Suppose $\{X_n, n > 1\}$ are independent non-negative random variables satis­fying $E(X_n)= \mu_n \> Var(X_n)=\sigma^2$

Question

Suppose $\{X_n, n > 1\}$ are independent non-negative random variables satis­fying $$E(X_n)= \mu_n \> Var(X_n)=\sigma_n^2$$ Define for $n \geq 1$, $S_n = \sum_{i=1}^{n} X_i$ and suppose $\sum_{n=1}^{\infty} \mu_n = \infty$ and $\sigma_{n}^2 \leq c \mu_n$ for some $c > 0$ and all $n$. Show $\frac{S_n} {E(S_n)}\rightarrow^{P} 1$.

Answer 1

I'm using markov's inequality below - $P(X \ge a) \le \frac{\mathbb{E}[X]}{a}$

\begin{align}\mathbb{P}\left\{ \left|\frac{S_n}{\mathbb{E}[S_n]} - 1\right| \ge \epsilon\right\} &= \mathbb{P}\left\{ \left|{S_n} - {\mathbb{E}[S_n]}\right| \ge {\mathbb{E}[S_n]}\epsilon\right\}\\ &= \mathbb{P}\left\{ \left|{S_n} - {\mathbb{E}[S_n]}\right|^2 \ge (\mathbb{E}[S_n]\epsilon)^2\right\}\\ & \le \frac{\text{Var}[S_n]}{\epsilon^2 (\mathbb{E}[S_n])^{2}}\\ &\le \frac{c}{\epsilon^2 (\mathbb{E}[S_n])} \underset{n \uparrow \infty}{\longrightarrow} 0\end{align} Since $\mathbb{E}[S_n] = \sum \mu_n \to \infty$