Stochastic ` where we know that ancient history is still relevent for the future predictions, even if we know the recent history.

In my own mental map this is near-synonymous with stateful models but where we ignore the state, where we ignore the state, which I suppose is a kind of coarse graining. In this formulation, we have a Markov process but because we do not observe the whole state it looks non-Markov. This is reasonably consistent with reality, where we believe the current state of reality determines the future, but we don’t know the whole current state.

(Sorta related: hidden variable quantum mechanics.)

But sure, maybe there is an explit long-time dependency in your process which is *not* nediated through a hidden state;
can we statistically distingish these cases? Sometimes, e.g. Küns86, but generally you probably want a model, or how will you do it?

There are some obvious process model which have a long-memeory in this sense, such as stack automata, or Hawkes processes with non-exponential kernels.

Note “long memory” can work with not only time series by but any random field, e.g. spatial, or random fields indexed by any number of dimensions, causal or non-causal.

Questions:

- Can we use the “memory length” of a system to infer the number of hidden states for some class of interesting systems, or vice versa?

- When?

- Can we infer the memory length
*alone*as a parameter of interest in some model classes? (need making precise). Information criteria don’t do this model order selection consistently; what does?

## Reading

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- Berkes, I., Horváth, L., Kokoszka, P., & Shao, Q.-M. (2006) On discriminating between long-range dependence and changes in mean.
*The Annals of Statistics*, 34(3), 1140–1165. DOI. - BrCL98
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*Bernoulli*, 5(2), 209–224. DOI. - DiIn01
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- FGLM07
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*Quantitative Finance*, 6(2), 107–112. DOI. - GiSu99
- Giraitis, L., & Surgailis, D. (1999) Central limit theorem for the empirical process of a linear sequence with long memory.
*Journal of Statistical Planning and Inference*, 80(1–2), 81–93. - Gnei00
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*Journal of Applied Probability*, 37(4), 1104–1109. - GrJo80
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*Journal of Time Series Analysis*, 1(1), 15–29. DOI. - Horv01
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*Journal of Multivariate Analysis*, 78(2), 218–234. - Küns86
- Künsch, H. R.(1986) Discrimination between monotonic trends and long-range dependence.
*Journal of Applied Probability*, 23(4), 1025–1030. - Lahi93
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*The European Physical Journal B*, 75(3), 343–355. DOI.