lln_clt.ipynb

{
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# LLN and CLT\n",
    "\n",
    "\n",
    "<a id='index-3'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contents\n",
    "\n",
    "- [LLN and CLT](#LLN-and-CLT)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [Relationships](#Relationships)  \n",
    "  - [LLN](#LLN)  \n",
    "  - [CLT](#CLT)  \n",
    "  - [Exercises](#Exercises)  \n",
    "  - [Solutions](#Solutions)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "This lecture illustrates two of the most important theorems of probability and statistics: The\n",
    "law of large numbers (LLN) and the central limit theorem (CLT)\n",
    "\n",
    "These beautiful theorems lie behind many of the most fundamental results in econometrics and quantitative economic modeling\n",
    "\n",
    "The lecture is based around simulations that show the LLN and CLT in action\n",
    "\n",
    "We also demonstrate how the LLN and CLT break down when the assumptions they are based on do not hold\n",
    "\n",
    "In addition, we examine several useful extensions of the classical theorems, such as\n",
    "\n",
    "- The delta method, for smooth functions of random variables  \n",
    "- The multivariate case  \n",
    "\n",
    "\n",
    "Some of these extensions are presented as exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Relationships\n",
    "\n",
    "The CLT refines the LLN\n",
    "\n",
    "The LLN gives conditions under which sample moments converge to population moments as sample size increases\n",
    "\n",
    "The CLT provides information about the rate at which sample moments converge to population moments as sample size increases\n",
    "\n",
    "\n",
    "<a id='lln-mr'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## LLN\n",
    "\n",
    "\n",
    "<a id='index-4'></a>\n",
    "We begin with the law of large numbers, which tells us when sample averages\n",
    "will converge to their population means\n",
    "\n",
    "\n",
    "<a id='lln-ksl'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Classical LLN\n",
    "\n",
    "The classical law of large numbers concerns independent and\n",
    "identically distributed (IID) random variables\n",
    "\n",
    "Here is the strongest version of the classical LLN, known as *Kolmogorov’s strong law*\n",
    "\n",
    "Let $ X_1, \\ldots, X_n $ be independent and identically\n",
    "distributed scalar random variables, with common distribution $ F $\n",
    "\n",
    "When it exists, let $ \\mu $ denote the common mean of this sample:\n",
    "\n",
    "$$\n",
    "\\mu := \\mathbb E X = \\int x F(dx)\n",
    "$$\n",
    "\n",
    "In addition, let\n",
    "\n",
    "$$\n",
    "\\bar X_n := \\frac{1}{n} \\sum_{i=1}^n X_i\n",
    "$$\n",
    "\n",
    "Kolmogorov’s strong law states that, if $ \\mathbb E |X| $ is finite, then\n",
    "\n",
    "\n",
    "<a id='equation-lln-as'></a>\n",
    "$$\n",
    "\\mathbb P \\left\\{ \\bar X_n \\to \\mu \\text{ as } n \\to \\infty \\right\\} = 1 \\tag{1}\n",
    "$$\n",
    "\n",
    "What does this last expression mean?\n",
    "\n",
    "Let’s think about it from a simulation perspective, imagining for a moment that\n",
    "our computer can generate perfect random samples (which of course [it can’t](https://en.wikipedia.org/wiki/Pseudorandom_number_generator))\n",
    "\n",
    "Let’s also imagine that we can generate infinite sequences, so that the\n",
    "statement $ \\bar X_n \\to \\mu $ can be evaluated\n",
    "\n",
    "In this setting, [(1)](#equation-lln-as) should be interpreted as meaning that the\n",
    "probability of the computer producing a sequence where $ \\bar X_n \\to \\mu $ fails to occur\n",
    "is zero"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proof\n",
    "\n",
    "\n",
    "<a id='index-5'></a>\n",
    "The proof of Kolmogorov’s strong law is nontrivial – see, for example, theorem 8.3.5 of [[Dud02]](https://lectures.quantecon.org/py/zreferences.html#dudley2002)\n",
    "\n",
    "On the other hand, we can prove a weaker version of the LLN very easily and\n",
    "still get most of the intuition\n",
    "\n",
    "The version we prove is as follows: If $ X_1, \\ldots, X_n $ is IID with $ \\mathbb E X_i^2 < \\infty $,\n",
    "then, for any $ \\epsilon > 0 $, we have\n",
    "\n",
    "\n",
    "<a id='equation-lln-ip'></a>\n",
    "$$\n",
    "\\mathbb P \\left\\{ | \\bar X_n - \\mu | \\geq \\epsilon \\right\\} \\to 0\n",
    "\\quad \\text{as} \\quad\n",
    "n \\to \\infty \\tag{2}\n",
    "$$\n",
    "\n",
    "(This version is weaker because we claim only [convergence in probability](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_probability) rather than [almost sure convergence](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Almost_sure_convergence), and assume a finite second moment)\n",
    "\n",
    "To see that this is so, fix $ \\epsilon > 0 $, and let $ \\sigma^2 $ be the variance of each $ X_i $\n",
    "\n",
    "Recall the [Chebyshev inequality](https://en.wikipedia.org/wiki/Chebyshev%27s_inequality), which tells us that\n",
    "\n",
    "\n",
    "<a id='equation-lln-cheb'></a>\n",
    "$$\n",
    "\\mathbb P \\left\\{ | \\bar X_n - \\mu | \\geq \\epsilon \\right\\}\n",
    "\\leq \\frac{\\mathbb E [ (\\bar X_n - \\mu)^2]}{\\epsilon^2} \\tag{3}\n",
    "$$\n",
    "\n",
    "Now observe that\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\mathbb E [ (\\bar X_n - \\mu)^2 ]\n",
    "    & = \\mathbb E \\left\\{ \\left[\n",
    "    \\frac{1}{n} \\sum_{i=1}^n (X_i - \\mu)\n",
    "    \\right]^2 \\right\\}\n",
    "    \\\\\n",
    "    & = \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n \\mathbb E (X_i - \\mu)(X_j - \\mu) \\nonumber\n",
    "    \\\\\n",
    "    & = \\frac{1}{n^2} \\sum_{i=1}^n \\mathbb E (X_i - \\mu)^2 \\nonumber\n",
    "    \\\\\n",
    "    & = \\frac{\\sigma^2}{n} \\nonumber\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Here the crucial step is at the third equality, which follows from\n",
    "independence\n",
    "\n",
    "Independence means that if $ i \\not= j $, then the covariance term $ \\mathbb E (X_i - \\mu)(X_j - \\mu) $ drops out\n",
    "\n",
    "As a result, $ n^2 - n $ terms vanish, leading us to a final expression that goes to zero in $ n $\n",
    "\n",
    "Combining our last result with [(3)](#equation-lln-cheb), we come to the estimate\n",
    "\n",
    "\n",
    "<a id='equation-lln-cheb2'></a>\n",
    "$$\n",
    "\\mathbb P \\left\\{ | \\bar X_n - \\mu | \\geq \\epsilon \\right\\}\n",
    "\\leq \\frac{\\sigma^2}{n \\epsilon^2} \\tag{4}\n",
    "$$\n",
    "\n",
    "The claim in [(2)](#equation-lln-ip) is now clear\n",
    "\n",
    "Of course, if the sequence $ X_1, \\ldots, X_n $ is correlated, then the cross-product terms\n",
    "$ \\mathbb E (X_i - \\mu)(X_j - \\mu) $ are not necessarily zero\n",
    "\n",
    "While this doesn’t mean that the same line of argument is impossible, it does mean\n",
    "that if we want a similar result then the covariances should be “almost zero”\n",
    "for “most” of these terms\n",
    "\n",
    "In a long sequence, this would be true if, for example, $ \\mathbb E (X_i - \\mu)(X_j - \\mu) $\n",
    "approached zero when the difference between $ i $ and $ j $ became\n",
    "large\n",
    "\n",
    "In other words, the LLN can still work if the sequence $ X_1, \\ldots, X_n $ has a kind of “asymptotic independence”, in the sense that correlation falls to zero as variables become further apart in the sequence\n",
    "\n",
    "This idea is very important in time series analysis, and we’ll come across it again soon enough"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Illustration\n",
    "\n",
    "\n",
    "<a id='index-6'></a>\n",
    "Let’s now illustrate the classical IID law of large numbers using simulation\n",
    "\n",
    "In particular, we aim to generate some sequences of IID random variables and plot the evolution\n",
    "of $ \\bar X_n $ as $ n $ increases\n",
    "\n",
    "Below is a figure that does just this (as usual, you can click on it to expand it)\n",
    "\n",
    "It shows IID observations from three different distributions and plots $ \\bar X_n $ against $ n $ in each case\n",
    "\n",
    "The dots represent the underlying observations $ X_i $ for $ i = 1, \\ldots, 100 $\n",
    "\n",
    "In each of the three cases, convergence of $ \\bar X_n $ to $ \\mu $ occurs as predicted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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      "text/plain": [
       "<Figure size 1440x1440 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import random\n",
    "import numpy as np\n",
    "from scipy.stats import t, beta, lognorm, expon, gamma, poisson\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "n = 100\n",
    "\n",
    "# == Arbitrary collection of distributions == #\n",
    "distributions = {\"student's t with 10 degrees of freedom\": t(10),\n",
    "                 \"β(2, 2)\": beta(2, 2),\n",
    "                 \"lognormal LN(0, 1/2)\": lognorm(0.5),\n",
    "                 \"γ(5, 1/2)\": gamma(5, scale=2),\n",
    "                 \"poisson(4)\": poisson(4),\n",
    "                 \"exponential with λ = 1\": expon(1)}\n",
    "\n",
    "# == Create a figure and some axes == #\n",
    "num_plots = 3\n",
    "fig, axes = plt.subplots(num_plots, 1, figsize=(20, 20))\n",
    "\n",
    "# == Set some plotting parameters to improve layout == #\n",
    "bbox = (0., 1.02, 1., .102)\n",
    "legend_args = {'ncol': 2,\n",
    "               'bbox_to_anchor': bbox,\n",
    "               'loc': 3,\n",
    "               'mode': 'expand'}\n",
    "plt.subplots_adjust(hspace=0.5)\n",
    "\n",
    "for ax in axes:\n",
    "    # == Choose a randomly selected distribution == #\n",
    "    name = random.choice(list(distributions.keys()))\n",
    "    distribution = distributions.pop(name)\n",
    "\n",
    "    # == Generate n draws from the distribution == #\n",
    "    data = distribution.rvs(n)\n",
    "\n",
    "    # == Compute sample mean at each n == #\n",
    "    sample_mean = np.empty(n)\n",
    "    for i in range(n):\n",
    "        sample_mean[i] = np.mean(data[:i+1])\n",
    "\n",
    "    # == Plot == #\n",
    "    ax.plot(list(range(n)), data, 'o', color='grey', alpha=0.5)\n",
    "    axlabel = '$\\\\bar X_n$ for $X_i \\sim$' + name\n",
    "    ax.plot(list(range(n)), sample_mean, 'g-', lw=3, alpha=0.6, label=axlabel)\n",
    "    m = distribution.mean()\n",
    "    ax.plot(list(range(n)), [m] * n, 'k--', lw=1.5, label='$\\mu$')\n",
    "    ax.vlines(list(range(n)), m, data, lw=0.2)\n",
    "    ax.legend(**legend_args)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The three distributions are chosen at random from a selection stored in the dictionary `distributions`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Infinite Mean\n",
    "\n",
    "What happens if the condition $ \\mathbb E | X | < \\infty $ in the statement of the LLN is not satisfied?\n",
    "\n",
    "This might be the case if the underlying distribution is heavy tailed — the best\n",
    "known example is the Cauchy distribution, which has density\n",
    "\n",
    "$$\n",
    "f(x) = \\frac{1}{\\pi (1 + x^2)} \\qquad (x \\in \\mathbb R)\n",
    "$$\n",
    "\n",
    "The next figure shows 100 independent draws from this distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import cauchy\n",
    "\n",
    "n = 100\n",
    "distribution = cauchy()\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "data = distribution.rvs(n)\n",
    "\n",
    "ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5)\n",
    "ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
    "ax.set_title(f\"{n} observations from the Cauchy distribution\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how extreme observations are far more prevalent here than the previous figure\n",
    "\n",
    "Let’s now have a look at the behavior of the sample mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 1000\n",
    "distribution = cauchy()\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "data = distribution.rvs(n)\n",
    "\n",
    "# == Compute sample mean at each n == #\n",
    "sample_mean = np.empty(n)\n",
    "\n",
    "for i in range(1, n):\n",
    "    sample_mean[i] = np.mean(data[:i])\n",
    "\n",
    "# == Plot == #\n",
    "ax.plot(list(range(n)), sample_mean, 'r-', lw=3, alpha=0.6,\n",
    "        label='$\\\\bar X_n$')\n",
    "ax.plot(list(range(n)), [0] * n, 'k--', lw=0.5)\n",
    "ax.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we’ve increased $ n $ to 1000, but the sequence still shows no sign\n",
    "of converging\n",
    "\n",
    "Will convergence become visible if we take $ n $ even larger?\n",
    "\n",
    "The answer is no\n",
    "\n",
    "To see this, recall that the [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29) of the Cauchy distribution is\n",
    "\n",
    "\n",
    "<a id='equation-lln-cch'></a>\n",
    "$$\n",
    "\\phi(t) = \\mathbb E e^{itX} = \\int e^{i t x} f(x) dx = e^{-|t|} \\tag{5}\n",
    "$$\n",
    "\n",
    "Using independence, the characteristic function of the sample mean becomes\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    \\mathbb E e^{i t \\bar X_n }\n",
    "    & = \\mathbb E \\exp \\left\\{ i \\frac{t}{n} \\sum_{j=1}^n X_j \\right\\}\n",
    "    \\\\\n",
    "    & = \\mathbb E \\prod_{j=1}^n \\exp \\left\\{ i \\frac{t}{n} X_j \\right\\}\n",
    "    \\\\\n",
    "    & = \\prod_{j=1}^n \\mathbb E \\exp \\left\\{ i \\frac{t}{n} X_j \\right\\}\n",
    "    = [\\phi(t/n)]^n\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "In view of [(5)](#equation-lln-cch), this is just $ e^{-|t|} $\n",
    "\n",
    "Thus, in the case of the Cauchy distribution, the sample mean itself has the very same Cauchy distribution, regardless of $ n $\n",
    "\n",
    "In particular, the sequence $ \\bar X_n $ does not converge to a point"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CLT\n",
    "\n",
    "\n",
    "<a id='index-7'></a>\n",
    "Next we turn to the central limit theorem, which tells us about the distribution of the deviation between sample averages and population means"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Statement of the Theorem\n",
    "\n",
    "The central limit theorem is one of the most remarkable results in all of mathematics\n",
    "\n",
    "In the classical IID setting, it tells us the following:\n",
    "\n",
    "\n",
    "<a id='statement-clt'></a>\n",
    "If the sequence $ X_1, \\ldots, X_n $ is IID, with common mean\n",
    "$ \\mu $ and common variance $ \\sigma^2 \\in (0, \\infty) $, then\n",
    "\n",
    "\n",
    "<a id='equation-lln-clt'></a>\n",
    "$$\n",
    "\\sqrt{n} ( \\bar X_n - \\mu ) \\stackrel { d } {\\to} N(0, \\sigma^2)\n",
    "\\quad \\text{as} \\quad\n",
    "n \\to \\infty \\tag{6}\n",
    "$$\n",
    "\n",
    "Here $ \\stackrel { d } {\\to} N(0, \\sigma^2) $ indicates [convergence in distribution](https://en.wikipedia.org/wiki/Convergence_of_random_variables#Convergence_in_distribution) to a centered (i.e, zero mean) normal with standard deviation $ \\sigma $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Intuition\n",
    "\n",
    "\n",
    "<a id='index-8'></a>\n",
    "The striking implication of the CLT is that for **any** distribution with\n",
    "finite second moment, the simple operation of adding independent\n",
    "copies **always** leads to a Gaussian curve\n",
    "\n",
    "A relatively simple proof of the central limit theorem can be obtained by\n",
    "working with characteristic functions (see, e.g., theorem 9.5.6 of [[Dud02]](https://lectures.quantecon.org/py/zreferences.html#dudley2002))\n",
    "\n",
    "The proof is elegant but almost anticlimactic, and it provides surprisingly little intuition\n",
    "\n",
    "In fact all of the proofs of the CLT that we know are similar in this respect\n",
    "\n",
    "Why does adding independent copies produce a bell-shaped distribution?\n",
    "\n",
    "Part of the answer can be obtained by investigating addition of independent Bernoulli\n",
    "random variables\n",
    "\n",
    "In particular, let $ X_i $ be binary, with $ \\mathbb P\\{X_i = 0\\} = \\mathbb P\\{X_i =\n",
    "1 \\} = 0.5 $, and let $ X_1, \\ldots, X_n $ be independent\n",
    "\n",
    "Think of $ X_i = 1 $ as a “success”, so that $ Y_n = \\sum_{i=1}^n X_i $ is the number of successes in $ n $ trials\n",
    "\n",
    "The next figure plots the probability mass function of $ Y_n $ for $ n = 1, 2, 4, 8 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import binom\n",
    "\n",
    "fig, axes = plt.subplots(2, 2, figsize=(10, 6))\n",
    "plt.subplots_adjust(hspace=0.4)\n",
    "axes = axes.flatten()\n",
    "ns = [1, 2, 4, 8]\n",
    "dom = list(range(9))\n",
    "\n",
    "for ax, n in zip(axes, ns):\n",
    "    b = binom(n, 0.5)\n",
    "    ax.bar(dom, b.pmf(dom), alpha=0.6, align='center')\n",
    "    ax.set(xlim=(-0.5, 8.5), ylim=(0, 0.55),\n",
    "           xticks=list(range(9)), yticks=(0, 0.2, 0.4),\n",
    "           title=f'$n = {n}$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $ n = 1 $, the distribution is flat — one success or no successes\n",
    "have the same probability\n",
    "\n",
    "When $ n = 2 $ we can either have 0, 1 or 2 successes\n",
    "\n",
    "Notice the peak in probability mass at the mid-point $ k=1 $\n",
    "\n",
    "The reason is that there are more ways to get 1 success (“fail then succeed”\n",
    "or “succeed then fail”) than to get zero or two successes\n",
    "\n",
    "Moreover, the two trials are independent, so the outcomes “fail then succeed” and “succeed then\n",
    "fail” are just as likely as the outcomes “fail then fail” and “succeed then succeed”\n",
    "\n",
    "(If there was positive correlation, say, then “succeed then fail” would be less likely than “succeed then succeed”)\n",
    "\n",
    "Here, already we have the essence of the CLT: addition under independence leads probability mass to pile up in the middle and thin out at the tails\n",
    "\n",
    "For $ n = 4 $ and $ n = 8 $ we again get a peak at the “middle” value (halfway between the minimum and the maximum possible value)\n",
    "\n",
    "The intuition is the same — there are simply more ways to get these middle outcomes\n",
    "\n",
    "If we continue, the bell-shaped curve becomes ever more pronounced\n",
    "\n",
    "We are witnessing the [binomial approximation of the normal distribution](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulation 1\n",
    "\n",
    "Since the CLT seems almost magical, running simulations that verify its implications is one good way to build intuition\n",
    "\n",
    "To this end, we now perform the following simulation\n",
    "\n",
    "1. Choose an arbitrary distribution $ F $ for the underlying observations $ X_i $  \n",
    "1. Generate independent draws of $ Y_n := \\sqrt{n} ( \\bar X_n - \\mu ) $  \n",
    "1. Use these draws to compute some measure of their distribution — such as a histogram  \n",
    "1. Compare the latter to $ N(0, \\sigma^2) $  \n",
    "\n",
    "\n",
    "Here’s some code that does exactly this for the exponential distribution\n",
    "$ F(x) = 1 - e^{- \\lambda x} $\n",
    "\n",
    "(Please experiment with other choices of $ F $, but remember that, to conform with the conditions of the CLT, the distribution must have finite second moment)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "\n",
    "\n",
    "# == Set parameters == #\n",
    "n = 250                  # Choice of n\n",
    "k = 100000               # Number of draws of Y_n\n",
    "distribution = expon(2)  # Exponential distribution, λ = 1/2\n",
    "μ, s = distribution.mean(), distribution.std()\n",
    "\n",
    "# == Draw underlying RVs. Each row contains a draw of X_1,..,X_n == #\n",
    "data = distribution.rvs((k, n))\n",
    "# == Compute mean of each row, producing k draws of \\bar X_n == #\n",
    "sample_means = data.mean(axis=1)\n",
    "# == Generate observations of Y_n == #\n",
    "Y = np.sqrt(n) * (sample_means - μ)\n",
    "\n",
    "# == Plot == #\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "xmin, xmax = -3 * s, 3 * s\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.hist(Y, bins=60, alpha=0.5, density=True)\n",
    "xgrid = np.linspace(xmin, xmax, 200)\n",
    "ax.plot(xgrid, norm.pdf(xgrid, scale=s), 'k-', lw=2, label='$N(0, \\sigma^2)$')\n",
    "ax.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice the absence of for loops — every operation is vectorized, meaning that the major calculations are all shifted to highly optimized C code\n",
    "\n",
    "The fit to the normal density is already tight, and can be further improved by increasing `n`\n",
    "\n",
    "You can also experiment with other specifications of $ F $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulation 2\n",
    "\n",
    "Our next simulation is somewhat like the first, except that we aim to track the distribution of $ Y_n := \\sqrt{n} ( \\bar X_n - \\mu ) $ as $ n $ increases\n",
    "\n",
    "In the simulation we’ll be working with random variables having $ \\mu = 0 $\n",
    "\n",
    "Thus, when $ n=1 $, we have $ Y_1 = X_1 $, so the first distribution is just\n",
    "the distribution of the underlying random variable\n",
    "\n",
    "For $ n=2 $, the distribution of $ Y_2 $ is that of $ (X_1 + X_2) / \\sqrt{2} $, and so on\n",
    "\n",
    "What we expect is that, regardless of the distribution of the underlying\n",
    "random variable, the distribution of $ Y_n $ will smooth out into a bell\n",
    "shaped curve\n",
    "\n",
    "The next figure shows this process for $ X_i \\sim f $, where $ f $ was\n",
    "specified as the convex combination of three different beta densities\n",
    "\n",
    "(Taking a convex combination is an easy way to produce an irregular shape for $ f $)\n",
    "\n",
    "In the figure, the closest density is that of $ Y_1 $, while the furthest is that of\n",
    "$ Y_5 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import gaussian_kde\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.collections import PolyCollection\n",
    "\n",
    "\n",
    "beta_dist = beta(2, 2)\n",
    "\n",
    "def gen_x_draws(k):\n",
    "    \"\"\"\n",
    "    Returns a flat array containing k independent draws from the\n",
    "    distribution of X, the underlying random variable.  This distribution is\n",
    "    itself a convex combination of three beta distributions.\n",
    "    \"\"\"\n",
    "    bdraws = beta_dist.rvs((3, k))\n",
    "    # == Transform rows, so each represents a different distribution == #\n",
    "    bdraws[0, :] -= 0.5\n",
    "    bdraws[1, :] += 0.6\n",
    "    bdraws[2, :] -= 1.1\n",
    "    # == Set X[i] = bdraws[j, i], where j is a random draw from {0, 1, 2} == #\n",
    "    js = np.random.randint(0, 2, size=k)\n",
    "    X = bdraws[js, np.arange(k)]\n",
    "    # == Rescale, so that the random variable is zero mean == #\n",
    "    m, sigma = X.mean(), X.std()\n",
    "    return (X - m) / sigma\n",
    "\n",
    "nmax = 5\n",
    "reps = 100000\n",
    "ns = list(range(1, nmax + 1))\n",
    "\n",
    "# == Form a matrix Z such that each column is reps independent draws of X == #\n",
    "Z = np.empty((reps, nmax))\n",
    "for i in range(nmax):\n",
    "    Z[:, i] = gen_x_draws(reps)\n",
    "# == Take cumulative sum across columns\n",
    "S = Z.cumsum(axis=1)\n",
    "# == Multiply j-th column by sqrt j == #\n",
    "Y = (1 / np.sqrt(ns)) * S\n",
    "\n",
    "# == Plot == #\n",
    "\n",
    "fig = plt.figure(figsize = (10, 6))\n",
    "ax = fig.gca(projection='3d')\n",
    "\n",
    "a, b = -3, 3\n",
    "gs = 100\n",
    "xs = np.linspace(a, b, gs)\n",
    "\n",
    "# == Build verts == #\n",
    "greys = np.linspace(0.3, 0.7, nmax)\n",
    "verts = []\n",
    "for n in ns:\n",
    "    density = gaussian_kde(Y[:, n-1])\n",
    "    ys = density(xs)\n",
    "    verts.append(list(zip(xs, ys)))\n",
    "\n",
    "poly = PolyCollection(verts, facecolors=[str(g) for g in greys])\n",
    "poly.set_alpha(0.85)\n",
    "ax.add_collection3d(poly, zs=ns, zdir='x')\n",
    "\n",
    "ax.set(xlim3d=(1, nmax), xticks=(ns), ylabel='$Y_n$', zlabel='$p(y_n)$',\n",
    "       xlabel=(\"n\"), yticks=((-3, 0, 3)), ylim3d=(a, b),\n",
    "       zlim3d=(0, 0.4), zticks=((0.2, 0.4)))\n",
    "ax.invert_xaxis()\n",
    "ax.view_init(30, 45)  # Rotates the plot 30 deg on z axis and 45 deg on x axis\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the distribution smooths out into a bell curve as $ n $\n",
    "increases\n",
    "\n",
    "We leave you to investigate its contents if you wish to know more\n",
    "\n",
    "If you run the file from the ordinary IPython shell, the figure should pop up in a\n",
    "window that you can rotate with your mouse, giving different views on the\n",
    "density sequence\n",
    "\n",
    "\n",
    "<a id='multivariate-clt'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Multivariate Case\n",
    "\n",
    "\n",
    "<a id='index-10'></a>\n",
    "The law of large numbers and central limit theorem work just as nicely in multidimensional settings\n",
    "\n",
    "To state the results, let’s recall some elementary facts about random vectors\n",
    "\n",
    "A random vector $ \\mathbf X $ is just a sequence of $ k $ random variables $ (X_1, \\ldots, X_k) $\n",
    "\n",
    "Each realization of $ \\mathbf X $ is an element of $ \\mathbb R^k $\n",
    "\n",
    "A collection of random vectors $ \\mathbf X_1, \\ldots, \\mathbf X_n $ is called independent if, given any $ n $ vectors $ \\mathbf x_1, \\ldots, \\mathbf x_n $ in $ \\mathbb R^k $, we have\n",
    "\n",
    "$$\n",
    "\\mathbb P\\{\\mathbf X_1 \\leq \\mathbf x_1,\\ldots, \\mathbf X_n \\leq \\mathbf x_n \\}\n",
    "= \\mathbb P\\{\\mathbf X_1 \\leq \\mathbf x_1 \\}\n",
    "\\times \\cdots \\times \\mathbb P\\{ \\mathbf X_n \\leq \\mathbf x_n \\}\n",
    "$$\n",
    "\n",
    "(The vector inequality $ \\mathbf X \\leq \\mathbf x $ means that $ X_j \\leq x_j $ for $ j = 1,\\ldots,k $)\n",
    "\n",
    "Let $ \\mu_j := \\mathbb E [X_j] $ for all $ j =1,\\ldots,k $\n",
    "\n",
    "The expectation $ \\mathbb E [\\mathbf X] $ of $ \\mathbf X $ is defined to be the vector of expectations:\n",
    "\n",
    "$$\n",
    "\\mathbb E [\\mathbf X] :=\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "    \\mathbb E [X_1] \\\\\n",
    "    \\mathbb E [X_2] \\\\\n",
    "    \\vdots \\\\\n",
    "    \\mathbb E [X_k]\n",
    "\\end{array}\n",
    "\\right) =\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "    \\mu_1 \\\\\n",
    "    \\mu_2\\\\\n",
    "    \\vdots \\\\\n",
    "    \\mu_k\n",
    "\\end{array}\n",
    "\\right) =: \\boldsymbol \\mu\n",
    "$$\n",
    "\n",
    "The *variance-covariance matrix* of random vector $ \\mathbf X $ is defined as\n",
    "\n",
    "$$\n",
    "\\mathop{\\mathrm{Var}}[\\mathbf X]\n",
    ":= \\mathbb E\n",
    "[ (\\mathbf X - \\boldsymbol \\mu) (\\mathbf X - \\boldsymbol \\mu)']\n",
    "$$\n",
    "\n",
    "Expanding this out, we get\n",
    "\n",
    "$$\n",
    "\\mathop{\\mathrm{Var}}[\\mathbf X] =\n",
    "\\left(\n",
    "\\begin{array}{ccc}\n",
    "    \\mathbb E [(X_1 - \\mu_1)(X_1 - \\mu_1)]\n",
    "        & \\cdots & \\mathbb E [(X_1 - \\mu_1)(X_k - \\mu_k)] \\\\\n",
    "    \\mathbb E [(X_2 - \\mu_2)(X_1 - \\mu_1)]\n",
    "        & \\cdots & \\mathbb E [(X_2 - \\mu_2)(X_k - \\mu_k)] \\\\\n",
    "    \\vdots & \\vdots & \\vdots \\\\\n",
    "    \\mathbb E [(X_k - \\mu_k)(X_1 - \\mu_1)]\n",
    "        & \\cdots & \\mathbb E [(X_k - \\mu_k)(X_k - \\mu_k)] \\\\\n",
    "\\end{array}\n",
    "\\right)\n",
    "$$\n",
    "\n",
    "The $ j,k $-th term is the scalar covariance between $ X_j $ and\n",
    "$ X_k $\n",
    "\n",
    "With this notation we can proceed to the multivariate LLN and CLT\n",
    "\n",
    "Let $ \\mathbf X_1, \\ldots, \\mathbf X_n $ be a sequence of independent and\n",
    "identically distributed random vectors, each one taking values in\n",
    "$ \\mathbb R^k $\n",
    "\n",
    "Let $ \\boldsymbol \\mu $ be the vector $ \\mathbb E [\\mathbf X_i] $, and let $ \\Sigma $\n",
    "be the variance-covariance matrix of $ \\mathbf X_i $\n",
    "\n",
    "Interpreting vector addition and scalar multiplication in the usual way (i.e., pointwise), let\n",
    "\n",
    "$$\n",
    "\\bar{\\mathbf X}_n := \\frac{1}{n} \\sum_{i=1}^n \\mathbf X_i\n",
    "$$\n",
    "\n",
    "In this setting, the LLN tells us that\n",
    "\n",
    "\n",
    "<a id='equation-lln-asmv'></a>\n",
    "$$\n",
    "\\mathbb P \\left\\{ \\bar{\\mathbf X}_n \\to \\boldsymbol \\mu \\text{ as } n \\to \\infty \\right\\} = 1 \\tag{7}\n",
    "$$\n",
    "\n",
    "Here $ \\bar{\\mathbf X}_n \\to \\boldsymbol \\mu $ means that $ \\| \\bar{\\mathbf X}_n - \\boldsymbol \\mu \\| \\to 0 $, where $ \\| \\cdot \\| $ is the standard Euclidean norm\n",
    "\n",
    "The CLT tells us that, provided $ \\Sigma $ is finite,\n",
    "\n",
    "\n",
    "<a id='equation-lln-cltmv'></a>\n",
    "$$\n",
    "\\sqrt{n} ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu ) \\stackrel { d } {\\to} N(\\mathbf 0, \\Sigma)\n",
    "\\quad \\text{as} \\quad\n",
    "n \\to \\infty \\tag{8}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises\n",
    "\n",
    "\n",
    "<a id='lln-ex1'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "One very useful consequence of the central limit theorem is as follows\n",
    "\n",
    "Assume the conditions of the CLT as [stated above](#statement-clt)\n",
    "\n",
    "If $ g \\colon \\mathbb R \\to \\mathbb R $ is differentiable at $ \\mu $ and $ g'(\\mu) \\not= 0 $, then\n",
    "\n",
    "\n",
    "<a id='equation-lln-dm'></a>\n",
    "$$\n",
    "\\sqrt{n} \\{ g(\\bar X_n) - g(\\mu) \\}\n",
    "\\stackrel { d } {\\to} N(0, g'(\\mu)^2 \\sigma^2)\n",
    "\\quad \\text{as} \\quad\n",
    "n \\to \\infty \\tag{9}\n",
    "$$\n",
    "\n",
    "This theorem is used frequently in statistics to obtain the asymptotic distribution of estimators — many of which can be expressed as functions of sample means\n",
    "\n",
    "(These kinds of results are often said to use the “delta method”)\n",
    "\n",
    "The proof is based on a Taylor expansion of $ g $ around the point $ \\mu $\n",
    "\n",
    "Taking the result as given, let the distribution $ F $ of each $ X_i $ be uniform on $ [0, \\pi / 2] $ and let $ g(x) = \\sin(x) $\n",
    "\n",
    "Derive the asymptotic distribution of $ \\sqrt{n} \\{ g(\\bar X_n) - g(\\mu) \\} $ and illustrate convergence in the same spirit as the program `illustrate_clt.py` discussed above\n",
    "\n",
    "What happens when you replace $ [0, \\pi / 2] $ with $ [0, \\pi] $?\n",
    "\n",
    "What is the source of the problem?\n",
    "\n",
    "\n",
    "<a id='lln-ex2'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "\n",
    "Here’s a result that’s often used in developing statistical tests, and is connected to the multivariate central limit theorem\n",
    "\n",
    "If you study econometric theory, you will see this result used again and again\n",
    "\n",
    "Assume the setting of the multivariate CLT [discussed above](#multivariate-clt), so that\n",
    "\n",
    "1. $ \\mathbf X_1, \\ldots, \\mathbf X_n $ is a sequence of IID random vectors, each taking values in $ \\mathbb R^k $  \n",
    "1. $ \\boldsymbol \\mu := \\mathbb E [\\mathbf X_i] $, and $ \\Sigma $ is the variance-covariance matrix of $ \\mathbf X_i $  \n",
    "1. The convergence  \n",
    "\n",
    "\n",
    "\n",
    "<a id='equation-lln-cltmv2'></a>\n",
    "$$\n",
    "\\sqrt{n} ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu ) \\stackrel { d } {\\to} N(\\mathbf 0, \\Sigma) \\tag{10}\n",
    "$$\n",
    "\n",
    "is valid\n",
    "\n",
    "In a statistical setting, one often wants the right hand side to be **standard** normal, so that confidence intervals are easily computed\n",
    "\n",
    "This normalization can be achieved on the basis of three observations\n",
    "\n",
    "First, if $ \\mathbf X $ is a random vector in $ \\mathbb R^k $ and $ \\mathbf A $ is constant and $ k \\times k $, then\n",
    "\n",
    "$$\n",
    "\\mathop{\\mathrm{Var}}[\\mathbf A \\mathbf X]\n",
    "= \\mathbf A \\mathop{\\mathrm{Var}}[\\mathbf X] \\mathbf A'\n",
    "$$\n",
    "\n",
    "Second, by the [continuous mapping theorem](https://en.wikipedia.org/wiki/Continuous_mapping_theorem), if $ \\mathbf Z_n \\stackrel{d}{\\to} \\mathbf Z $ in $ \\mathbb R^k $ and $ \\mathbf A $ is constant and $ k \\times k $, then\n",
    "\n",
    "$$\n",
    "\\mathbf A \\mathbf Z_n\n",
    "\\stackrel{d}{\\to} \\mathbf A \\mathbf Z\n",
    "$$\n",
    "\n",
    "Third, if $ \\mathbf S $ is a $ k \\times k $ symmetric positive definite matrix, then there\n",
    "exists a symmetric positive definite matrix $ \\mathbf Q $, called the inverse\n",
    "[square root](https://en.wikipedia.org/wiki/Square_root_of_a_matrix) of $ \\mathbf S $, such that\n",
    "\n",
    "$$\n",
    "\\mathbf Q \\mathbf S\\mathbf Q' = \\mathbf I\n",
    "$$\n",
    "\n",
    "Here $ \\mathbf I $ is the $ k \\times k $ identity matrix\n",
    "\n",
    "Putting these things together, your first exercise is to show that if\n",
    "$ \\mathbf Q $ is the inverse square root of $ \\mathbf \\Sigma $, then\n",
    "\n",
    "$$\n",
    "\\mathbf Z_n := \\sqrt{n} \\mathbf Q ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu )\n",
    "\\stackrel{d}{\\to}\n",
    "\\mathbf Z \\sim N(\\mathbf 0, \\mathbf I)\n",
    "$$\n",
    "\n",
    "Applying the continuous mapping theorem one more time tells us that\n",
    "\n",
    "$$\n",
    "\\| \\mathbf Z_n \\|^2\n",
    "\\stackrel{d}{\\to}\n",
    "\\| \\mathbf Z \\|^2\n",
    "$$\n",
    "\n",
    "Given the distribution of $ \\mathbf Z $, we conclude that\n",
    "\n",
    "\n",
    "<a id='equation-lln-ctc'></a>\n",
    "$$\n",
    "n \\| \\mathbf Q ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu ) \\|^2\n",
    "\\stackrel{d}{\\to}\n",
    "\\chi^2(k) \\tag{11}\n",
    "$$\n",
    "\n",
    "where $ \\chi^2(k) $ is the chi-squared distribution with $ k $ degrees\n",
    "of freedom\n",
    "\n",
    "(Recall that $ k $ is the dimension of $ \\mathbf X_i $, the underlying random vectors)\n",
    "\n",
    "Your second exercise is to illustrate the convergence in [(11)](#equation-lln-ctc) with a simulation\n",
    "\n",
    "In doing so, let\n",
    "\n",
    "$$\n",
    "\\mathbf X_i :=\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "    W_i \\\\\n",
    "    U_i + W_i\n",
    "\\end{array}\n",
    "\\right)\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- each $ W_i $ is an IID draw from the uniform distribution on $ [-1, 1] $  \n",
    "- each $ U_i $ is an IID draw from the uniform distribution on $ [-2, 2] $  \n",
    "- $ U_i $ and $ W_i $ are independent of each other  \n",
    "\n",
    "\n",
    "Hints:\n",
    "\n",
    "1. `scipy.linalg.sqrtm(A)` computes the square root of `A`.  You still need to invert it  \n",
    "1. You should be able to work out $ \\Sigma $ from the proceding information  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "Here is one solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Illustrates the delta method, a consequence of the central limit theorem.\n",
    "\"\"\"\n",
    "\n",
    "from scipy.stats import uniform\n",
    "\n",
    "\n",
    "# == Set parameters == #\n",
    "n = 250\n",
    "replications = 100000\n",
    "distribution = uniform(loc=0, scale=(np.pi / 2))\n",
    "μ, s = distribution.mean(), distribution.std()\n",
    "\n",
    "g = np.sin\n",
    "g_prime = np.cos\n",
    "\n",
    "# == Generate obs of sqrt{n} (g(X_n) - g(μ)) == #\n",
    "data = distribution.rvs((replications, n))\n",
    "sample_means = data.mean(axis=1)  # Compute mean of each row\n",
    "error_obs = np.sqrt(n) * (g(sample_means) - g(μ))\n",
    "\n",
    "# == Plot == #\n",
    "asymptotic_sd = g_prime(μ) * s\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "xmin = -3 * g_prime(μ) * s\n",
    "xmax = -xmin\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.hist(error_obs, bins=60, alpha=0.5, density=True)\n",
    "xgrid = np.linspace(xmin, xmax, 200)\n",
    "lb = \"$N(0, g'(\\mu)^2  \\sigma^2)$\"\n",
    "ax.plot(xgrid, norm.pdf(xgrid, scale=asymptotic_sd), 'k-', lw=2, label=lb)\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens when you replace $ [0, \\pi / 2] $ with\n",
    "$ [0, \\pi] $?\n",
    "\n",
    "In this case, the mean $ \\mu $ of this distribution is\n",
    "$ \\pi/2 $, and since $ g' = \\cos $, we have $ g'(\\mu) = 0 $\n",
    "\n",
    "Hence the conditions of the delta theorem are not satisfied"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "\n",
    "First we want to verify the claim that\n",
    "\n",
    "$$\n",
    "\\sqrt{n} \\mathbf Q ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu )\n",
    "\\stackrel{d}{\\to}\n",
    "N(\\mathbf 0, \\mathbf I)\n",
    "$$\n",
    "\n",
    "This is straightforward given the facts presented in the exercise\n",
    "\n",
    "Let\n",
    "\n",
    "$$\n",
    "\\mathbf Y_n := \\sqrt{n} ( \\bar{\\mathbf X}_n - \\boldsymbol \\mu )\n",
    "\\quad \\text{and} \\quad\n",
    "\\mathbf Y \\sim N(\\mathbf 0, \\Sigma)\n",
    "$$\n",
    "\n",
    "By the multivariate CLT and the continuous mapping theorem, we have\n",
    "\n",
    "$$\n",
    "\\mathbf Q \\mathbf Y_n\n",
    "\\stackrel{d}{\\to}\n",
    "\\mathbf Q \\mathbf Y\n",
    "$$\n",
    "\n",
    "Since linear combinations of normal random variables are normal, the\n",
    "vector $ \\mathbf Q \\mathbf Y $ is also normal\n",
    "\n",
    "Its mean is clearly $ \\mathbf 0 $, and its variance covariance\n",
    "matrix is\n",
    "\n",
    "$$\n",
    "\\mathrm{Var}[\\mathbf Q \\mathbf Y]\n",
    "= \\mathbf Q \\mathrm{Var}[\\mathbf Y] \\mathbf Q'\n",
    "= \\mathbf Q \\Sigma \\mathbf Q'\n",
    "= \\mathbf I\n",
    "$$\n",
    "\n",
    "In conclusion,\n",
    "$ \\mathbf Q \\mathbf Y_n \\stackrel{d}{\\to} \\mathbf Q \\mathbf Y \\sim N(\\mathbf 0, \\mathbf I) $,\n",
    "which is what we aimed to show\n",
    "\n",
    "Now we turn to the simulation exercise\n",
    "\n",
    "Our solution is as follows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import chi2\n",
    "from scipy.linalg import inv, sqrtm\n",
    "\n",
    "# == Set parameters == #\n",
    "n = 250\n",
    "replications = 50000\n",
    "dw = uniform(loc=-1, scale=2)  # Uniform(-1, 1)\n",
    "du = uniform(loc=-2, scale=4)  # Uniform(-2, 2)\n",
    "sw, su = dw.std(), du.std()\n",
    "vw, vu = sw**2, su**2\n",
    "Σ = ((vw, vw), (vw, vw + vu))\n",
    "Σ = np.array(Σ)\n",
    "\n",
    "# == Compute Σ^{-1/2} == #\n",
    "Q = inv(sqrtm(Σ))\n",
    "\n",
    "# == Generate observations of the normalized sample mean == #\n",
    "error_obs = np.empty((2, replications))\n",
    "for i in range(replications):\n",
    "    # == Generate one sequence of bivariate shocks == #\n",
    "    X = np.empty((2, n))\n",
    "    W = dw.rvs(n)\n",
    "    U = du.rvs(n)\n",
    "    # == Construct the n observations of the random vector == #\n",
    "    X[0, :] = W\n",
    "    X[1, :] = W + U\n",
    "    # == Construct the i-th observation of Y_n == #\n",
    "    error_obs[:, i] = np.sqrt(n) * X.mean(axis=1)\n",
    "\n",
    "# == Premultiply by Q and then take the squared norm == #\n",
    "temp = Q @ error_obs\n",
    "chisq_obs = np.sum(temp**2, axis=0)\n",
    "\n",
    "# == Plot == #\n",
    "fig, ax = plt.subplots(figsize=(10, 6))\n",
    "xmax = 8\n",
    "ax.set_xlim(0, xmax)\n",
    "xgrid = np.linspace(0, xmax, 200)\n",
    "lb = \"Chi-squared with 2 degrees of freedom\"\n",
    "ax.plot(xgrid, chi2.pdf(xgrid, 2), 'k-', lw=2, label=lb)\n",
    "ax.legend()\n",
    "ax.hist(chisq_obs, bins=50, density=True)\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "filename": "lln_clt.rst",
  "kernelspec": {
   "display_name": "Python",
   "language": "python3",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.0"
  },
  "title": "LLN and CLT"
 },
 "nbformat": 4,
 "nbformat_minor": 2
}