The acquisition and interpretation of increasingly high-resolution climate data from polar ice and firn cores motivates the question: What is the finest depth or timescale on which measurements on cores arrayed over a given area correlate? We analyze dated depth series of electrical and oxygen isotope measurements from a spatial array of firn cores with 3.5--7 km spacing in Dronning Maud Land, Antarctica, each with a temporal span of approximately 200 years. We use wavelet analysis to decompose the series into components associated with changes of averages on different scales, and thus deduce which scales are dominated by environmental noise, and which may contain a common signal. We find that common signals in electrical records have timescales of approximately 1--3 years. We identify only one electrical signal which rises significantly above the background in our 200-year records, evidently corresponding to the Tambora eruption. Several smaller signals correlate in a few of pairs of cores, one of which may correspond to a known volcanic event, but the others appear to be spurious. We present a simulation-based method for testing the significance of apparent electrical signal correlations, and highlight the importance of accurate relative dating between cores. In the case of oxygen-isotope records, we find, surprisingly, no significant correlation on any scale in the records, for any of the pairs of cores. There is, however, a weak trend toward positive correlation at longer timescales (up to 16 years). Statistical theory for the relevant confidence intervals and the observed statistics of the records permit estimation of the length of a data series necessary to reliably detect a hypothetical correlation equal to that observed. For the highest correlation observed on 16-year scales, core records of about 380 years (approximately 30 m at the Dronning Maud Land site) would be necessary to establish significance.