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  "body": "\n## Definition\n\nLet $(X_t)_{t \\ge 0}$ be a real-valued [[stochastic process|0QH5ENKZM1]] and let $H \u003e 0$.\nThe process is *self-similar with Hurst index $H$* if for every $a \u003e 0$ the processes $(X_{at})_{t \\ge 0}$ and $(a^H X_t)_{t \\ge 0}$ have the same [[finite-dimensional distributions|Y0CN7V3B7E]].\nThat is, for all $n \\ge 1$ and $t_1, \\dots, t_n \\ge 0$, the vectors $(X_{a t_1}, \\dots, X_{a t_n})$ and $(a^H X_{t_1}, \\dots, a^H X_{t_n})$ have the same law.\n\n## Remarks\n\nThe [[Wiener process|22735FGJV6]] $W$ is self-similar with $H = 1/2$.\nCheck: $(W_{at})_{t \\ge 0}$ and $(\\sqrt{a}\\, W_t)_{t \\ge 0}$ are both centred Gaussian processes with covariance $a \\min(s, t)$, so they have the same finite-dimensional distributions.\nBrownian motion with drift, $X_t = W_t + t$, is not self-similar for any $H$.\nIndeed $E[X_{at}] = at$ and $E[a^H X_t] = a^H t$ force $H = 1$, while $\\operatorname{Var}(X_{at}) = at$ and $\\operatorname{Var}(a^H X_t) = a^{2H} t$ force $H = 1/2$.\nOne often writes $X_{at} \\overset{d}{=} a^H X_t$, where equality in law is meant for the whole process.\nTaking $t = 0$ gives $X_0 \\overset{d}{=} a^H X_0$ for all $a \u003e 0$, which forces $X_0 = 0$ almost surely.\n",
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