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  "body": "\n## Definition\n\nLet $X_1, X_2, \\dots$ be [[random variables|0DKRR22AEQ]] observed one at a time, and let $\\mathcal{F}_n = \\sigma(X_1, \\dots, X_n)$ be the information after $n$ observations, with $\\mathcal{F}_0$ trivial.\nLet $D$ be a set of possible decisions, such as \"accept $H_0$\" and \"reject $H_0$\".\nA *sequential procedure* is a pair $(N, \\delta)$ such that:\n\n1. $N$ is a [[stopping time|WND87FBAKC]] with respect to $(\\mathcal{F}_n)_{n \\ge 0}$, called the stopping rule.\n2. $\\delta$ is a terminal decision rule: on the event $\\set{N = n}$, the decision $\\delta$ is a function of $(X_1, \\dots, X_n)$ with values in $D$.\n\n*Sequential analysis* is the study of sequential procedures, in particular of their error probabilities and of the distribution of the sample size $N$.\n\n## Remarks\n\nExample: in a [[test|087P03NEND]] about the mean of $X_i \\in \\set{0, 1}$, the rule \"stop at the first $n$ with $\\abs{\\sum_{i \\le n} (2X_i - 1)} = 5$, and reject $H_0$ if the sum is positive\" is a sequential procedure.\nThe event $\\set{N = n}$ depends only on $X_1, \\dots, X_n$, so $N$ is a stopping time.\nNon-example: \"stop at the $n$ for which $X_n$ is the largest of the first $100$ observations\" is not a sequential procedure.\nDeciding whether $N = n$ needs future observations, so clause 1 fails.\nA procedure with $N = m$ constant is the usual fixed-sample procedure, so fixed-sample statistics is a special case.\n",
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