# Does the Shannon entropy of stock returns change over time?

Shannon entropy, $$H(X) = -\sum_{i=1}^n p(x) \ln p(x)$$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram.

If we take rolling window segments, or snapshots, of a full-sample time series of stock returns $$X$$, can we expect the Shannon entropy of these windows to consistently increase or decrease as $$t\rightarrow \infty$$? i.e. each window has its own pdf.

or would their entropy merely be a function of the volatility (clustering) in each window, given that the spread of a distribution (volatility) and its randomness (entropy) are inextricably linked?

In thermodynamics, time entropy naturally grows with time, so mustn't there be a connection between the Shannon entropy and time entropy of a stock's returns?

How about the entropy of stock returns using expanding windows instead?

• Just note, in thermodynamics, the $\Delta S \ge 0$, i.e. it grows or stay same. The entropy is same in case of reversible processes. I would expect something similar for stock. For example, for well established companies, $\Delta S \approx 0$. However, as uncertainty on market increases or decreases, also entropy would increases or decreases. Very nice question! – Martin Vesely Oct 19 '20 at 4:54
• This can be of interest for you: quant.stackexchange.com/questions/879/… – Martin Vesely Oct 19 '20 at 4:57
• Last comment to entropy in thermodynamics, it can also decrease but you have to do some work. The decrease in entropy has to be compensanted by its increase somewhere else. – Martin Vesely Oct 19 '20 at 5:01
• This is closely related to whether stock returns are stationary. If they are, the unconditional distribution is not a function of time. Then the true entropy, defined using unconditional distribution should be constant. – fesman Oct 19 '20 at 5:42
• @MartinVesely what is $\Delta S$ in thermodynamics? – develarist Oct 19 '20 at 14:49