Consider a sequence of IID random variables $\{Y_n\}$ taking value in $(c,\infty)$ with $c>0$. $\mathbb{E}[Y_n^2]$ and $\mathbb{E}[1/Y_n^2]$ being finite.
I want to prove that $\frac{1}{n}\sum_{k=1}^n\log(c+Y_k) =: \frac{S_n}{n}$ converges almost surely to some finite number as $n\to\infty$.
I couldn't fully figure out how to show that the given assumptions about $Y_n$ satisfies the assumption of the Strong Law of Large Number, which requires $\mathbb{E}[|\log(c+Y_1)|]<\infty$; this is the part I couldn't show.
As of now, I'm only able to show that $\mu:=\mathbb{E}[\log(c+Y_1)]<\infty$ using Jensen's inequality and the assumption of finite second moment to bound $\mu$. I want to show that $\frac{S_n}{n}\to\mu$ almost surely, but I need $\mathbb{E}[|\log(c+Y_1)|]<\infty$ in order to invoke SLLN. I'd appreciate any help.
The $Z_k=\log(c+Y_k)$ are mutually independent, further $Z_k\ge\log(2c)$ almost surely and \begin{aligned} \mathbb E[Z_k^2]&\le\mathbb E[\unicode{120793}\{Z_k\le 0\}Z_k^2]+\mathbb E[\unicode{120793}\{Z_k\ge 0\}(c+Y_k-1)^2]\\ &\le\max(0,-\log(2c))+(c-1)^2+\mathbb E[Y_k^2]+2(c-1)\mathbb E[Y_k]<\infty. \end{aligned} Hence, the strong law of large numbers applies.