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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?

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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.

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