12.3 Types of convergence

Statistical convergence is a key concept in probability theory. It helps us understand how random variables behave as we collect more data or increase sample sizes. There are three main types: convergence in probability, almost sure convergence, and convergence in distribution.

Each type of convergence has unique properties and applications. Understanding their relationships and examples can help us analyze limiting behavior of random variables and make useful approximations in real-world scenarios. This knowledge is crucial for statistical inference and modeling.

Types of Convergence

Types of statistical convergence

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• Convergence in probability occurs when a sequence of random variables $X_1, X_2, \ldots$ converges to a random variable $X$ if for any $\epsilon > 0$, the probability that the absolute difference between $X_n$ and $X$ is greater than $\epsilon$ approaches 0 as $n$ approaches infinity ($\lim_{n \to \infty} P(|X_n - X| > \epsilon) = 0$), denoted as $X_n \xrightarrow{p} X$
• Almost sure convergence (a.s. convergence) is stronger than convergence in probability and happens when a sequence of random variables $X_1, X_2, \ldots$ converges to a random variable $X$ with probability 1 ($P(\lim_{n \to \infty} X_n = X) = 1$), denoted as $X_n \xrightarrow{a.s.} X$
• Convergence in distribution takes place when a sequence of random variables $X_1, X_2, \ldots$ converges to a random variable $X$ if the limit of the cumulative distribution functions of $X_n$ equals the cumulative distribution function of $X$ at all continuity points $x$ of $F_X$ ($\lim_{n \to \infty} F_{X_n}(x) = F_X(x)$), denoted as $X_n \xrightarrow{d} X$

Examples of convergent sequences

• Convergence in probability: Let $X_n \sim \text{Uniform}(0, \frac{1}{n})$. Then, $X_n \xrightarrow{p} 0$ as $n \to \infty$ (uniform distribution with decreasing interval width)
• Almost sure convergence: Let $X_n = \frac{Y_1 + Y_2 + \ldots + Y_n}{n}$, where $Y_1, Y_2, \ldots$ are independent and identically distributed random variables with finite mean $\mu$. By the Strong Law of Large Numbers, $X_n \xrightarrow{a.s.} \mu$ as $n \to \infty$ (sample mean converging to population mean)
• Convergence in distribution: Let $X_n \sim \text{Binomial}(n, p)$. By the Central Limit Theorem, $\frac{X_n - np}{\sqrt{np(1-p)}} \xrightarrow{d} N(0, 1)$ as $n \to \infty$, where $N(0, 1)$ is the standard normal distribution (binomial distribution converging to normal distribution)

Relationships between convergence types

• Almost sure convergence implies convergence in probability ($X_n \xrightarrow{a.s.} X \Rightarrow X_n \xrightarrow{p} X$), but the converse is not true in general
• Convergence in probability does not imply almost sure convergence, as there exist sequences that converge in probability but not almost surely (Bernoulli random variables with $p_n = \frac{1}{n}$)
• Convergence in distribution does not imply convergence in probability or almost sure convergence, as there exist sequences that converge in distribution but not in probability or almost surely (Cauchy random variables)
• Convergence in probability or almost sure convergence implies convergence in distribution ($X_n \xrightarrow{p} X$ or $X_n \xrightarrow{a.s.} X \Rightarrow X_n \xrightarrow{d} X$)

Applications of convergence concepts

• Determine the limiting behavior of a given sequence of random variables by checking if the sequence satisfies the conditions for convergence in probability, almost sure convergence, or convergence in distribution
• Draw conclusions about the limiting behavior using the relationships between different types of convergence:
  1. If a sequence converges almost surely, it also converges in probability and distribution
  2. If a sequence converges in probability, it also converges in distribution
• Approximate the distribution of a random variable using convergence results: If $X_n \xrightarrow{d} X$, the distribution of $X_n$ can be approximated by the distribution of $X$ for large $n$ (Central Limit Theorem)