We examine distance record setting by a random walker in the presence of a measurement error $\delta$ and additive noise $\gamma$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $⟨R_n⟩\sim n^{1/2}$ for large $n$ for all jump densities, including Lévy distributions, and for all $\delta$ and $\gamma$. In contrast, the pace of record setting, measured by the amplitude of the $n^{1/2}$ growth, depends on $\delta$ and $\gamma$. In the absence of noise ($\gamma =0$), the amplitude $S(\delta )$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $\delta$ whereas, in the case of perfect measurement ($\delta =0$), the corresponding amplitude $T(\gamma )$ increases with $\gamma$. The exact results for $S(\delta )$ offer a new perspective for characterizing instrumental precision by means of record counting. Our analytical results are supported by extensive numerical simulations.

• Received 3 February 2013

DOI:https://doi.org/10.1103/PhysRevLett.110.180602