Editing Logistic map (section)

====Upper bound when {{math|0 ≤ ''r'' ≤ 1}}====
Although exact solutions to the recurrence relation are only available in a small number of cases, a closed-form upper bound on the logistic map is known when {{math|0 ≤ ''r'' ≤ 1}}.<ref>{{cite arXiv|eprint=1710.05053|class=stat.ML|first1=Trevor|last1=Campbell|first2=Tamara|last2=Broderick|title=Automated scalable Bayesian inference via Hilbert coresets|date=2017}}</ref> There are two aspects of the behavior of the logistic map that should be captured by an upper bound in this regime: the asymptotic geometric decay with constant {{mvar|r}}, and the fast initial decay when {{math|''x''<sub>0</sub>}} is close to 1, driven by the {{math|(1 − ''x<sub>n</sub>'')}} term in the recurrence relation. The following bound captures both of these effects:

<math display="block"> \forall n \in \{0, 1, \ldots \} \quad \text{and} \quad x_0, r \in [0, 1], \quad x_n \le \frac{x_0}{r^{-n} + x_0n}. </math>