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  "body": "\n## Summary\n\nFractional Brownian motion is a continuous centred [[Gaussian process|22WEM6RTW0]] with stationary increments that is [[self-similar|RASDTDEKP7]] with a Hurst index $H \\in (0, 1)$.\nFor $H = 1/2$ it is the [[Wiener process|22735FGJV6]].\nFor $H \\ne 1/2$ its increments are correlated.\nFor $H \u003e 1/2$ this gives long memory, which is used to model data in hydrology, finance and network traffic.\n\n## Explanations\n\n### L1 human\n\nOrdinary Brownian motion forgets its past: each new step is independent of the earlier ones.\nFractional Brownian motion has a memory controlled by one number $H$ between $0$ and $1$.\nFor $H \u003e 1/2$, a rise tends to be followed by a rise, so paths look smoother and trend more.\nFor $H \u003c 1/2$, a rise tends to be followed by a fall, so paths look rougher and more jagged.\n",
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