In orthogonal frequency division multiple access (OFDMA)-based cellular networks subject to the distancedependent path loss and shadow fading (SF), the downlink inter-cell interference (ICI) for a given user equipment (UE) is essentially a sum of several lognormal random variables (RVs). So far, no method of approximating the lognormal sum distribution is explicitly accurate when the component lognormal RVs with different logarithmic means and logarithmic variances are correlated. In this paper, the Normal Inverse Gaussian (NIG) distribution is proposed to approximate the downlink ICI for a given UE with the correlated SF. First, the downlink ICI is modelled as a sum of several correlated lognormal RVs. Then original moments of the lognormal sum in the logarithmic domain are obtained analytically. Finally the estimated parameters of the NIG distribution are computed explicitly by the mean, variance, skewness and kurtosis of the lognormal sum in the logarithmic domain through moment matching. Numerical results verify the accuracy of the NIG approximation when the correlated component lognormal RVs have different logarithmic means and logarithmic variances, and show that the NIG approximation outperforms the MGF-based lognormal approximation in various scenarios.