Record-breaking statistics for random walks in the presence of measurement error and noise

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