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The post states (without qualification) that Monte Carlo has no mathematical guarantees.

However, standard Monte Carlo estimation is explicitly supported by formal probabilistic guarantees:

1. Law of Large Numbers (LLN) provides a convergence guarantee

In Los Alamos National Laboratory lecture material “Fundamentals of Monte Carlo,” the LLN is stated in a form that directly describes Monte Carlo sample-mean convergence: as the number of samples increases, the sample mean converges (in probability) to the true mean. The slides include the weak LLN statement:

• (\lim_{N\to\infty} \Pr(|m_N-\mu|>\varepsilon)=0)

This is a mathematical guarantee (probabilistic convergence) for Monte Carlo mean estimation, given the stated assumptions (IID sampling and finite moments).

2. Central Limit Theorem (CLT) provides a distribution/error guarantee enabling confidence intervals

In “A New Method to Assess Monte Carlo Convergence” (Los Alamos report), the abstract explicitly notes that when CLT conditions hold (finite mean/variance; large number of independent observations), a confidence interval with specified coverage probability can be formed. This is another mathematical guarantee routinely used to quantify Monte Carlo error (e.g., standard error decreasing like (1/\sqrt{N}) under typical conditions).

Because Monte Carlo methods do have these established theoretical guarantees (with stated assumptions/conditions), the blanket claim that they have no mathematical guarantees is contradicted by standard probability theory and by the cited technical references.