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  "body": "\n## Definition\n\nLet $(x_n)_{n \\ge 1}$ be a [[sequence|0QE8STGJH2]] of real or complex numbers and let $L$ be a number.\nThe sequence *converges statistically* to $L$ if for every $\\varepsilon \u003e 0$ the set\n$$\\set{n \\ge 1 : \\abs{x_n - L} \\ge \\varepsilon}$$\nhas [[natural density|RVG9Z8NYA6]] $0$.\n\n## Remarks\n\nEvery [[convergent sequence|0W4CN9E2W5]] with limit $L$ converges statistically to $L$, since each of the sets above is finite and finite sets have density $0$.\nThe statistical limit is unique when it exists.\n\nExample: let $x_n = 1$ if $n$ is a perfect square and $x_n = 0$ otherwise.\nFor $0 \u003c \\varepsilon \\le 1$, the set of $n$ with $\\abs{x_n} \\ge \\varepsilon$ is the set of squares, which has at most $\\sqrt{N}$ elements up to $N$, so its density is $0$.\nHence $(x_n)$ converges statistically to $0$, although it does not converge.\n\nNon-example: $x = (1, 0, 1, 0, \\dots)$ does not converge statistically.\nFor any $L$, one of $\\abs{1 - L}$ and $\\abs{L}$ is at least $1/2$.\nSo for $\\varepsilon = 1/2$ the set above contains all odd or all even $n$, and its density is not $0$.\n\nA sequence converges statistically to $L$ if and only if there is a set $K$ of density $1$ such that $x_n \\to L$ as $n \\to \\infty$ through $K$.\nA bounded sequence converges statistically to $L$ if and only if $\\frac{1}{n} \\sum_{k=1}^{n} \\abs{x_k - L} \\to 0$, which is the condition of [[strong Cesàro summability|750MCA763X]] with exponent $1$ applied to the sequence $(x_n)$.\n",
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