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  "body": "\n## Summary\n\nA self-similar process is a [[stochastic process|0QH5ENKZM1]] whose law is unchanged when time is scaled by $a$ and space by $a^H$.\nThe number $H$ is the Hurst index.\nThe [[Wiener process|22735FGJV6]] and [[fractional Brownian motion|VC6R7XSKRQ]] are the main examples.\nSelf-similar processes model random signals that look alike at every scale, in physics, hydrology, networks and finance.\n\n## Explanations\n\n### L1 human\n\nZoom into the graph of a self-similar process and rescale the vertical axis in the right way: the picture you get is statistically the same as the original.\nThe vertical rescaling is a power $a^H$ of the horizontal zoom $a$.\nFor Brownian motion $H = 1/2$: watch it for four times as long and its typical size doubles.\nA straight line $t \\mapsto t$ is self-similar with $H = 1$ in a trivial, non-random way.\n",
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