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  "body": "\n## Statement\n\nLet $(B_t)_{t \\ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$, and let $(\\mathcal{F}_t)$ be the filtration generated by $B$.\nLet $\\mu$ be a [[probability measure|059TJN0NZ1]] on $\\R$ with $\\int x^2 \\, d\\mu(x) \u003c \\infty$ and $\\int x \\, d\\mu(x) = 0$.\nThen there is a [[stopping time|WND87FBAKC]] $T$ for $(\\mathcal{F}_t)$ such that $B_T$ has law $\\mu$ and\n$$E[T] = \\int_{\\R} x^2 \\, d\\mu(x).$$\n\n## Remarks\n\nFor a law on two points $a \u003c 0 \u003c b$, the exit time of $B$ from $(a, b)$ works.\nThe [[optional stopping theorem|DC9HZT2HVJ]] for the martingales $B_t$ and $B_t^2 - t$ gives the right weights and $E[T] = -ab$.\nThe general case writes $\\mu$ as a mixture of such two-point laws, or builds $T$ by a sequence of exit times from nested intervals.\nApplied repeatedly, the theorem gives stopping times $0 = T_0 \\le T_1 \\le T_2 \\le \\cdots$ with i.i.d. increments such that $(B_{T_n})_{n \\ge 0}$ has the law of a [[random walk|04VZE3FZXF]] with step law $\\mu$.\nThis embedding is one route to [[Donsker's theorem|818832PMSV]] and to limit laws for random walks.\nThe mean zero condition is necessary: if $E[T] \u003c \\infty$, then Wald's identities give $E[B_T] = 0$ and $E[B_T^2] = E[T]$.\n\n## Explanations\n\n### L1 human\n\nAny centred distribution with finite variance can be produced by watching a Brownian path and stopping it at a cleverly chosen moment.\nThe decision to stop uses only the path seen so far.\nFor a fair coin with values $\\pm 1$, one stops the first time the path reaches $1$ or $-1$.\n\n### L2 human\n\nThe theorem lets one transfer questions about sums of independent variables to questions about a single Brownian path.\nA random walk becomes Brownian motion sampled at random times $T_n$, and the strong law of large numbers gives $T_n / n \\to \\int x^2 \\, d\\mu$.\nSo the walk and the path stay close on the scale $\\sqrt{n}$, which yields functional limit theorems.\nMany explicit constructions of $T$ exist, with different extra properties.\n",
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