= Atomic compound Poisson process with drift
{title2=$X_t=\gamma t+\sum_j z_jN_t^j$}
With finitely many nonzero marks $z_j$ and independent <Poisson processes> $N^j$ of rates $\lambda_j$, the displayed process is a <Lévy process> with no Gaussian component. Its <Lévy measure> is $\sum_j\lambda_j\delta_{z_j}$, its mean is $t(\gamma+\sum_j\lambda_jz_j)$, and its <variance> is $t\sum_j\lambda_jz_j^2$. Its paths have finitely many jumps on every bounded interval and deterministic slope $\gamma$ between jumps. Coincident mark sizes combine their rates; zero marks have no effect. The drift here uses the uncompensated finite-sum convention, which differs from the drift parameter in a compensated <Lévy–Khintchine formula>.
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