Editing Random walk (section)

===Information rate===
The [[information rate]] of a Gaussian random walk with respect to the squared error distance, i.e. its quadratic [[rate distortion function]], is given parametrically by<ref>{{Cite journal |doi = 10.1109/TIT.1970.1054423|title = Information rates of Wiener processes|year = 1970|last1 = Berger|first1 = T.|journal = IEEE Transactions on Information Theory|volume = 16|issue = 2|pages = 134–139}}</ref>
<math display="block"> R(D_\theta) = \frac{1}{2} \int_0^1 \max\{0, \log_2\left(S(\varphi)/\theta \right) \} \, d\varphi, </math>
<math display="block"> D_\theta = \int_0^1 \min\{S(\varphi),\theta\} \, d\varphi, </math>
where <math>S(\varphi) = \left(2 \sin (\pi \varphi/2) \right)^{-2}</math>. Therefore, it is impossible to encode <math>{\{Z_n\}_{n=1}^N}</math> using a [[binary code]] of less than <math>NR(D_\theta)</math> [[bit]]s and recover it with expected mean squared error less than <math>D_\theta</math>. On the other hand, for any <math>\varepsilon>0</math>, there exists an <math>N \in \mathbb N</math> large enough and a [[binary code]] of no more than <math>2^{N R(D_{\theta})}</math> distinct elements such that the expected mean squared error in recovering <math>{\{Z_n\}_{n=1}^N}</math> from this code is at most <math>D_\theta - \varepsilon</math>.