Far-Ultraviolet and Far-Infrared Bivariate Luminosity Function of
  Galaxies: Complex Relation between Stellar and Dust Emission
  Far-ultraviolet (FUV) and far-infrared (FIR) luminosity functions (LFs) of
galaxies show a strong evolution from $z = 0$ to $z = 1$, but the FIR LF
evolves much stronger than the FUV one. The FUV is dominantly radiated from
newly formed short-lived OB stars, while the FIR is emitted by dust grains
heated by the FUV radiation field. It is known that dust is always associated
with star formation activity. Thus, both FUV and FIR are tightly related to the
star formation in galaxies, but in a very complicated manner. In order to
disentangle the relation between FUV and FIR emissions, we estimate the UV-IR
bivariate LF (BLF) of galaxies with {\sl GALEX} and {\sl AKARI} All-Sky Survey
datasets. Recently we invented a new mathematical method to construct the BLF
with given marginals and prescribed correlation coefficient. This method makes
use of a tool from mathematical statistics, so called "copula". The copula
enables us to construct a bivariate distribution function from given marginal
distributions with prescribed correlation and/or dependence structure. With
this new formulation and FUV and FIR univariate LFs, we analyze various FUV and
FIR data with {\sl GALEX}, {\sl Spitzer}, and {\sl AKARI} to estimate the UV-IR
BLF. The obtained BLFs naturally explain the nonlinear complicated relation
between FUV and FIR emission from star-forming galaxies. Though the faint-end
of the BLF was not well constrained for high-$z$ samples, the estimated linear
correlation coefficient $\rho$ was found to be very high, and is remarkably
stable with redshifts (from 0.95 at $z = 0$ to 0.85 at $z = 1.0$). This implies
the evolution of the UV-IR BLF is mainly due to the different evolution of the
univariate LFs, and may not be controlled by the dependence structure.
