Limit theorems are the backbone of understanding how random variables behave as sample sizes grow. They show us that even with unpredictable individual outcomes, patterns emerge when we look at the big picture.
These theorems help us make sense of real-world data. From predicting election outcomes to estimating financial risks, they give us tools to work with uncertainty and make informed decisions based on large-scale trends.
Limit Theorems
Fundamental Limit Laws
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describes behavior of sample averages as sample size increases
states sample mean converges in probability to expected value
states sample mean converges almost surely to expected value
Applies to independent, identically distributed random variables
establishes convergence of standardized sums to normal distribution
For large samples, distribution of sample mean approximates normal distribution
Applies even when underlying distribution is not normal
Requires finite mean and variance
provides more precise approximation for probabilities of specific values
Refines central limit theorem for discrete distributions
Approximates probability mass function rather than cumulative distribution function
Useful for estimating probabilities of rare events
Advanced Limit Concepts
studies probabilities of rare events in limit distributions
Focuses on tail probabilities that decay exponentially
Cramer's theorem provides exponential bounds for sums of independent random variables
Applications in risk analysis, queueing theory, and statistical physics
quantifies rate of convergence in central limit theorem
Provides upper bound on difference between cumulative distribution functions
Depends on third absolute moment of random variables
Useful for assessing accuracy of for finite samples
Convergence Concepts
Types of Convergence
(weak convergence) occurs when cumulative distribution functions converge
Denoted by XndX as n→∞
Equivalent to convergence of characteristic functions
Does not imply convergence of moments or other properties
measures likelihood of small differences between random variables
Denoted by XnPX as n→∞
For any ϵ>0, P(∣Xn−X∣>ϵ)→0 as n→∞
Stronger than convergence in distribution
(strong convergence) requires convergence with probability 1
Denoted by Xna.s.X as n→∞
Implies P(limn→∞Xn=X)=1
Strongest form of convergence among these three types
Relationships and Applications
: almost sure ⇒ in probability ⇒ in distribution
combines convergence results for sums and products of random variables
If XndX and YnPc, then Xn+YndX+c and XnYndcX
Useful for deriving limit distributions of transformed random variables
extends convergence to continuous functions of random variables
If XndX and g is continuous, then g(Xn)dg(X)
Applies to various types of convergence (in distribution, probability, almost sure)
Approximations
Poisson Approximation Techniques
estimates probabilities for rare events in large samples
Applies to sum of many independent, rare events
Approximates binomial distribution when n is large and p is small
Probability mass function given by P(X=k)=k!e−λλk, where λ=np
justifies Poisson approximation for certain limit processes
As n→∞ and p→0 with np→λ, binomial distribution converges to Poisson
Useful in modeling rare events (radioactive decay, website traffic spikes)
provides bounds on the accuracy of Poisson approximation
Total variation distance between binomial and Poisson distributions bounded by 2(1−e−λ)p
Helps assess when Poisson approximation is appropriate
Other Discrete Approximations
Normal approximation to binomial distribution improves for large n
Uses continuity correction for better accuracy
Applies when np and n(1−p) are both greater than 5
for negative binomial distribution
Useful when number of successes r is large
Approximates waiting time until rth success
for factorials in large discrete distributions
n!≈2πn(en)n
Improves accuracy of calculations involving large factorials (binomial coefficients)