Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. The functions $\varphi$ and $\Phi$ are defined by $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$ and $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$. Deduce that we have $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \int _ { u } ^ { v } \varphi ( x ) \mathrm { d } x$$ then that $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \right\} \right) = 1 - \Phi ( u )$$
Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. The functions $\varphi$ and $\Phi$ are defined by $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$ and $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$.
Deduce that we have
$$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \int _ { u } ^ { v } \varphi ( x ) \mathrm { d } x$$
then that
$$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \right\} \right) = 1 - \Phi ( u )$$