Answer to Question #339464 in Statistics and Probability for Mwangi

Question

Define the following theorems with respect to a random vector

i) Central limit theorem

ii) Weak law of large numbers

Answer 1

i). Central limit theorem: Let $X_1,X_2,X_3,\ldots,X_n$ be independent and identically distributed random variables with expected value $E[X_i]=\mu<\infty$ and variance $Var[X_i]=\sigma^2<\infty$. Denote $\bar{X}=\frac{X_1+X_2+\ldots+X_n}{n}$, $Z_n=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{X_1+X_2+\ldots+X_n-n\mu}{\sqrt{n}\sigma}$. Then, $\lim_{n\rightarrow\infty}P(Z_n\leq x)=F(x)$, where $F(x)$ is the cumulative distribution function of the standard normal distribution. I.e., $F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-\frac{t^2}{2}}dt.$

ii). Weak law of large numbers: Let $X_1,X_2,X_3,\ldots,X_n$ be independent and identically distributed random variables with expected value $E[X_i]=\mu<\infty$ and variance $Var[X_i]=\sigma^2<\infty$. Denote $\bar{X}=\frac{X_1+X_2+\ldots+X_n}{n}$ Then, for any $\varepsilon>0$, $\lim_{n\rightarrow\infty}P(|\bar{X}-\mu|\geq\varepsilon)=0.$