The Central Limit Theorem is a foundational mathematical principle in probability theory stating that the average of $$n$$ independent random samples drawn from any distribution (with mean $$\mu$$ and standard deviation $$\sigma$$) will approximately follow a normal distribution centered at the true mean $$\mu$$ with a standard deviation of $$\sigma/\sqrt{n}$$, as the sample size $$n$$ grows large. In machine learning, this theorem explains why the empirical error of a classifier evaluated on a test set converges to its true population error at an asymptotic rate of $$\mathcal{O}(1/\sqrt{n})$$.

Central Limit Theorem

The Law of Large Numbers is a mathematical theorem guaranteeing that for repeated independent events, as the number of repetitions grows, the empirical average of the outcomes will converge to the true underlying expected probability. This principle mathematically ensures that consistent estimators become perfectly accurate given infinite data.

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In statistics, consistency is a property of an estimator where its probabilistic limit converges to the true underlying parameter value as the sample size approaches infinity ($$	ext{plim}_{mightarrow\infty}\hat{	heta}_m=	heta$$). When an estimator is consistent, accumulating more data guarantees that empirical estimates will eventually match the true theoretical probabilities.

Consistency of an Estimator

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Strong Consistency or "Almost Sure Consistency" is defined as follows:
$$P(lim_{m\rightarrow\infty}\hat{\theta}_m = \theta) = 1$$
So where weak consistency is the ideal limit of a probability, strong consistency is the ideal probability of a limit.

Strong Consistency of an Estimator

The statistical efficiency of consistent estimators can vary based on the generalization error or the number of samples.

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