Proposition

Let X1,X2,...,XnX_1, X_2,...,X_n be a sequence of mutually independent and identically distributed random variables where each has finite mean (expected value) E[Xk]=η<∞E[X_k] = \eta < \infty, then, suppose YnY_n such that

Yn=X1+...+XnnY_n = \frac{X_1 + ... + X_n}{n}

then for any ε>0\varepsilon > 0,

P(limn→∞⁡|Yn−η|>ε)=0P(\lim_{n \to \infty} \lvert Y_n - \eta \rvert > \varepsilon) = 0

(w.p. 1)

See also

References

  1. https://www.sciencedirect.com/topics/mathematics/strong-law-of-large-number
  2. https://zhuanlan.zhihu.com/p/422520475