Prove that the functions

\[ f_{1}(t)=\sum_{k=0}^{\infty} a_{k} \cos k t, \quad f_{2}(t)=\sum_{k=0}^{\infty} a_{k} e^{i \lambda_{k} t} \]

where \(a_{k} \geqslant 0\) and \(\sum_{k=0}^{\infty} a_{k}=1\) are characteristic functions; determine the corresponding probability distributions.