Editing Lévy process (section)

=== Stationary increments ===
{{Main|Stationary increments}}
To call the increments stationary means that the [[probability distribution]] of any increment ''X''<sub>''t''</sub> − ''X''<sub>''s''</sub> depends only on the length ''t''&nbsp;−&nbsp;''s'' of the time interval; increments on equally long time intervals are identically distributed.

If <math>X</math> is a [[Wiener process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is [[normal distribution|normal]] with [[expected value]] 0 and [[variance]] ''t''&nbsp;−&nbsp;''s''.

If <math>X</math> is a [[Poisson process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is a [[Poisson distribution]] with expected value λ(''t''&nbsp;−&nbsp;''s''), where λ > 0 is the "intensity" or "rate" of the process.

If <math>X</math> is a [[Cauchy process]], the probability distribution of ''X''<sub>''t''</sub>&nbsp;−&nbsp;''X''<sub>''s''</sub> is a [[Cauchy distribution]] with density <math>f(x; t) = { 1 \over \pi } \left[ { \gamma \over x^2 + \gamma^2  } \right] </math> where <math>\gamma=t-s</math>.