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  "body": "\n## Definition\n\nLet $d \\ge 1$, let $x_0 \\in \\mathbb{R}^d$, and let $\\xi_1, \\xi_2, \\dots$ be [[i.i.d.|0AJN7WHBJZ]] random vectors in $\\mathbb{R}^d$.\nThe random walk with start $x_0$ and steps $\\xi_i$ is the [[stochastic process|0QH5ENKZM1]] $(S_n)_{n \\ge 0}$ with\n$$S_0 = x_0 \\quad \\text{and} \\quad S_n = x_0 + \\xi_1 + \\dots + \\xi_n \\quad \\text{for } n \\ge 1.$$\nThe simple random walk on $\\mathbb{Z}$ has $d = 1$, $x_0 = 0$ and $P(\\xi_i = 1) = P(\\xi_i = -1) = 1/2$.\n\n## Remarks\n\nFor the simple random walk on $\\mathbb{Z}$, $E[S_n] = 0$ and $\\operatorname{Var}(S_n) = n$. By the central limit theorem, $S_n/\\sqrt{n}$ converges in distribution to $\\mathcal{N}(0, 1)$.\nThe simple random walk on $\\mathbb{Z}^d$ returns to its start with probability $1$ if $d \\le 2$, and with probability less than $1$ if $d \\ge 3$. This is Pólya's theorem.\nA random walk is a Markov chain with transition probabilities that do not change under translation.\nAfter scaling of time and space, the simple random walk converges to the Wiener process.\n\n## Explanations\n\n### L1 human\n\nA random walk is the path of someone who, at each step, chooses a direction at random.\nIt models the price of a share, the motion of a particle in a fluid, and the fortune of a gambler.\n",
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