Central Limit Theorem

Formula

\[ \frac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma} \xrightarrow[]{\;d\;} \mathcal{N}(0,1) \]

Parameters

• \(X_i\): i.i.d. with mean \(\mu\) and std \(\sigma\)
• \(\bar{X}_n\): sample mean

What it means

Normalized sums of i.i.d. variables converge in distribution to a Gaussian.

What it's used for

• Approximating sums/means with a normal distribution.
• Justifying confidence intervals.

Key properties

• Explains why Gaussian approximations are common
• Many variants exist (Lyapunov, Lindeberg)

Common gotchas

• Convergence in distribution is not the same as convergence in probability.
• Heavy tails can violate conditions.

Example

For 100 fair coin flips, the sample mean is approximately \(\mathcal{N}(0.5, 0.25/100)\).

How to Compute (Pseudocode)

Input: assumptions/quantities required by the theorem or inequality (for example means, variances, sample size)
Output: bound, approximation, or theorem-based diagnostic

verify the theorem/inequality assumptions (at least approximately/in modeling terms)
compute the bound or approximation using the card formula
return the resulting bound/approximation and note its conditions

Complexity

• Time: Usually \(O(1)\) once the required summary quantities are available
• Space: \(O(1)\)
• Assumptions: This is a formula-application workflow; estimating required moments/parameters from data can dominate cost (often \(O(n)\))

See also

• Variance