🎲Intro to Probabilistic Methods Unit 7 – Limit Theorems in Probability

Key Concepts and Definitions

• Probability theory studies the behavior of random variables and their distributions
• Limit theorems describe the asymptotic behavior of sequences of random variables as the sample size approaches infinity
• Convergence refers to the tendency of a sequence of random variables to approach a specific value or distribution as the sample size increases
• Almost sure convergence occurs when a sequence of random variables converges to a value with probability 1
• Convergence in probability happens when the probability of the difference between the sequence and the limit being greater than any positive value approaches 0 as n tends to infinity
• Convergence in distribution takes place when the cumulative distribution functions of a sequence of random variables converge to the cumulative distribution function of a random variable
• Stochastic processes are collections of random variables indexed by time or space (Markov chains, Brownian motion)

Types of Convergence

• Almost sure convergence is the strongest form of convergence
  • Occurs when a sequence of random variables converges to a value with probability 1
  • Denoted as $X_n \xrightarrow{a.s.} X$
• Convergence in probability is weaker than almost sure convergence but stronger than convergence in distribution
  • Happens when $\lim_{n \to \infty} P(|X_n - X| > \epsilon) = 0$ for any $\epsilon > 0$
  • Denoted as $X_n \xrightarrow{p} X$
• Convergence in distribution describes the convergence of the cumulative distribution functions
  • Takes place when $\lim_{n \to \infty} F_{X_n}(x) = F_X(x)$ for all continuity points x of $F_X$
  • Denoted as $X_n \xrightarrow{d} X$
• Convergence in mean square implies convergence in probability
  • Occurs when $\lim_{n \to \infty} E[(X_n - X)^2] = 0$
• Convergence in $L^p$ norm generalizes convergence in mean square to the $p$-th moment
  • Happens when $\lim_{n \to \infty} E[|X_n - X|^p] = 0$ for $p \geq 1$

Law of Large Numbers

• States that the sample mean of a large number of independent and identically distributed (i.i.d.) random variables converges to their expected value
• Weak Law of Large Numbers (WLLN) asserts convergence in probability
  • If $X_1, X_2, \ldots$ are i.i.d. with finite mean $\mu$, then $\bar{X}_n \xrightarrow{p} \mu$ as $n \to \infty$
• Strong Law of Large Numbers (SLLN) asserts almost sure convergence
  • If $X_1, X_2, \ldots$ are i.i.d. with finite mean $\mu$, then $\bar{X}_n \xrightarrow{a.s.} \mu$ as $n \to \infty$
• Provides justification for using sample means to estimate population means
• Holds under weaker conditions than independence (ergodicity, stationarity)

Central Limit Theorem

• Describes the asymptotic distribution of the standardized sum of i.i.d. random variables with finite variance
• If $X_1, X_2, \ldots$ are i.i.d. with mean $\mu$ and variance $\sigma^2$, then $\frac{\sum_{i=1}^n X_i - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0, 1)$ as $n \to \infty$
• Implies that the distribution of the sample mean approaches a normal distribution, regardless of the underlying distribution of the random variables
• Allows for the construction of confidence intervals and hypothesis tests for large sample sizes
• Generalizations exist for non-i.i.d. random variables (Lindeberg-Feller CLT) and multivariate random variables (Multivariate CLT)

Chebyshev's Inequality

• Provides an upper bound on the probability that a random variable deviates from its mean by more than a specified amount
• For a random variable $X$ with mean $\mu$ and variance $\sigma^2$, $P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$ for any $k > 0$
• Useful for obtaining bounds on tail probabilities when the exact distribution is unknown
• Can be used to prove the Weak Law of Large Numbers
• Tighter bounds can be obtained using higher moments (Markov's inequality) or additional information about the distribution (Chernoff bounds)

Applications in Real-World Scenarios

• Finance: Portfolio theory, risk management, and option pricing rely on limit theorems to model asset returns and estimate probabilities of extreme events
• Insurance: Law of Large Numbers justifies the pooling of risks and the use of historical data to estimate future claims
• Quality control: Central Limit Theorem allows for the construction of control charts and hypothesis tests to monitor manufacturing processes
• Polling and surveys: Limit theorems provide a foundation for determining sample sizes and margins of error
• Machine learning: Convergence results are crucial for understanding the behavior of learning algorithms and the consistency of estimators

Common Misconceptions and Pitfalls

• Assuming that the Law of Large Numbers implies convergence of individual random variables rather than their average
• Misinterpreting the Central Limit Theorem as a statement about the distribution of individual random variables instead of their standardized sum
• Applying the Central Limit Theorem to dependent or non-identically distributed random variables without checking the necessary conditions
• Neglecting the assumptions of finite variance or higher moments when using limit theorems
• Overestimating the rate of convergence, especially for heavy-tailed distributions or in the presence of dependence

Practice Problems and Solutions

1. Let $X_1, X_2, \ldots$ be i.i.d. random variables with $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$. Show that $\frac{1}{n} \sum_{i=1}^n X_i \xrightarrow{a.s.} 0$ as $n \to \infty$.

   • Solution: Apply the Strong Law of Large Numbers with $\mu = E[X_i] = 0$.

2. Suppose $X_1, X_2, \ldots$ are i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. Find the asymptotic distribution of $\sqrt{n}(\bar{X}_n - \mu)$ as $n \to \infty$.

   • Solution: By the Central Limit Theorem, $\sqrt{n}(\bar{X}_n - \mu) \xrightarrow{d} N(0, \sigma^2)$.

3. Let $X$ be a random variable with mean $\mu = 2$ and variance $\sigma^2 = 4$. Use Chebyshev's inequality to bound $P(|X - 2| \geq 3)$.

   • Solution: Applying Chebyshev's inequality with $k = \frac{3}{2}$, we have $P(|X - 2| \geq 3) \leq \frac{4}{(\frac{3}{2})^2} = \frac{16}{9}$.

4. Suppose $X_1, X_2, \ldots$ are i.i.d. random variables with $E[X_i] = 1$ and $Var(X_i) = 2$. Find the limit of $P(\bar{X}_n \leq 1.5)$ as $n \to \infty$.

   • Solution: By the Central Limit Theorem, $\sqrt{n}(\bar{X}_n - 1) \xrightarrow{d} N(0, 2)$. Thus, $\lim_{n \to \infty} P(\bar{X}_n \leq 1.5) = P(Z \leq \frac{0.5}{\sqrt{2/n}}) \to P(Z \leq \infty) = 1$, where $Z \sim N(0, 1)$.