From eda12f755d09922c58a1e58c54a636cfb0291388 Mon Sep 17 00:00:00 2001 From: Kees van Kempen Date: Mon, 19 Sep 2022 20:57:12 +0200 Subject: [PATCH] 02: Finish the ugh derivation of the CLT --- Exercise sheet 2/exercise_sheet_02.ipynb | 28 +++++++++++------------- 1 file changed, 13 insertions(+), 15 deletions(-) diff --git a/Exercise sheet 2/exercise_sheet_02.ipynb b/Exercise sheet 2/exercise_sheet_02.ipynb index 430c4de..c924c1c 100644 --- a/Exercise sheet 2/exercise_sheet_02.ipynb +++ b/Exercise sheet 2/exercise_sheet_02.ipynb @@ -708,19 +708,18 @@ "\\begin{align}\n", "\\phi_{Z_n}(t) &= \\left( \\phi_X(cn^{-2/3}t) \\right)^n \\mathbb{E}\\left[ e^{itcn^{1/3}\\mathbb{E}[X])} \\right]\n", " \\\\ &= \\left( 1 + 3 i cn^{-2/3}t - (|cn^{-2/3}t|+i cn^{-2/3}t)\\,\\sqrt{2\\pi|cn^{-2/3}t|} + \\mathcal{O}(t^2) \\right)^n e^{3itcn^{1/3}}\n", + " \\\\ &= \\left( 1 + \\frac{1}{n} \\left[ 3 i cn^{1/3}t - (|ct|+i ct)\\,\\sqrt{2\\pi|ct|} \\right] + \\mathcal{O}(t^2) \\right)^n e^{3itcn^{1/3}}\n", "\\end{align}\n", "\n", - "Taking the limit $n \\to \\infty$, the " - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "id": "4e36d50e", - "metadata": {}, - "outputs": [], - "source": [ - "# TODO: Finish the derivation above and delete this cell." + "Taking the limit $n \\to \\infty$, the first set of parentheses can be rewritten in terms of an exponential using the identity $\\lim_{n\\to\\infty} (1 + \\frac{a}{n})^n = e^{a}$, we find a way to our desired expression.\n", + "\n", + "\\begin{align}\n", + "\\lim_{n\\to\\infty} \\phi_{Z_n}(t) &= \\lim_{n\\to\\infty} \\left( 1 + \\frac{1}{n} \\left[ 3 i cn^{1/3}t - (|ct|+i ct)\\,\\sqrt{2\\pi|ct|} \\right] + \\mathcal{O}(t^2) \\right)^n e^{-3itcn^{1/3}}\n", + " \\\\ &= \\lim_{n\\to\\infty} \\exp{({3 i cn^{1/3}t - (|ct|+i ct)\\,\\sqrt{2\\pi|ct|}})} \\exp{(e^{-3itcn^{1/3}})}\n", + " \\\\ &= \\exp{({-(|ct|+i ct)\\,\\sqrt{2\\pi|ct|}})}\n", + "\\end{align}\n", + "\n", + "This matches $\\phi_S(t) = \\exp\\big(-(|t|+it)\\sqrt{|t|}\\big)$ for $\\sqrt{2\\pi} c^{3/2} = 1 \\implies c = (2\\pi)^{\\frac{-1}{3}}$." ] }, { @@ -746,7 +745,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 11, "id": "b06896e5", "metadata": { "deletable": false, @@ -765,7 +764,7 @@ "outputs": [ { "data": { - "image/png": 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\n", 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\n", 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" ] @@ -789,8 +788,7 @@ "\n", "alpha = 3/2\n", "beta = 1\n", - "# TODO: Find proper c in the derivation above.\n", - "c = np.pi/6\n", + "c = (2*np.pi)**(-1/3)\n", "n = 1000\n", "pdf = lambda x: levy_stable.pdf(x, alpha, beta)\n", "samples = [sample_Zn(alpha, b, c, n) for _ in range(10000)]\n",