Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by: $$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$ For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$. Let $a , b \in \mathbf { R }$ such that $a \neq 0$ and $| b | \leq | a |$. Show that $$| a + b | = | a | + \operatorname { sign } ( a ) b$$ where $\operatorname { sign } ( x ) = x / | x |$ for nonzero real $x$. Deduce that: $$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right)$$
Let $\left( X _ { k } \right) _ { k \in \mathbf{N} ^ { * } }$ be independent random variables with the same distribution given by:
$$P \left( X _ { 1 } = - 1 \right) = P \left( X _ { 1 } = 1 \right) = \frac { 1 } { 2 }$$
For all $n \in \mathbf { N } ^ { * }$, we denote $S _ { n } = \sum _ { k = 1 } ^ { n } X _ { k }$.
Let $a , b \in \mathbf { R }$ such that $a \neq 0$ and $| b | \leq | a |$. Show that
$$| a + b | = | a | + \operatorname { sign } ( a ) b$$
where $\operatorname { sign } ( x ) = x / | x |$ for nonzero real $x$. Deduce that:
$$\forall n \in \mathbf { N } ^ { * } , \quad E \left( \left| S _ { 2 n } \right| \right) = E \left( \left| S _ { 2 n - 1 } \right| \right)$$