Collatz conjecture and stationarity of time series

Question

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?

Answer 1

I don't think statistics alone could be strong enough to resolve the Collatz conjecture, as statistics deals with expected properties of random processes, whereas Collatz concerns the actual properties of a deterministic process. There is a heuristic argument that assuming the iterates in the Collatz sequences are independent and uniformly distributed mod 2, the sequence must be decreasing in expectation. But that is far from a proof.

Terry Tao recently wrote a very nice blog post describing some of the obstacles that make the Collatz conjecture very difficult. It includes a better description of the heuristic argument I mentioned.

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).