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

We address the question of distance record-setting by a random walker in the presence of measurement error, $δ$, and additive noise, $γ$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $δ$ and $γ$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $δ$ and $γ$. In the absence of noise ($γ=0$), the pre-factor $S(δ)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $δ$ whereas, in case of perfect measurement $(δ=0)$, the corresponding pre-factor $T(γ)$ increases with $γ$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions.