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Let $X$ denote a random variable with values in $\mathbb{N}_0\cup\left\{\infty\right\}$. Let $r_X$ denote the radius of convergence of $\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)z^n$ with $z\in\mathbb{C}$. Then $g_X\colon B_{r_X}(0)\to\mathbb{C}$ given by $g_X(z)=\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)z^n$ is called the generating function of $X$.

Now there is one sentence I cannot understand.

Note that $r_X\geq 1$ since if $\lvert z\rvert <1$, then $$ \sum_{n\in\mathbb{N}_0}\lvert\mathbb{P}(X=n)z^n\rvert\leq\sum_{n\in\mathbb{N}_0}\mathbb{P}(X=n)\leq 1. $$

Is that really a reason why $r_X\geq 1$? It means that the series converges absolutely.

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The note says that if $|z| \leq 1$, then we have absolute convergence, since the sum of the probabilities is bounded above by 1. Therefore, the radius of convergence is at least 1, hence $r_X \geq 1$. We do not have enough information to conclude how much bigger (if at all) the radius of convergence is.

msteve
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