# Strong law of Large numbers (SLLN4)

I'm trying to prove SLLN(4): Let $$\{X_n : n\geq1\}$$ be a sequence of $$L^1-$$integrable independent random variables on a probability space and $$S_n = \sum_{j=1}^nX_{j}$$ for every $$n\geq1$$. Let $$\phi : \mathbb{R} \to \mathbb{R}$$ be a positive and continuous even function such that $$\frac{\phi(x)}{x}$$ is non-decreasing in $$x \in (0,\infty)$$ and $$\frac{\phi(x)}{x^2}$$ is non-increasing in $$x \in (0,\infty)$$. Assume that for some sequence $$\{b_n: n\geq1\}$$ of positive real numbers with $$b_n \to \infty$$ as $$n \to \infty$$, $$\sum_{n\geq1}\frac{\mathbb{E}[\phi(X_n)]}{\phi(b_n)} <\infty$$ Prove that $$\frac{S_n-\mathbb{E}[S_n]}{b_n} \to 0 \qquad \mbox{a.s.}$$

In order to finalize the proof I just need to show $$\frac{T_n-\mathbb{E}[T_n]}{b_n} \to 0 \quad \mbox{a.s.}$$ where $$T_n = \sum_{j=1}^{n}Y_j$$, and $$Y_j$$ is truncated version of $$X_j$$: $$Y_n = X_n \mathbb{1}_{|X_n|\le b_n},$$ which in turn, only suffices to show $$\sum_{n=1}^{\infty} \frac{\operatorname{Var}(Y_n)}{b_n^2} < \infty$$.

How can I prove this last inequality?

For a non negative random variable $$X$$, the following inequalities hold: $$X^2\mathbf 1\left\{X\leqslant b\right\}\leqslant bX\mathbf 1\left\{X\leqslant b\right\} =b\frac{X}{\phi\left(X\right)}\phi\left(X\right)\mathbf 1\left\{X\leqslant b\right\} \leqslant b\frac{b}{\phi\left(b\right)}\phi\left(X\right),$$ where we used the fact that $$x\mapsto \phi(x)/x$$ is non-decreasing. Apply this to $$X=\left\lvert X_n\right\rvert$$ and $$b=b_n$$ to get the wanted result.