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02: TODO finish the derivation for the CLT

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Kees van Kempen
2022-09-19 15:58:58 +02:00
parent 04c5e86ecd
commit f07d6a69d5
1 changed files with 14 additions and 2 deletions
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16
Exercise sheet 2/exercise_sheet_02.ipynb
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@ -707,8 +707,20 @@
"\n", "\n",
"\\begin{align}\n", "\\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", "\\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|} \\right)^n e^{3itcn^{1/3}}\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",
"\\end{align}" "\\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."
] ]
}, },
{ {