= Hurdle decomposition of a positive random sum
{title2=$\mathcal L(S)=p\delta_0+(1-p)\mathcal L(S\mid S>0)$}
For positive claim sizes, a <random sum of independent claims> is zero exactly when its count is zero. Its law is a <mixture distribution> of a zero atom of mass $p=\mathbb P(N=0)$ and, with weight $1-p$, a random sum whose count has the <zero-truncated claim-count distribution>. An independent <Bernoulli random variable> multiplying that positive component gives the same law. This is distinct from arbitrarily inserting additional zeros into an otherwise unchanged count law.
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