Asymptotic Analysis for Hidden Markovian Process with Quantum Hidden System
We focus on a data sequence produced by repetitive quantum measurement on an
internal hidden quantum system, and call it a hidden Markovian process. Using a
quantum version of the Perron-Frobenius theorem, we derive novel upper and
lower bounds for the cumulant generating function of the sample mean of the
data. Using these bounds, we derive the central limit theorem and large and
moderate deviations for the tail probability. Then, we give the asymptotic
variance is given by using the second derivative of the cumulant generating
function. We also derive another expression for the asymptotic variance by
considering the quantum version of the fundamental matrix. Further, we explain
how to extend our results to a general probabilistic system.Comment: 21 pages, 2 figures; version 14 July 201
Local Equivalence Problem in Hidden Markov Model
In the hidden Markov process, there is a possibility that two different
transition matrices for hidden and observed variables yield the same stochastic
behavior for the observed variables. Since such two transition matrices cannot
be distinguished, we need to identify them and consider that they are
equivalent, in practice. We address the equivalence problem of hidden Markov
process in a local neighborhood by using the geometrical structure of hidden
Markov process. For this aim, we introduce a mathematical concept to express
Markov process, and formulate its exponential family by using generators. Then,
the above equivalence problem is formulated as the equivalence problem of
generators. Taking this equivalence problem into account, we derive several
concrete parametrizations in several natural cases.Comment: This paper has an overlap with the version 1 of arXiv:1705.06040.
However, this overlap will be removed in the second version of
arXiv:1705.0604