Let (T_{a}^{*}=inf left{t:left|W_{t} ight|=a ight}). Prove that the r.v's (T_{a}^{*}) and (W_{T_{a}^{*}}) are independent and show that

Let \(T_{a}^{*}=\inf \left\{t:\left|W_{t}\right|=a\right\}\). Prove that the r.v's \(T_{a}^{*}\) and \(W_{T_{a}^{*}}\) are independent and show that \(W_{T_{a}^{*}}\) is symmetric with values \(\pm a\).