# Collatz conjecture and stationarity of time series

The Collatz conjecture is known to all. Has this question been approached by methods related to statistics? I think of Collatz iterates as a time series, and the question of whether we always get the number 1 in the end then becomes a question related to stationarity of corresponding time series.

So the question is twofold:

i) Have such methods ever been applied to Collatz?

ii) Can someone suggest some material related to diagnostics of discrete time series where each element in sequence is defined in terms of recurrence relations?

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arxiv.org/abs/math/0201102 –  S. Carnahan Sep 7 '11 at 8:19
thanks, I will see it and ask in case of doubt. –  Nikhil Bellarykar Sep 7 '11 at 8:44

As to part (ii) of your question, (apologies if you're already aware of this and were looking for something more advanced), but most time series analysis actually is phrased in a recurrence-type model such as $X_t = c + \varepsilon_t + \sum c_i X_{t-i}$, where $c, c_i$ are constants and $\varepsilon_t$ are independent random variables. As a starting point, you can look at e.g. the ARMA model on wikipedia. The standard intro text seems to be Time Series Analysis by JD Hamilton. But Collatz defies these models due to the number-theoretic element of the problem (i.e. it refers to divisibility by 2).