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I have some difficulties to really understand what does a partition function say about your data/observations.

For instance, say that we have a price series $P(t)$ on the time interval $[0,T]$ and the log-price $X(t)=lnP(t)-lnP(0)$, hence we have log returns. Partitioning $[0,T]$ into integer $N$ intervals of length $\Delta t$, we can define the following partition function:

\begin{equation} S_q(T,\Delta t)=\sum_{i=0}^{N-1}|X(i\Delta t+\Delta t)-X(i\Delta t)|^q \end{equation}

where $q$ is a moment.

So what does $S_q(T,\Delta t)$ say about my log returns? What is the use to know the partition function of my data? I really have a difficulty to grasp the significance of the use of partition functions. When we compare different partition function with different $q$, what does it say?

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