We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$ If $k$ and $n$ are positive integers such that $k \leq n$, show that $$P \left( \left( S _ { n } = 0 _ { d } \right) \cap ( R = k ) \right) = P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$ Deduce that $$\forall n \in \mathbb{N}^{*} , \quad P \left( S _ { n } = 0 _ { d } \right) = \sum _ { k = 1 } ^ { n } P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$
We consider the functions $F$ and $G$ defined by the formulas
$$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$
If $k$ and $n$ are positive integers such that $k \leq n$, show that
$$P \left( \left( S _ { n } = 0 _ { d } \right) \cap ( R = k ) \right) = P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$
Deduce that
$$\forall n \in \mathbb{N}^{*} , \quad P \left( S _ { n } = 0 _ { d } \right) = \sum _ { k = 1 } ^ { n } P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$