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The information content of the autocorrelation function (ACF) of intensity fluctuations of the X-ray background (XRB) is analyzed. The tight upper limits set by ROSAT deep survey data on the ACF at arcminute scales imply strong constraints on clustering properties of X-ray sources at cosmological distances and on their contribution to the soft XRB. If quasars have a clustering radius r0 = 12-20 Mpc (H-0 = 50), and their two-point correlation function, xi(r), is constant in comoving coordinates (epsilon = - 1.2), as indicated by optical data, they cannot make up more than 40%-50% of the soft XRB (higher contributions corresponding to lower r0); the maximum contribution may reach 80% in the case of stable clustering (epsilon = 0). If r0 greater-than-or-equal-to 12 Mpc, a slow decrease of the xi(r) of the AGNs with increasing redshift (epsilon congruent-to - 3) is ruled out since it would imply an implausibly low contribution to the XRB. Active star-forming (ASF) galaxies clustered like normal galaxies (r0 congruent-to 12 Mpc) can yield up to 20% or up to 40% of the soft XRB for epsilon = - 1.2 or epsilon = 0, respectively. The ACF on degree scales, typical of existing hard X-ray surveys, essentially reflects the clustering properties of local sources and is proportional to their volume emissivity. The upper limits on scales of a few degrees imply that hard X-ray-selected AGNs have r0 less-than-or-equal-to 25 Mpc if epsilon = 0 or r0 less-than-or-equal-to 20 Mpc if epsilon = - 1.2. No significant constraints are set on clustering of ASF galaxies, due to their low local volume emissivity. The possible signal on scales greater-than-or-equal-to 6-degrees, if real, may be due to AGNs with r0 congruent-to 20 Mpc; the contribution from clusters of galaxies with r0 congruent-to 50 Mpc is a factor congruent-to 2 lower. Other classes of sources clustered like normal galaxies could only account for such signal if their local volume emissivity is almost-equal-to 6 x 10(38) ergs s-1 Mpc-3. This limit is lower than the values indicated by recent, indirect, estimates and implies that the bulk of the XRB is not local. We have also computed the expected ACF in the 2-10 keV energy band on arcminute scales, that will be useful for comparison to the upcoming data from the Astro-D satellite. Although the nature of sources producing the bulk of the soft and of the hard XRB is likely to be different, their clustering properties appear to be not much different from those of normal galaxies

Angular correlations of the X-ray background and clustering of extragalactic X-ray sources