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tokens: needed, the sampling has been done over 10 6 randomized maps. To determine the random cases, we explored three different procedures: (1) randomization of the γ-ray sample coordinates by drawing an equal number of positions homogeneously distributed over the sky, making sure that only random sources with b II values in the same range as the original γ-ray catalogue are considered. This leaves untouched the IceCube positions, which are known to be not uniformly distributed, but might lose any large-scale structure present in the γ-ray sample; (2) to at least partially obviate this, we have also randomized only the γ-ray sample right ascensions; (3) randomization of the IceCube right ascensions, making sure that only random sources that can be associated with the γ-ray catalogue within the relevant error circles are considered (for example, a | b II | 10° IceCube random source can still be associated with a 2WHSP object provided it has a large enough error radius). This preserves any large-scale structure present in the γ-ray sample plus the IceCube declinations. However, the requirement above does not conserve the total area sampled by the IceCube error circles, resulting in a biased test statistics. As a result, we used procedures (1) and (2), conservatively taking as our best estimate of the probability of random association the largest of the two probabilities. Since our three catalogues are somewhat overlapping, in particular at large fluxes and FoM (see Section 2.2.2 ), the number of bins per scan is 10, and the points are not independent, we do not correct the probabilities for the ‘look elsewhere effect’ (which would take into account the artificial p-value reduction due to the application of multiple tests). We have also fixed the catalogues studied at the beginning of this work as well as the bins in flux and FoM used for the scans. 4 RESULTS 4.1 <italic>Fermi</italic> 2FHL Fig. 1 shows the chance probability of association of 2FHL high Galactic latitude sources with IceCube events for objects having F (>50 GeV) larger than the value on the x -axis and a randomization of the γ-ray sample on right ascension, which gives the most conservative result. Red squares indicate the whole high Galactic latitude sample, black circles denote HBL only, and blue triangles represent non-HBL sources. The numbers give the observed (above the points) and average random value (below the points) of N ν for HBL. Fig. 1 shows the following: for the whole sample and for HBL the chance probability is strongly dependent on γ-ray flux, with an anti-correlation between probability and flux up to F (>50 GeV) ∼ 1.8 × 10 −11 photon cm −2 s −1 with a p-value 10 −4 according to a Spearman test. We attribute the turn-over in p-value at very large γ-ray fluxes to small number statistics; p-values ∼1.8 per cent are reached for F ( > 50 GeV) ≳ 1.8 × 10 − 11 photon cm −2 s −1 for the whole sample; p-values ∼0.4 per cent are reached for F ( > 50 GeV) ≳ 1.8 × 10 − 11