This is an exercise of Feller Probability vol.2 that I'm trying to prove it without much progress.

Suppose that $c_k>0$, $\sum c_k=1$ but $\sum c_k2^k=\infty.$ Let $u$ be an even continuous density concentrated on $(-1,1)$ and let $w$ be its characteristic function. Then $$f(x)=\sum c_k2^ku(2^kx)$$ defines a density that is continuous except at the origin and has the characteristic function $$\phi(\zeta)=\sum c_{k}2^kw(2^{-k}\zeta)$$ Show that $|\phi|^n$ is not integrable for any $n.$

As a hint that gives the book is if $x\neq 0$ the serie determinated by $f$ is finite. Consider the inequality $(\sum c_kp_k)^n\geq\sum c_k^np_k^n$ valid for $p_k\geq 0.$

Any kind of idea or help is welcome.


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