Question

State and prove kolmogorov's SLLN theorem.​

Answer 1

Suppose ξiξi are i.i.d. and E(|ξ1|)<∞E(|ξ1|)<∞

Let Xn=∑ni=1ξiXn=∑i=1nξi Then we have Xnn→E(|ξ1|)Xnn→E(|ξ1|)a.s.

In the proof of this theorem:

Xnn=1n∑i=1nE[ξi|Xn]

Xnn=1n∑i=1nE[ξi|Xn]

I don't know why this equality holds since I don't think ξi∈σ(Xn)