Question

Let $\{X_{n}\}$ be a sequence of $r \cdot v$'s and $Y_{n}=\left(\frac{S_{n}-E(S_{n})}{n}\right)$ where $S_{n}=\sum_{i=1}^{n} X_{i}$ then the necessary and sufficient condition for the sequence $\{X_{n}\}$ to satisfy W.L.L.N is

• A. $E\left(\frac{Y_{n}}{1+Y_{n}}\right) \rightarrow 0$ as $n \rightarrow \infty$
• B. $E\left(\frac{Y_{n}^{2}}{1+Y_{n}^{2}}\right) \rightarrow 0$ as $n \rightarrow \infty$
• C. $E\left(\frac{Y_{n}}{1+Y_{n}^{2}}\right) \rightarrow 0$ as $n \rightarrow \infty$
• D. $E\left(\frac{Y_{n}^{2}}{1+Y_{n}}\right) \rightarrow 0$ as $n \rightarrow \infty$

Hint:

Convergence in probability can always be expressed as the expectation of a bounded function. The form $\frac{x^2}{1+x^2}$ is the most common "distance" function used to prove WLLN for variables without finite second moments.

Answer 1

The Correct Option is B