Editing Wiener process (section)

=== Complex-valued Wiener process ===
The complex-valued Wiener process may be defined as a complex-valued random process of the form <math>Z_t = X_t + i Y_t</math> where <math>X_t</math> and <math>Y_t</math> are [[Independence (probability theory)|independent]] Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify <math>\R^2</math> with <math>\mathbb C</math>.<ref>{{Citation|title = Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory| journal = IEEE Transactions on Information Theory | pages =  2859–2867  | volume = 55 | issue = 6 | doi = 10.1109/TIT.2009.2018329 | last1 = Navarro-moreno | first1 =  J. | last2 =  Estudillo-martinez | first2 =  M.D | last3 =  Fernandez-alcala | first3 =  R.M. | last4 =  Ruiz-molina | first4 =  J.C. |year = 2009 | s2cid = 5911584 }}</ref>

==== Self-similarity ====
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.

Rotation invariance: for every complex number <math>c</math> such that <math>|c|=1</math> the process <math>c \cdot Z_t</math> is another complex-valued Wiener process.

==== Time change ====
If <math>f</math> is an [[entire function]] then the process <math> f(Z_t) - f(0) </math> is a time-changed complex-valued Wiener process.

'''Example:''' <math> Z_t^2 = \left(X_t^2 - Y_t^2\right) + 2 X_t Y_t i = U_{A(t)} </math> where
<math display="block">A(t) = 4 \int_0^t |Z_s|^2 \, \mathrm{d} s </math>
and <math>U</math> is another complex-valued Wiener process.

In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale <math>2 X_t + i Y_t</math> is not (here <math>X_t</math> and <math>Y_t</math> are independent Wiener processes, as before).