I am trying to solve the problem 10.22 from Folland's book "Real analysis: Modern Techniques and Their Applications".
If $\{X_j\}$ is a sequence of independent random variables in the unitary circle $\mathbb{T}$ with a common distribution $\lambda$ supported on a finite subgroup $Z_m=\{e^{2\pi i \frac{j}{m}}: 0\leq j <m \}$, then the distribution of $\prod_{i=1}^n X_j$ converges vaguely to $\frac{1}{m}\sum_{z \in Z_m} \delta_z$ on $Z_m$.
The hint is to use the following exercise: If $\mu$ is a positive Borel measure on $\mathbb{T}$ with $\mu(\mathbb{T})=1$ that is a linear combination, with positive coefficientes, of the point masses at $0, \frac{1}{m}, \dots, \frac{m-1}{m}$ for some $m \in \mathbb{N}$, then $\hat{\mu}(jm)=1$ for all $j \in \mathbb{Z}$.
I think that the idea is to prove that the Fourier transform of the convolution $\lambda*\cdots *\lambda$ converges pointwise to the Fourier transform of $\frac{1}{m}\sum_{z \in Z_m} \delta_z$ but I can't find a way to solve this. Can any one help me?
Edit: We can use that $P_{X_1\cdot X_2\cdot X\cdots X_n}=\lambda *\cdots *\lambda$, meaning the distribution of the product is the convolution of $\lambda$'s.
I will represent $\mathbb{T}$ as $[0, 1]$ instead of the circle in the complex plane. So $Z_m = \{0, 1/m, \ldots, (m-1)/m\}$
The Fourier transform of $\lambda * \cdots * \lambda$ is $\hat{\lambda}^n$. We have $$\hat{\lambda}(\kappa) = \sum_{j=0}^{m-1} e^{-2 \pi i \kappa j/m}\lambda(\{j/m\})$$ for integers $\kappa$. By the triangle inequality, $|\hat{\lambda}(\kappa)| \le 1$. Equality occurs if and only if $e^{-2\pi i \kappa j/m} = 1$ for all $j$, that is, if $\kappa$ is an integer multiple of $m$. [This is Exercise 8.39, where you have forgotten to quote an important part of the exercise: that we have the strict inequality $|\hat{\lambda}(\kappa)| < 1$ when $\kappa$ is not an integer multiple of $m$.]
Thus, $$(\hat{\lambda}(\kappa))^n \to \begin{cases} 0 & \text{$\kappa$ not integer multiple of $m$,} \\ 1 & \text{$\kappa$ integer mulitple of $m$.} \end{cases}$$
The Fourier transform of $\nu := \frac{1}{m} \sum_{z \in Z_m} \delta_z$ is $$\hat{\nu}(\kappa) = \frac{1}{m}\sum_{j=0}^{m-1} e^{-2\pi i \kappa j/m} = \begin{cases} 0 & \text{$\kappa$ not integer multiple of $m$,} \\ 1 & \text{$\kappa$ integer mulitple of $m$.} \end{cases}$$ for integers $\kappa$.