Characteristic functions of bernoulli binomial and poisson laws

All displayed masses are nonnegative. The Bernoulli masses sum to one. The binomial sum is $(p+1-p{)}^n=1$, including $n=0$, when there is exactly one term of value one. The Poisson sum is $e^{-\lambda }{\sum }_{k\ge 0}{\lambda }^k\mathrm{/}k!=e^{-\lambda }{e}^{\lambda }=1$. Weighted Dirac sums therefore define Borel probabilities on $\mathbb{R}$. For any such countably supported law with masses $b_k$ at distinct integers, $f_N(x)={\sum }_{k=0}^N{e}^{itk}{1}_{\{k\}}(x)$ tends to $e^{itx}$ almost everywhere for that law and satisfies $\mathrm{\mid }{f}_N\mathrm{\mid }\le 1$. Its integral is the finite sum of values times singleton masses. DCT yields $\phi (t)={\sum }_{k\ge 0}{b}_k{e}^{itk}$, with absolute sum $\sum b_k=1$. Finite supports are the same calculation with zero masses afterwards.

Substitute the Bernoulli masses to get $(1-p)+p{e}^{it}$. For the binomial law the finite sum is ${\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)(p{e}^{it}{)}^k(1-p{)}^{n-k}=(1-p+p{e}^{it}{)}^n$. This is also the product supplied for any already-given family of $n$ independent Bernoulli variables; no existence of an infinite family is needed. For Poisson, absolute convergence allows recognition of the defining series: $e^{-\lambda }{\sum }_{k\ge 0}(\lambda e^{it}{)}^k\mathrm{/}k!=e^{-\lambda }\exp (\lambda e^{it})=\exp (\lambda (e^{it}-1))$. At $p=0$ the first two laws are ${\delta }_0$; at $p=1$ they are ${\delta }_1$ and ${\delta }_n$; $n=0$ and $\lambda =0$ give ${\delta }_0$. The stated formulas give precisely their constant-point transforms, and all values at $t=0$ equal one.

For the finite mixture, the weighted-sum theorem gives a measure of total mass ${\sum }_j{a}_j=1$. For a simple complex function $s={\sum }_{\mathrm{\ell }=1}^q{c}_{\mathrm{\ell }}{1}_{E_{\mathrm{\ell }}}$ on a disjoint measurable partition, the integral definition and finite sums give $\int sd\rho ={\sum }_j{a}_j\int sd{\mu }_j$. Approximate $e^{itx}$ by rounding its real and imaginary parts down to multiples of $2^{-N}$; each approximation is Borel, simple, uniformly bounded by three and converges pointwise. DCT for $\rho$ and for each of the finitely many ${\mu }_j$ passes the simple identity to the limit, giving the mixture formula. Zero weights contribute zero, a one-component mixture returns that component, and no empty mixture has weights summing to one. No AC is used in these explicit sums and approximations.