  Recent gravitational wave (GW) observations by LIGO/Virgo show evidence for
hierarchical mergers, where the merging BHs are the remnants of previous BH
merger events. These events may carry important clues about the astrophysical
host environments of the GW sources. In this paper, we present the
distributions of the effective spin parameter ($\chi_\mathrm{eff}$), the
precession spin parameter ($\chi_\mathrm{p}$), and the chirp mass
($m_\mathrm{chirp}$) expected in hierarchical mergers. Under a wide range of
assumptions, hierarchical mergers produce (i) a monotonic increase of the
average of the typical total spin for merging binaries, which we characterize
with ${\bar \chi}_\mathrm{typ}\equiv
\overline{(\chi_\mathrm{eff}^2+\chi_\mathrm{p}^2)^{1/2}}$, up to roughly the
maximum $m_\mathrm{chirp}$ among first-generation (1g) BHs, and (ii) a plateau
at ${\bar \chi}_\mathrm{typ}\sim 0.6$ at higher $m_\mathrm{chirp}$. We suggest
that the maximum mass and typical spin magnitudes for 1g BHs can be estimated
from ${\bar \chi}_\mathrm{typ}$ as a function of $m_\mathrm{chirp}$. The GW
data observed in LIGO/Virgo O1--O3a prefers an increase in ${\bar
\chi}_\mathrm{typ}$ at low $m_\mathrm{chirp}$, which is consistent with the
growth of the BH spin magnitude by hierarchical mergers, at $\sim 2 \sigma$
confidence. A Bayesian analysis suggests that 1g BHs have the maximum mass of
$\sim 15$--$30\,M_\odot$ if the majority of mergers are of high-generation BHs
(not among 1g-1g BHs), which is consistent with mergers in active galactic
nucleus disks and/or nuclear star clusters, while if mergers mainly originate
from globular clusters, 1g BHs are favored to have non-zero spin magnitudes of
$\sim 0.3$. We also forecast that signatures for hierarchical mergers in the
${\bar \chi}_\mathrm{typ}$ distribution can be confidently recovered once the
number of GW events increases to $\gtrsim O(100)$.
