State the Law of Large Numbers, precisely

Question

State the law of large numbers precisely, including its assumptions, and name the most common ways it is misused.

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Statement. Let X1, X2, \dots be i.i.d. random variables with finite mean \mu = \mathbb{E}[X1]. Let \bar{X}n = 	frac1n\sum{i=1}^n Xi be the sample mean. Then: - Weak LLN: \bar{X}n \xrightarrow{P} \mu, the sample mean converges to \mu in probability (for any \varepsilon 0, \mathbb{P}(\bar{X}n - \mu \varepsilon) 	o 0). - Strong LLN: \bar{X}n \xrightarrow{a.s.} \mu, the sample mean converges to \mu almost surely (the sequence of averages converges to \mu with probability 1). The one essential assumption is a finite mean; finite variance makes the proofs easy but is not required for the strong l…

State the Law of Large Numbers, precisely

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