Let $(X_n, n \geq 1)$ be an i.i.d.\ sequence of positive random variables with distribution function $H$. Let $\Phi_H := \{ (n,X_n),\;n \geq 1\}$ be the associated observation process. We view $\Phi_H$ as a measure on $E := [0,\infty) \times (0,\infty]$ where $\Phi_H(A)$ is the number of points of $\Phi_H$ which lie in $A$. A family $(V_s, s > 0)$ of transformations is defined on $E$ in such a way that for suitable $H$ the distributions of $(V_s\Phi_H, s > 0)$ satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some related large deviation principles and loglog laws are then derived for extreme values and partial sums processes. Similar results are proved for $\Phi_H$ replaced by certain planar Poisson processes, with parallel applications to extremal processes and spectrally positive Levy processes.