  We investigate the nature of correlations in the small-scale flux statistics
of the Lyman-$\alpha$ (Ly$\alpha$) forest across redshift bins. Understanding
these correlations is important for unbiased cosmological and astrophysical
parameter inference using the Ly$\alpha$ forest. We focus on the 1-dimensional
flux power spectrum (FPS) and mean flux ($\bar F$) simulated using the
semi-numerical lognormal model we developed in earlier work. The lognormal
model can capture the effects of long wavelength modes with relative ease as
compared to full smoothed particle hydrodynamical (SPH) simulations that are
limited by box volume. For a single redshift bin of size $\Delta z\simeq 0.1$,
we show that the lognormal model predicts positive cross-correlations between
$k$-bins in the FPS, and a negative correlation for $\bar F\times$ FPS, in
qualitative agreement with SPH simulations and theoretical expectations. For
measurements across two neighbouring redshift bins of width $\Delta z$ each
(obtained by 'splitting' skewers of length $2\Delta z$ in half), the lognormal
model predicts an anti-correlation for FPS $\times$ FPS and a positive
correlation for $\bar F\times$ FPS, caused by long wavelength modes. This is in
contrast to SPH simulations which predict a negligible magnitude for
cross-redshift correlations derived from such `split' skewers, and we discuss
possible reasons for this difference. Finally, we perform a preliminary test of
the impact of neglecting long wavelength modes on parameter inference, finding
that whereas the correlation structure of neighbouring redshift bins has
relatively little impact, the absence of long wavelength modes in the model can
lead to $\gtrsim2-\sigma$ biases in the inference of astrophysical parameters.
Our results motivate a more careful treatment of long wavelength modes in
analyses that rely on the small scale Ly$\alpha$ forest for parameter
inference.
