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 $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$