Strong law for adjacent products
= Strong law for adjacent products
For <independent and identically distributed random variables> $X_j$ with $\mathbb E|X_1|<\infty$ and $m=\mathbb EX_1$, the <sample mean> of $X_jX_{j+1}$ tends to $m^2$ <almost surely>. Split into odd and even indexed products; each subsequence consists of <independent and identically distributed random variables> that are <integrable random variables>. Apply the <strong law of large numbers> separately.