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02: Draft generalized central limit theorem proof for the Perato distribution

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Kees van Kempen
2022-09-19 15:34:37 +02:00
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Exercise sheet 2/exercise_sheet_02.ipynb
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@ -692,7 +692,23 @@
} }
}, },
"source": [ "source": [
"YOUR ANSWER HERE" "\\begin{align}\n",
"\\phi_{Z_n}(t) &= \\mathbb{E}\\left[ e^{itZ_n} \\right]\n",
" \\\\ &= \\mathbb{E}\\left[ e^{itcn^{1/3}(\\bar{X_n} - \\mathbb{E}[X])} \\right]\n",
" \\\\ &= \\mathbb{E}\\left[ e^{itcn^{1/3}(\\frac{1}{n}\\sum_{i=1}^n X_i - \\mathbb{E}[X])} \\right]\n",
" \\\\ &= \\mathbb{E}\\left[ \\left( \\prod_{i=1}^n e^{itcn^{-2/3}X_i} \\right) e^{itcn^{1/3}\\mathbb{E}[X])} \\right]\n",
" \\\\ &= \\left( \\prod_{i=1}^n \\mathbb{E}\\left[ e^{itcn^{-2/3}X_i} \\right] \\right)\\mathbb{E}\\left[ e^{itcn^{1/3}\\mathbb{E}[X])} \\right]\n",
" \\\\ &= \\left( \\prod_{i=1}^n \\phi_X(cn^{-2/3}t) \\right)\\mathbb{E}\\left[ e^{itcn^{1/3}\\mathbb{E}[X])} \\right]\n",
" \\\\ &= \\left( \\phi_X(cn^{-2/3}t) \\right)^n \\mathbb{E}\\left[ e^{itcn^{1/3}\\mathbb{E}[X])} \\right]\n",
"\\end{align}\n",
"where we used the identity for products of indepedent expectation values <https://hef.ru.nl/~tbudd/mct/lectures/probability_random_variables.html#equation-product-expectation>, and the definition of $\\phi_X(t) := \\mathbb{E}\\left[ e^{itX} \\right]$.\n",
"\n",
"Next, we will use the Taylor expansion around $t = 0$ as is given above, and, for the latter exponential, $\\mathbb{E}(X) = 3$ for $\\alpha = 3/2, b = 1$ as given.\n",
"\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",
" \\\\ &= \\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",
"\\end{align}"
] ]
}, },
{ {