  The recent discovery of a Galactic fast radio burst (FRB) associated with a
hard X-ray burst from the soft gamma-ray repeater (SGR) J1935+2154 has
established the magnetar origin of at least some FRBs. In this work, we study
the statistical properties of soft gamma-/hard X-ray bursts from SGRs 1806--20
and J1935+2154 and of radio bursts from the repeating FRB 121102. For SGRs, we
show that the probability density functions for the differences of fluences,
fluxes, and durations at different times have fat tails with a $q$-Gaussian
form. The $q$ values in the $q$-Gaussian distributions are approximately steady
and independent of the temporal interval scale adopted, implying a
scale-invariant structure of SGRs. These features indicate that SGR bursts may
be governed by a self-organizing criticality (SOC) process, confirming previous
findings. Very recently, 1652 independent bursts from FRB 121102 have been
detected by the Five-hundred-meter Aperture Spherical radio Telescope (FAST).
Here we also investigate the scale-invariant structure of FRB 121102 based on
the latest observations of FAST, and show that FRB 121102 and SGRs share
similar statistical properties. Given the bimodal energy distribution of FRB
121102 bursts, we separately explore the scale-invariant behaviors of low- and
high-energy bursts of FRB 121102. We find that the $q$ values of low- and
high-energy bursts are different, which further strengthens the evidence of the
bimodality of the energy distribution. Scale invariance in both the high-energy
component of FRB 121102 and SGRs can be well explained within the same physical
framework of fractal-diffusive SOC systems.
