I stumbled upon what seems like an interesting bifurcation. I wonder if anyone has an explanation for this. Here are the details:
Let $a_n\in\mathbb{R}$, $n \in \mathbb{N}$ be a sequence. Define a new sequence for $n\in\mathbb{N}$:
$$ \sigma_n = \begin{cases} a_1 & \text{if } n = 1 \\\\ \sigma_{n-1}+a_n & \text{if } n > 1 \text{ and } \sigma_{n-1}-a_n<0 \\\\ \sigma_{n-1}-a_n & \text{otherwise} \end{cases} $$
Now lets look at all indexes $n$ such that satisfy the condition $\sigma_{n-1}<\sigma_{n}<\sigma_{n+1}$. Let's look at the case where there are infinitely many such indexes, hence they form a sequence, which we denote by $i_m$. So, $i_1$ is the first index $n$ which satisfies the condition, $i_2$ is the second, and so on.
Next, we look at these three sequences:
$I_m=\frac{i_{m+1}}{i_m}$
$A_m=\frac{a_{i_{m+1}}}{a_{i_m}}$
$\Sigma_m=\frac{\sigma_{i_{m+1}}}{\sigma_{i_m}}$
Each of these sequences is associated with a set of its accumulation points, that is, the set of all finite limits of subsequences.
I checked empirically what are the accumulation points of these three sequences, for two cases: a power law $a_n=n^r$, $r>0$; and an exponential law $a_n=r^n$, $r>1$.
I plot the two cases, as a function of $r$ (Note: the calculations have been done in double precision float format; and accumulation points were found by calculating many sequence elements and then simply plotting the last four elements of the sequence, I understand that this is not always entirely correct, but it seemed to work in many cases):
Assuming numerical errors aren't misleading us, we see interesting empirical properties in the $A$ and $\Sigma$ panels:
The power law case has a (single) limit of 3 for any $r$ provided that $r$ is large enough. (by the way, if I use the prime numbers as $a_n$, I get this same limit of 3 as well).
The exponential law has a (single) limit of $r$, provided that $r$ is at least 2. (I think this is the easiest case to understand).
For small values of $r$, we see multiple accumulation points per $r$, and they jointly form continuous line segments (note: I zoomed in to verify that indeed we see multiple accumulation points for every $r$). The line segments end points have a smooth envelope. In the power law case, we see an "intersection" of different such patterns around $r=0.45$.
The aspects I would like to understand are:
- Where does the universal limit of 3 come from?
- What determines when the "lines" in the plot break, and their envelopes?
- Any other aspect/application.