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Proof of the Weak Law of Large Numbers (WLLN)

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The Formal Theorem

Let X1,X2,…,Xn X_1, X_2, \dots, X_n be a sequence of independent and identically distributed (i.i.d.) random variables with a finite mean E[Xi]=μ E[X_i] = \mu and a finite variance Var(Xi)=σ2<∞ Var(X_i) = \sigma^2 < \infty . Let the sample average be defined as Xˉn=1n∑i=1nXi \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i . For any arbitrary positive constant ϵ>0 \epsilon > 0 , the following convergence in probability holds:
lim⁡n→∞P(∣Xˉn−μ∣≥ϵ)=0 \lim_{n \to \infty} P(|\bar{X}_n - \mu| \geq \epsilon) = 0

Analytical Intuition.

Visualize a vast, chaotic sea of data points, where each individual observation Xi X_i is a flickering candle in a hurricane of randomness. At small sample sizes, the average Xˉn \bar{X}_n dances wildly, unpredictable and sensitive to every gust of chance. However, the Weak Law of Large Numbers acts as a cosmic dampener. As the number of observations n n scales toward infinity, the collective weight of these independent signals begins to anchor the mean. The variance of the sample average, defined by σ2/n \sigma^2/n , decays relentlessly. In this cinematic transition from chaos to order, the probability distribution of the sample mean effectively 'crushes' itself into a singular point—the true population mean μ \mu . The WLLN ensures that the probability of the average straying even a hair's breadth ϵ \epsilon from the truth vanishes. It is the mathematical bridge between the subjective individual and the objective collective, proving that in the limit, noise is conquered by sheer volume and the truth emerges from the static.
CAUTION

Institutional Warning.

Students frequently conflate Convergence in Probability (WLLN) with Convergence Almost Surely (SLLN). While the WLLN guarantees that the probability of a 'bad' average goes to zero for a fixed, large n n , it does not guarantee that the sequence of averages will never deviate again as n n increases further.

Academic Inquiries.

01

Why is finite variance typically assumed in the BSc proof?

Assuming finite variance allows the use of Chebyshev's Inequality, where P(∣Xˉn−μ∣≥ϵ)≤σ2nϵ2 P(|\bar{X}_n - \mu| \geq \epsilon) \leq \frac{\sigma^2}{n\epsilon^2} . As n→∞ n \to \infty , the right side clearly goes to zero.

02

What happens if the variance is infinite?

If the variance is infinite but the mean is finite, the WLLN still holds (Khinchin's Theorem), but the proof requires characteristic functions instead of the simpler Chebyshev approach.

03

Does the WLLN apply to dependent variables?

Not necessarily. The standard WLLN requires independence, though versions for dependent sequences exist if the correlation between variables decays sufficiently fast.

Standardized References.

  • Definitive Institutional SourceCasella, G., & Berger, R. L., Statistical Inference.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of the Weak Law of Large Numbers (WLLN): Visual Proof & Intuition. Retrieved from https://nicefa.org/library/applied-statistics/proof-of-the-weak-law-of-large-numbers--wlln-

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