can be clearly seen, the upper envelope of Scryst(33$\mu$m) decreases towards
a higher neon ratio with no distinction between mid-infrared galaxy class (as
shown by color coding). Above [Ne iii]/[Ne ii]=6.5 there is only one more
galaxy with a detection of the 33$\mu$m forsterite band: the low-metallicity
galaxy SHOC 391 at [Ne iii]/[Ne ii]=22. This is, however, not the only galaxy
above [Ne iii]/[Ne ii]=6.5 with a detection of forsterite. As shown by thick
upper limit symbols, there are two sources with detections of the 23$\mu$m
forsterite band, in absorption: the low-metallicity galaxies SBS 1533+574B and
IRAS 11485–2018. A similar trend can be seen by using [O iv]/[Ne ii] or [O
iv]/[S iii] 33 $\mu$m instead, with one important difference: since all low-
metallicity galaxies have upper limits for the [O iv] line, as is true for all
galaxies powered by stellar sources of ionizing radiation (see e.g. Fig. 18),
these galaxies appear along with other star forming galaxies towards the
extreme left in those diagrams.

Since a silicate strength is measured relative to the local continuum, it is
possible that the observed decrease in Scryst(33$\mu$m) with increasing [Ne
iii]/[Ne ii] can be fully attributed to an increase in the 33 $\mu$m flux
density. The comparison of the panchromatic SED modeling of Galliano et al.
(2008a) to the mid-IR spectral decomposition of Galliano et al. (2008b), for
30 nearby galaxies, indeed shows that an increase in [Ne iii]/[Ne ii] is
accompanied by an increase in 33 $\mu$m dust emission relative to the total IR
emission. Since it is beyond the scope of the current paper to include far-
infrared data, we cannot determine whether the crystallinity decreases as the
hardness of the ionizing SED increases, or whether the 33 $\mu$m crystalline
silicate feature gets overwhelmed by the 33 $\mu$m dust continuum.

Attempts to find correlations of Scryst(33$\mu$m) with the 30 $\mu$m to 5.5
$\mu$m continuum slope and with EQW(PAH62) have come up empty. In the absence
of a mid-infrared diagnostic that allows us to distinguish between quiescent
and vigorous star formation among IDEOS galaxies we also cannot address the
question whether the higher input of stardust in a galaxy would have a
measurable effect on the global dust content.

For the IDEOS sample as a whole it thus appears that crystalline silicates are
a common component of the ISM of galaxies — or, taking into account the non-
detection of crystalline silicates in the spectra of the disks of NGC 4945 and
M 101 (Sect. 9.3), of galactic nuclei.

##  10 IDEOS portal

The results of our spectral fitting and decomposition can be examined in
detail via the IDEOS web portal ideos.astro.cornell.edu. The website offers
several important functionalities beyond what machine-readable tables of
measured quantities offer, such as, for individual sources:

  * •

The extraction positions and positions angles for the individual SL and LL nod
spectra are overlaid on an optical survey map.

  * •

The nod1 and nod2 spectra are plotted together to allow assessment of the
significance of spectral structure found in the final nod-averaged spectrum.

  * •

Continuum, line and feature fits can be visually inspected by choosing the SED
fit of the appropriate wavelength range.

  * •

Amorphous and crystalline silicate band fits are plotted side-by-side.

  * •

All fitted emission lines can be compared in one plot.

  * •

The excitation diagram of warm molecular hydrogen, and postage stamp plots of
the individual H2 line fits are shown together.

  * •

Synthetic photometric points are overlaid on the SED.

  * •

The spectral decomposition into PAH and blackbody components from either
PAHFIT or QUESTFIT is visualized.

The portal also provides a way to rank and examine sources by any of the
measured quantities, or to select a galaxy sample based on constraints built
on inequalities of observables conjoined by logical operators. E.g. irs15flux
$>$ 2 $*$ mips24flux, ne3_15flux $>$ ne2flux, or redshift $>$ 0.3. The
observables and spectra can also be downloaded as machine readable tables.

Within the portal we also link to auxilary pages that explain the
characteristics of CHUNKFIT or how we came to adopt our spectral resolution
table.

Table 6: IDEOS observables: emission and absorption features

Column | Observable | Unit | Description
---|---|---|---
1 | Name |  | IDEOS identifier
2 | Galaxy |  | NED galaxy identifier
3 | RA | degrees | Right Ascension J2000.0
4 | Dec | degrees | Declination J2000.0
5 | z |  | Redshift (Paper i)
6 | MirClass |  | Mid-IR class designation
7–9 | Pah62Flux | W m-2 | 6.2 $\mu$m PAH flux measured using spline method
10–12 | Pah62EQW |  $\mu$m | Ice-corrected equivalent width of the 6.2 $\mu$m PAH feature measured using spline method
13–15 | Pah11Flux | W m-2 | 11.2 $\mu$m PAH flux (sum of PAH111 and PAH112) measured using spline method
16–18 | Pah11EQW |  $\mu$m | Equivalent width of the 11.2 $\mu$m PAH feature measured using spline method
19–21 | Pah127Flux | W m-2 | 12.7 $\mu$m PAH flux measured using spline method
22–24 | Pah127EQW |  $\mu$m | Equivalent width of the 12.7 $\mu$m PAH feature measured using spline method
25–27 | DrudePAH62FLUX | W m-2 | 6.2 $\mu$m PAH flux measured using drude method
28–30 | DrudePAH62EQW |  $\mu$m | Ice-corrected equivalent width of the 6.2 $\mu$m PAH feature measured using drude method
31–33 | DrudePAH77Flux | W m-2 | 7.7 $\mu$m PAH flux measured using drude method
34–36 | DrudePAH77EQW |  $\mu$m | Equivalent width of the 7.7 $\mu$m PAH feature measured using drude method
37–39 | DrudePAH11Flux | W m-2 | 11.2 $\mu$m PAH flux measured using drude method
40–42 | DrudePAH11EQW |  $\mu$m | Euivalent width of the 11.2 $\mu$m PAH feature measured using drude method
43–45 | DrudeTOTPahFlux | W m-2 | Integrated flux of all 5–19 $\mu$m PAH feature measured using drude method
46–48 | DrudeMirFlux | W m-2 | Integrated 5–25 $\mu$m continuum flux using drude method
49 | DrudeOrigin |  | Decomposition method: PAHFIT or QUESTFIT
50–52 | SilStrength |  | Strength of the 9.8 $\mu$m amorphous silicate feature
53 | SilFitType |  | Interpolation method used to create local continuum: powerlaw or spline
54 | SilAnchorType |  | Origin of anchors for continuum interpolation
55–57 | Cryst19Strength |  | Strength of the 19 $\mu$m crystalline silicate feature
58–60 | Cryst23Strength |  | Strength of the 23 $\mu$m crystalline silicate feature
61–63 | Cryst28Strength |  | Strength of the 28 $\mu$m crystalline silicate feature
64–66 | Cryst33Strength |  | Strength of the 33 $\mu$m crystalline silicate feature
67–68 | TauIce |  | Optical depth of the 6 $\mu$m ice feature
69–70 | Tau685 |  | Optical depth of the 6.85 $\mu$m aliphatic feature

Table 7: IDEOS observables: emission lines

Column | Observable | Unit | Description
---|---|---|---
71–73 | Ar2Flux | W m-2 | 6.99 $\mu$m [Ar ii] line flux
74–76 | Ar3Flux | W m-2 | 8.99 $\mu$m [Ar iii] line flux
77–79 | Cl2Flux | W m-2 | 14.37 $\mu$m [Cl ii] line flux
80–82 | Fe2Flux | W m-2 | 25.99 $\mu$m [Fe ii] line flux
83–85 | H2S0Flux | W m-2 | 28.22 $\mu$m H2 0–0 S(0) line
86–88 | H2S1Flux | W m-2 | 17.03 $\mu$m H2 0–0 S(1) line
89–91 | H2S2Flux | W m-2 | 12.28 $\mu$m H2 0–0 S(2) line
92–94 | H2S3Flux | W m-2 | 9.66 $\mu$m H2 0–0 S(3) line
95–97 | H2S5Flux | W m-2 | 6.91 $\mu$m H2 0–0 S(5) line
98–100 | H2S7Flux | W m-2 | 5.51 $\mu$m H2 0–0 S(7) line
101-103 | Ne2Flux | W m-2 | 12.81 $\mu$m [Ne ii] line flux
104–106 | Ne3_15Flux | W m-2 | 15.56 $\mu$m [Ne iii] line flux
107–109 | Ne3_36Flux | W m-2 | 36.01 $\mu$m [Ne iii] line flux
110–112 | Ne5_14Flux | W m-2 | 14.32 $\mu$m [Ne v] line flux
113–115 | Ne5_24Flux | W m-2 | 24.32 $\mu$m [Ne v] line flux
116–118 | O4Flux | W m-2 | 25.89 $\mu$m [O iv] line flux
119–121 | S3_18Flux | W m-2 | 18.71 $\mu$m [S iii] line flux
122–124 | S3_33Flux | W m-2 | 33.48 $\mu$m [S iii] line flux
125–127 | S4Flux | W m-2 | 10.51 $\mu$m [S iv] line flux
128–130 | Si2_Flux | W m-2 | 34.82 $\mu$m [Si ii] line flux
131–133 | Ar3Ar2Ratio |  | 8.99 $\mu$m [Ar iii] / 6.99 $\mu$m [Ar ii]
134–136 | Ne3_15Ne2Ratio |  | 15.56 $\mu$m [Ne iii] / 12.81 $\mu$m [Ne ii]
137–139 | Ne5_14Ne2Ratio |  | 14.32 $\mu$m [Ne v] / 12.81 $\mu$m [Ne ii]
140–142 | Ne5_24Ne2Ratio |  | 24.32 $\mu$m [Ne v] / 12.81 $\mu$m [Ne ii]
143–145 | O4Ne2Ratio |  | 25.89 $\mu$m [O iv] / 12.81 $\mu$m [Ne ii]
146–148 | O4S3_33Ratio |  | 25.89 $\mu$m [O iv] / 33.48 $\mu$m [S iii]
149–151 | O4S3_18Ratio |  | 25.89 $\mu$m [O iv] / 18.71 $\mu$m [S iii]
152–154 | S3_33S3_18Ratio |  | 33.48 $\mu$m [S iii] / 18.71 $\mu$m [S iii]
155–157 | S4S3_18Ratio |  | 10.51 $\mu$m [S iv] / 18.71 $\mu$m [S iii]
158–160 | Si2S3_33Ratio |  | 34.82 $\mu$m [Si ii] / 33.48 $\mu$m [S iii]

Table 8: IDEOS observables: synthetic photometry and rest frame continuum flux densities Column | Observable | Unit | Description
---|---|---|---
161–162 | IRAC8Flux | mJy | Synthetic IRAC 8 $\mu$m photometry
163–164 | IRS15Flux | mJy | IRS blue peak-up photometry
165–166 | IRS22Flux | mJy | IRS red peak-up photometry
167–168 | MIPS24Flux | mJy | MIPS 24 $\mu$m photometry
169–170 | MIRI56Flux | mJy | Synthetic MIRI 5.6 $\mu$m photometry
171–172 | MIRI77Flux | mJy | Synthetic MIRI 7.7 $\mu$m photometry
173–174 | MIRI10Flux | mJy | Synthetic MIRI 10 $\mu$m photometry
175–176 | MIRI11Flux | mJy | Synthetic MIRI 11.3 $\mu$m photometry
177–178 | MIRI13Flux | mJy | Synthetic MIRI 12.8 $\mu$m photometry
179–180 | MIRI15Flux | mJy | Synthetic MIRI 15 $\mu$m photometry
181–182 | MIRI18Flux | mJy | Synthetic MIRI 18 $\mu$m photometry
183–184 | MIRI21Flux | mJy | Synthetic MIRI 21 $\mu$m photometry
185–186 | MIRI25Flux | mJy | Synthetic MIRI 25.5 $\mu$m photometry
187–188 | WISE12Flux | mJy | WISE band 3 photometry
189–190 | WISE22Flux | mJy | WISE band 4 photometry
191–193 | CONT37Flux | mJy | Rest frame continuum flux density at 3.7 $\mu$m
194–196 | CONT42Flux | mJy | Rest frame continuum flux density at 4.2 $\mu$m
197–199 | CONT55Flux | mJy | Rest frame continuum flux density at 5.5 $\mu$m
200–202 | CONT97Flux | mJy | Rest frame continuum flux density at 9.7 $\mu$m
203–205 | CONT15Flux | mJy | Rest frame continuum flux density at 15 $\mu$m
206–208 | CONT24Flux | mJy | Rest frame continuum flux density at 24 $\mu$m
209–211 | CONT30Flux | mJy | Rest frame continuum flux density at 30 $\mu$m
212–214 | C15C55Ratio |  | Rest frame continuum slope 15 $\mu$m/5.5 $\mu$m
215–217 | C24C55Ratio |  | Rest frame continuum slope 24 $\mu$m/5.5 $\mu$m
218–220 | C30C55Ratio |  | Rest frame continuum slope 30 $\mu$m/5.5 $\mu$m
221 | SNR66 |  | Continuum S/N at 6.6 $\mu$m
222 | SNR9 |  | Continuum S/N at 9.0 $\mu$m
223 | SNR1125 |  | Continuum S/N at 11.25 $\mu$m
224 | SNR1325 |  | Continuum S/N at 13.25 $\mu$m
225 | SNR14 |  | Continuum S/N at 14 $\mu$m
226 | SNR17 |  | Continuum S/N at 17 $\mu$m
227 | SNR24 |  | Continuum S/N at 24 $\mu$m
228 | SNR30 |  | Continuum S/N at 30 $\mu$m
229 | SL2ScalingFactor |  | Factor by which the SL2 spectral segment has been scaled
230 | SL1ScalingFactor |  | Factor by which the SL1 spectral segment has been scaled
231 | LL2ScalingFactor |  | Factor by which the LL2 spectral segment has been scaled
232 | LL1ScalingFactor |  | Factor by which the LL1 spectral segment has been scaled

##  11 Discussion

An unexpected by-product of the spectral fitting of the IDEOS spectra has been
the discovery of crystalline silicate emission and absorption features in the
23–34 $\mu$m range of almost all good S/N spectra with coverage of these
features. The large majority of these spectra show these forsterite features
in emission: 417 times in emission and 9 times in absorption for the 33 $\mu$m
feature; 97 times in emission and 76 times in absorption for the 28 $\mu$m
feature; 320 times in emission and 235 times in absorption for the 23 $\mu$m
feature. Only the shorter wavelength 16 and 19 $\mu$m forsterite features are
fitted exclusively in absorption in spectra that satisfy Ssil$<$-0.5, where
they are detected 149 times.

We have found that the strength of the five fitted crystalline features is
correlated with the strength of the 9.8 $\mu$m amorphous silicate feature
(Fig. 20) and that the longer wavelength crystalline silicate bands require a
deeper 9.8 $\mu$m silicate absorption feature for them to be seen in
absorption than their shorter wavelength counterparts (Figs. 22 & 23). These
findings are reminiscent of the structure seen in spectra of evolving AGB
stars. The silicate bands in these sources are found to be in emission or
absorption (see Fig. 33) depending on the thickness of the dust shell
(Sylvester et al., 1999; Dijkstra et al., 2003; de Vries et al., 2014; Golriz
et al., 2014).

For galaxies in our sample for which all silicate features are seen in
emission (105 sources) in theory another kind of dust geometry may apply: that
of a class i or ii protoplanetary disc (Meeus et al., 2001; Watson et al.,
2009; Furlan et al., 2011). In these flared discs the dust temperature
increases away from the mid-plane, as the surface layer of the disc is
directly exposed to the radiation of the protostar. As a result all silicate
features are seen in emission. Most of the IDEOS sources that share these
spectral properties are AGN, whose dust geometries may have many similarities
to those of protoplanetary disks.

In analogy to the evolving AGB stars discussed above, IDEOS galaxies with at
least one forsterite band in absorption (323 sources) should host a centrally
heated dust structure, which we identify with the galaxy’s nucleus. As
illustrated in Fig. 22, the 23 and 33 $\mu$m forsterite bands can be used to
classify galaxies as hosting a deeply obscured, semi obscured and unobscured
nucleus. With the 28 $\mu$m forsterite feature as a proxy for the 33 $\mu$m
forsterite feature in sources for which the 33 $\mu$m forsterite band is
redshifted out of the IRS wavelength range, we are able to identify a total of
40 deeply obscured galactic nuclei. As can be seen in Fig. 24, not all of
these sources (color-coded as blue) are found in the upper left part of the
Fork Diagram (class 3A), which is home to the most deeply obscured galactic
nuclei. As demonstrated in Sect. 9.3, circumnuclear star formation shifts
these sources along the diagonal branch of the Fork Diagram towards class 1C.
Class 3A is hence the locus of ’naked’ buried nuclei: those that are not
’dressed up’ with circumnuclear star formation. We predict that millimetre
HCN-vib studies looking for Compact Obscured Nuclei (CONs; Falstad et al.,
2021) will expose most of our 40 deeply obscured galaxies as CONs. We are
especially confident about this since all CONs and non-CONs in the sample of
Falstad et al. (2021) align with our crystalline silicate-based
classifications for the sources that we have in common.

The case of Mrk 231 offers another lesson. Our crystalline diagnostics
classify this ULIRG as containing a semi-obscured nucleus (the 33 $\mu$m
feature is in emission, the 23 $\mu$m feature in absorption). A keyhole view
through the obscuring dust shell (Marshall et al., 2018) of this BAL QSO
results in a significant reduction of the depth of the 9.8 $\mu$m silicate
feature, resulting in a significant downward shift in the Fork Diagram. The
silicate strength is hence not a good proxy for the level of nuclear
obscuration if: 1\. There are light leaks (e.g. Mrk 231). 2\. Circumnuclear
star formation strongly contributes to the integrated SED (e.g. NGC 4945). 3\.
The dust distribution is clumpy (Nenkova et al., 2002). Sources well below the
diagonal branch of the Fork Diagram may thus be affected by these AGN
orientation effects or by light leaks.

For sources with a centrally heated dust geometry one can attempt to infer the
crystallinity of the ISM, provided that both the amorphous and the crystalline
silicate band used to probe the column density are in the optically thick
regime. The shorter the wavelengths of the features used, the more easily this
precondition is met. Kemper et al. (2004) exploited the proximity of the 11.2
$\mu$m forsterite band to the 9.8 $\mu$m amorphous silicate band to infer an
upper limit of CISM(11.2 $\mu$m)=0.011 to the crystallinity of the Galactic
Center line of sight. Contamination by 11.2 $\mu$m PAH emission forced Spoon
et al. (2006) to use a longer wavelength forsterite band, at 16.3 $\mu$m, for
their crystallinity determinations in a sample of 12 ULIRGs. They found
CISM(16 $\mu$m)252525Using our definition of crystallinity: Eq. 2 to range
from 0.07 to 0.13. For the centrally heated sources in the IDEOS sample we
have used the more commonly studied 19 $\mu$m forsterite band and were able to
infer crystallinities for 254 sources. We found an average crystallinity
CISM(19 $\mu$m)=0.027 with no apparent dependence on the silicate strength
(see Fig. 27). These crystallinities are far lower than previously reported by
Spoon et al. (2006) for their ULIRG sample and can be entirely explained by
our different assumption for the mass absorption coefficient and our different
choice of forsterite band for the measurement (see Appendix C). X-ray
absorption spectroscopy probing the dense ISM towards bright X-ray binaries in
the Galactic Center region (Rogantini et al., 2020; Zeegers et al., 2019) has
revealed average crystallinities to be markedly higher (on the order of 10%)
than the upper limit (1.1%) derived from the non-detection of the 11.2 $\mu$m
forsterite band Kemper et al. (2004) towards Sgr A∗. Our IR-derived
crystallinities may thus be equally underestimating the true forsterite
abundance in galactic nuclei.

An intriguing result of our study is that it appears that crystalline silicate
features are present in all galaxy spectra obtained at sufficient S/N.
Depending on the dust configuration and the S/N at 16–34 $\mu$m the detections
may be limited to just the strong 33 $\mu$m forsterite feature in emission, a
23 $\mu$m feature in absorption, or a bucket-shaped 18 $\mu$m silicate
absorption feature carved out by forsterite absorption at 16 and 19 $\mu$m.
Fact is that our analysis shows that only one source, 3C 390.3, positively
shows no crystalline silicate bands to the strict limits we set for non-
detection of the forsterite bands. Even though we cannot exclude the existence
of more sources like 3C 390.3 hiding among lower S/N spectra in our sample,
the results found based on 960 galaxies with coverage of all five crystalline
silicate bands makes it plausible to assume that crystalline silicates are a
common component of the ISM of all galactic nuclei in the Local Universe.

We can only speculate on why we find hardly any exceptions. Small number
statistics may be one reason. 3C 390.3 is part of a sample of only 158 very
high262626This sample has SNR$\geq$100 at 24 and 30 $\mu$m and coverage of all
23–33 $\mu$m forsterite bands. S/N spectra, with 802 other galaxies making up
the rest of our study sample272727The median SNR of our study sample is 48 and
61 at 24 and 30 $\mu$m, respectively.. On the other hand, of the 960 galaxies
with coverage of all crystalline bands we detect forsterite in 538 and only in
4/960 we do not. Likely the large majority of the remaining 418 lower S/N
sources will be like the 538 that do have detections. Another possible reason
is more astrophysical in nature. The dust content of metal-rich galaxies is
believed to be dominated by ISM dust production, which will form amorphous
silicates (Asano et al., 2013). The stardust contribution (from evolved stars
and supernovae) is minor. In addition, both the input of stardust and its
destruction by supernovae depend on the star formation rate (SFR), which may
result in a net weak dependence of the stardust abundance on the SFR,
especially if a starburst occurs in a dusty galaxy. This may result in the
observed apparent lack of correlation between galaxy type and detection rate
of crystalline silicates.

In this context it would be interesting to study the dust content in low-
metallicity galaxies that have an ongoing starburst. In such systems the
freshly made stardust may dominate the total dust content, and would provide a
clean measure of the stardust destruction rate by supernovae.

With the launch of JWST just a few days away, a new era of mid-infrared
spectroscopy will soon arrive, providing the opportunity to disentangle the
nuclear and circumnuclear spectral properties for galaxies beyond the next few
Mpc. While this should remove some of the current guesswork, the long
wavelength cut-off at 28 $\mu$m will also limit the number of available
diagnostic features, the crystalline silicates in particular.

##  12 Conclusions

The homogeneous analysis of the homogeneously extracted, stitched, and curated
Spitzer-IRS low-resolution spectra in IDEOS has produced a wealth of
observables for more than 3300 galaxies and galactic nuclei. Among the 74
observables in IDEOS are PAH fluxes and their equivalent widths, the strength
of the 9.8 $\mu$m silicate feature, emission line fluxes, solid-state
features, rest frame continuum fluxes, synthetic JWST, Spitzer and WISE
photometry, and a mid-infrared spectral galaxy classification. Our web portal
allows each of our fits to be inspected. Our spectral fits allowed us to
identify 388 active galactic nuclei (AGN) based on the detection of emission
lines of [Ne v] at 14.3 or 24.3 $\mu$m.

Emission line diagnostic diagrams based on the ratio of a high-ionization
tracer like [O iv] or [Ne v] to a low-ionization tracer like [Ne ii] or [S
iii] on one axis plotted against the ratio of a medium-ionization tracer like
[Ne iii] or [Ar iii] to a low-ionization tracer like [Ne ii] or [Ar ii] on the
other axis are effective tools to separate out low-metallicity galaxies from
galaxies on the AGN/starburst scale. The former lack $>$54eV photons needed to
create O3+ and Ne4+, but are able to drive ratios of medium-to-low ionization
tracers like [Ne iii]/[Ne ii] and [Ar iii]/[Ar ii] to higher values than AGN-
dominated galaxies do.

We have detected weak emission and absorption features of crystalline
silicates at 16, 19, 23, 28 and 33 $\mu$m in the spectra of 786 IDEOS
galaxies. Quantitative analysis of the detected crystalline and amorphous
silicate bands in this large galaxy sample reveals several trends. First, the
23, 28 and 33 $\mu$m crystalline silicate bands can de detected both in
emission and in absorption. The 16 and 19 $\mu$m bands are fitted only in
absorption. Second, the strengths of the various crystalline silicate bands
show a positive correlation with the amorphous silicate strength. Third,
crystalline silicate bands switch from absorption to emission at a more
negative silicate strength as the wavelength of the crystalline band is
longer. These observed characteristics are consistent with an origin for the
amorphous and crystalline silicate features in a centrally heated dust shell –
a dusty or buried nucleus. Based on the detection of one or more crystalline
silicate bands in absorption, we find at least 323 galaxies in the IDEOS
sample that host a centrally heated dust structure. For a galaxy showing the
9.8 $\mu$m silicate feature in emission our observations may also be
consistent with an origin of the crystalline silicate detections in an AGN
torus geometry.

Exploiting the nature of centrally heated dust structures, we have used the
sign of the 23 and 33 $\mu$m crystalline silicate strengths – emission or
absorption – to classify galaxies as deeply obscured, semi obscured or
unobscured. Based on this diagnostic, we identify buried nuclei in a total of
40 galaxies. Interestingly, the 5–20 $\mu$m spectra of these galaxies range
from fully absorption-dominated to fully PAH-dominated, which is likely the
result of varying contributions of circumnuclear star formation to their
galaxy-integrated spectra. In the Fork Diagram these same galaxies span the
full length of the diagonal branch, with the most extreme member shifted all
the way to the locus of dusty starburst galaxies by drowning the spectral
signatures of its buried nucleus in PAH emission. Above 20 $\mu$m the
circumnuclear spectrum is featureless, thus preserving the crystalline
silicate signatures of the buried nucleus in the galaxy-integrated spectrum.

We have used the strength of the 9.8 $\mu$m amorphous and the 19 $\mu$m
crystalline silicate absorption bands to infer the crystallinity of the ISM in
obscured galactic nuclei (Ssil$<$-1). For the 114 sources with a detection of
the 19 $\mu$m band crystallinities range from 0.7% to 6% with an average level
of 2.7%, with no dependence on the level of obscuration. Based on our
detection statistics across all five forsterite bands in the 16–34 $\mu$m
range, we conclude that crystalline silicates are a common component of the
ISM of local (z$<$0.068) galactic nuclei.

We would like to thank the anonymous referee for constructive comments which
helped to improve this paper. We further would like to thank John Miles for
help with setting up the precursor to CHUNKFIT, Jack Gallimore for sharing his
adaptation of PAHFIT, Lee Armus, Vassilis Charmandaris, Daniel Dale, Andreas
Efstathiou, Bill Forrest, Frédéric Galliano, Ismael Garcia-Bernete, Dimitra
Rigopoulou, and J.-D. Smith for discussions, B. Sargent for providing Galactic
comparison spectra, Jason Marshall for providing a galaxy disk template
spectrum, Pedro Beirao, Karl Gordon, Els Peeters and Miguel Pereira-Santaella
for galaxy disk spectra, and Margaret Meixner for suggesting to add JWST-MIRI
bands to our suite of synthetic photometric fluxes. HS would also want to
acknowledge his friend Jack Waas for designing the logo of the IDEOS project
for the web portal. The IDEOS project is funded by NASA under awards
NNX13AE69G and NNX16AF26G. This research has made use of the NASA/IPAC
Extragalactic Database (NED), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.

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##  Appendix A 5.5–8 $\mu$m absorption profile template

We have searched for a template to fit the absorption features commonly seen
in the 5.5–8 $\mu$m spectra of strongly obscured galactic nuclei. To qualify,
the spectra needed to be free of PAH contamination in the 5.5–8 $\mu$m range,
have a silicate strength S${}_{\rm sil}<$-2.5, and be detected at high S/N.
The resulting sample consists of just three galaxies, NGC 4418 (Spoon et al.,
2001), IRAS F00183–7111 (Spoon et al., 2004), and IRAS 08572+3915 (Spoon et
al., 2006). We infered their optical depth spectra by performing a spline fit
as described in Sect. 7.2 and illustrated in Fig. 13. The optical depth
spectra for the three galaxies are shown in Fig. 29 together with two
laboratory spectra of pure water ice282828Sackler Laboratory Ice Database:
http://icedb.strw.leidenuniv.nl/, and one of an a:C–H analog (Dartois et al.,
2005).

The three galaxy spectra clearly show resemblance to features seen in the
laboratory profiles: the optical depth peaks at 6.85 and 7.25 $\mu$m agree in
peak position and width with the structure in the spectrum of the a:C–H
hydrogenated amorphous carbon analog (Dartois et al., 2007), while the
position and width of the broad 6 $\mu$m feature agrees with the profile of
120 K water ice. Neither compound, nor a combination of the two, will,
however, reproduce the empirical profiles to a degree that they can be used as
templates in the fitting of the IDEOS spectra. The true composition will
likely be far more complex and diverse than can be infered from these spectra.

Figure 29: Comparison of the normalized 5.5–8 $\mu$m optical depth profiles of
three galactic nuclei. Overlaid with dotted lines are the profiles of pure
water ice and a a:C-H hydrogenated amorphous carbon analog.  Figure 30:
Absorption features in the 5–8 $\mu$m spectrum of NGC 4418. Upper panel: the
Spitzer-IRS low-resolution spectrum of NGC 4418 (Spoon et al., 2007) with the
local 5.6–7.8 $\mu$m absorption-corrected continuum shown as a dashed line.
Lower panel: Optical depth spectrum resulting from adopting the local 5.6–7.8
$\mu$m spline continuum shown in the upper panel. Overlaid in red is the 6
$\mu$m water ice profile and overlaid in blue is the aliphatic CH deformation
mode feature to be used for fitting IDEOS spectra.

We therefore resorted to using an empirical optical depth profile as our
template, and separated the contribution of the CH deformation modes at 6.85 &
7.25 $\mu$m from the rest of the optical depth profile (6 $\mu$m profile
hereafter) to maximise the ability to fit these features in other IDEOS
spectra. We chose the spectrum of NGC 4418 as our template292929Note that the
spectral decomposition codes CAFE (Marshall et al., 2007) and QUESTFIT
(Veilleux et al., 2009) prefer the optical depth profiles of IRAS F00183–7111
and IRAS 08572+3915, respectively, for their spectral fitting., over the
spectra of IRAS F00183–7111 and IRAS 08572+3915, because its more symmetric 6
$\mu$m profile provides better fits to most spectra than the highly asymmetric
6 $\mu$m profiles of IRAS F00183–7111 and IRAS 08572+3915 (Fig. 29).

Following Dartois et al. (2007), we fit the 6.85 and 7.25 $\mu$m aliphatic CH
deformation absorption bands with gaussian profiles and subtract these from
the optical depth spectrum. The remaining optical depth profile in the 5.6–6.7
$\mu$m range is then attributed to 6 $\mu$m water ice and is smoothed so that
it can be used as a water ice template for fitting IDEOS spectra. This profile
is overlaid in red in the lower panel of Fig. 30. The blue profile in the same
panel is the combined aliphatic CH deformation mode profile.

##  Appendix B PAH profile templates

The PAH emission features at 6.22, 11.25 and 12.65 $\mu$m (PAH62, PAH112, and
PAH127) have asymmetric profiles, requiring either a combination of gaussian
probability functions, or a single asymmetric probability function to
accurately describe the observed shapes.

We chose the type IV probability function of Pearson (1895), used to model
pull distributions with non-gaussian tails, for our purposes. The functional
form for this Pearson-IV profile, P(x), is as follows:

$P(x)=k\ [1+(\frac{x-\lambda}{a})^{2}\ ]^{-m}\ {\rm exp}[\ -\nu\ {\rm
tan}^{-1}(\frac{x-\lambda}{a})\ ]$ (B1)

where the normalization coefficient k is defined by Heinrich (2004) for
m$>$1/2 as:

$k=\frac{1}{\sqrt{\pi}\ a}\ \frac{\Gamma(m)}{\Gamma(m-1/2)}\
\left|\frac{\Gamma(m\ +\ i\nu/2)}{\Gamma(m)}\right|^{2}$ (B2)

where $\Gamma$ is defined as

$\Gamma(x)=\int_{0}^{\infty}t^{x-1}\ e^{-t}\ dt$ (B3)

The last term in Eq. B2 can be computed using a piece of computer code,
GAMMAR2, provided in Sect. 5.1 of Heinrich (2004).

Table 9 shows for each of the three PAH profiles the allowed parameter ranges
for the four parameters in the Pearson-IV function. Only $a$ and $\lambda$ are
allowed to vary. The former to accommodate the variation in width of the PAH
feature resulting from the factor $\sim$2 variation in spectral resolution.
The latter to allow compensation for the shift of the peak as a result of a
change in $a$. See Fig. 31.

t  PAH62 PAH112 PAH127 $\lambda$ 6.14–6.152 11.07–11.14 13.02–13.04 a
0.095–0.11 0.13–0.155 0.345–0.375 m 2.0 2.0 5.0 $\nu$ -2.5 -4.0 10.0

Table 9: Pearson IV parameters used to define shapes of PAH profiles Figure
31: Normalized pearson-IV profiles used to model the PAH62, PAH112 and PAH127
features in the SED fitting of the 5.39–7.25 $\mu$m and 9.8–13.5 $\mu$m
ranges. Different colors are used to show the effect of varying the A
parameter within the allowed ranges.

##  Appendix C Crystalline silicate templates

Following Spoon et al. (2006), we attribute the 16.1, 18.5, 23.3, 27.6, and
33.2 $\mu$m absorption and emission bands seen in the IDEOS galaxy sample to
forsterite crystalline silicates and fit these bands with the empirical
profiles described below and as shown in Fig. 32.

Figure 32: Normalized optical depth profiles of crystalline silicates found in
spectra of IDEOS galaxies. The panel of the left shows the profiles of the
16.1 and 18.5 $\mu$m forsterite bands, which are fitted together. The other
forsterite bands, centered at 23.3, 27.6 and 33.2 $\mu$m, are fitted
individually.

In Galactic sources the 16.1 $\mu$m forsterite band can be easily identified
in the spectra of select young and old stars (Fig. 33), both in emission (e.g.
T Tauri stars IRAS F12571–7657 and SSTc2d J033035.9+303024; see the spectra in
Olofsson et al. (2009)) and in absorption (e.g. OH/IR stars IRAS 17347–2319,
IRAS 17276–2846, and OH 21.5+0.5; see the spectra in Jones et al. (2012)). The
band profile looks the same as in deeply obscured galactic nuclei (NGC 4418,
IRAS 08572+3915, and IRAS 04454–4838), as can be seen in the inset of Fig. 33,
after removal of the spline-fitted amorphous silicate component (Spoon et al.,
2006). We therefore pick the 16.1 $\mu$m profile of the highest S/N galactic
nucleus, IRAS 08572+3915, to serve as the profile to be used in the spectral
fitting described in Sect. 3.5.

For some sources in Fig. 33 the strength of the 16.1 $\mu$m band is almost
equal to the strength of the broader forsterite band peaking at 18.5–19
$\mu$m. The profile of the latter shows some variation among the galactic
nuclei, with some sources peaking near 18.5 $\mu$m (e.g. IRAS 08572+3915 and
IRAS 04454–4838) and others near 19.0 $\mu$m (e.g. NGC 4418). The latter is
where the profiles of the stellar sources peak. Since the 16.1 and 18.5 $\mu$m
forsterite features are fitted jointly with a fixed profile ratio, we choose
the high S/N spectrum of IRAS 08572+3915 to also provide the template for the
18.5 $\mu$m forsterite profile.

Figure 33: Silicate optical depth profiles of three deeply obscured galactic
nuclei (NGC 4418, IRAS 08572+3915, and IRAS 04454-4838), five stellar sources
(SSTc2d J033035.9+303024, IRAS F12571–7657, IRAS 17347–2319, IRAS 17276–2846,
and OH 21.5+0.5 (Sloan et al., 2003)), and the Galactic Center Source GCS-3
(Chiar&Tielens, 2006). The silicate absorption spectra have been scaled to a
9.8 $\mu$m optical depth of 4. The silicate emission sources have not. The
inset shows the residual optical depth features in the 14.5–26 $\mu$m range
after subtraction of the spline-fitted amorphous silicate component (Spoon et
al., 2006). GCS-3 is not included as it does not show crystalline signatures.
For clarity, the residuals have been scaled to a common 16.1 $\mu$m absolute
value.

The peak-to-peak profile ratio for the 16.1 and 18.5 $\mu$m bands in IRAS
08572+3915 is 1.0:0.83 (Fig. 32). This is far higher than calculated opacity
curves for forsterite under various assumptions of grain size, grain shape,
and optical constants predict. Take for example the forsterite opacity curves
shown in Fig. 2 of de Vries et al. (2015), computed using the DHS grain shape
distributions of Min et al. (2005) using optical constants of Suto et al.
(2006); or the forsterite opacity curve shown in Fig. 2 of Spoon et al.
(2006), which uses the optical constants of Fabian et al. (2001) and the
assumption of a CDS grain shape distribution (Bohren et al., 1983). Invariably
the 16.1 $\mu$m forsterite band is much weaker than the 19 $\mu$m band.

Figure 34: Optical depth profile for ESO 374-IG 032 NED02 in the 18 $\mu$m
silicate absorption band (black). Dashed lines trace the amorphous 18 $\mu$m
profile resulting from a spline fit to pivots at 14.5, 15.0, 17.0, 17.5, 20.0,
and 21.0 $\mu$m. Residual optical depth features centered at 16.1, 18.5, and
23.3 $\mu$m are attributed to forsterite absorption. The profiles shown in red
and in shades of gray are the result of subtracting the noise-free pure PAH
template #3 of Smith et al. (2007) in increasing amounts.

This leads us to explore an alternative origin for the 16.1 and 18.5 $\mu$m
absorption bands in galaxies, despite strong empirical evidence provided by
the stellar spectra shown in Fig. 33. Could the appearance of 16.1 and 18.5
$\mu$m absorption features in the spectra of buried galactic nuclei be
ascribed to the presence of PAH emission from the 17 $\mu$m PAH complex (Smith
et al., 2007) instead? Fig. 34 shows the 14–25 $\mu$m spectrum of ESO 374-IG
032 NED02, a deeply obscured galactic nucleus with clear signs of PAH emission
in the 6–13 $\mu$m range and at 16.4 and 17.4 $\mu$m. Using the noise-free PAH
template #3 of Smith et al. (2007), we subtracted as much of the 17 $\mu$m PAH
complex emission from the galaxy spectrum as is consistent with the amount of
PAH emission found in the 6–13 $\mu$m range. As can be seen in Fig. 34, the
16.4 and 17.4 $\mu$m PAH bands disappear as the amount of subtracted PAH
template emission increases, yet the 16.1 and 18.5 $\mu$m absorption
structures remain even in the red spectrum, which represents moderate
oversubtraction of 6–13 $\mu$m PAH emission. A full fit to the latter spectrum
(the red dashed line in Fig. 34) would measure almost the same forsterite
strength as a fit to the original spectrum would. We thus conclude that the
16.1 and 18.5 $\mu$m forsterite absorption bands cannot be mimicked by
emission from the 17 $\mu$m PAH complex.

The profiles used to fit the 23.2 $\mu$m and the 27.6 $\mu$m forsterite bands
in the 21.5–36 $\mu$m spectral range (Sect. 3.6) have been adapted from the
average crystalline silicate optical depth spectrum of 12 ULIRGs obtained by
Spoon et al. (2006). This average profile has higher S/N in the 21–30 $\mu$m
range than the profile of IRAS 08572+3915. The resulting profiles,
characteristic of forsterite, are shown in the middle panels of Fig. 32.

The profile of the 33.2 $\mu$m crystalline silicate band is best determined
from high S/N spectra of galaxies in mid-IR classes 1A–C and 2A, which show
the feature in emission. The galaxies we selected have the following IDEOS
IDs: 10510592_0, 10511872_0, 10870272_0, 10871808_0, 11290112_1, 12432128_3,
14839552_0, 14840320_0, 14842112_0, and 14869504_0. To obtain the feature
profile we first averaged the continuum-normalized spectra, and then removed
the profile of the 33.48 $\mu$m [S iii] line. The resulting forsterite profile
is shown in the right panel of Fig. 32.


Comments on “Testing Conditional Independence of Discrete Distributions”

Ilmun Kim

Department of Statistics and Data Science

Yonsei University

###### Abstract

In this short note, we identify and address an error in the proof of Theorem
1.3 in Canonne et al., (2018), a recent breakthrough in conditional
independence testing. After correcting the error, we show that the general
sample complexity result established in Canonne et al., (2018) remains the
same.

##  1 A Closer Look at Canonne et al., (2018)

Let $(X,Y,Z)$ follow a distribution $p_{X,Y,Z}$ on a discrete domain
$[\ell_{1}]\times[\ell_{2}]\times[n]$. Given $m$ i.i.d. copies of $(X,Y,Z)$,
the work of Canonne et al., (2018) is concerned with the problem of testing
conditional independence between $X$ and $Y$ given $Z$. Let $\mathcal{P}$
denote the set of all discrete distributions over
$[\ell_{1}]\times[\ell_{2}]\times[n]$ and $\mathcal{P}_{0}\subset\mathcal{P}$
denote the set of null distributions such that $X\perp\\!\\!\\!\\!\perp
Y\,|\,Z$. Consider the set of alternative distributions, denoted by
$\mathcal{P}_{1}(\varepsilon)$, which are $\varepsilon$ far way from
$\mathcal{P}_{0}$ in the total variation distance:

$\displaystyle\mathcal{P}_{1}(\varepsilon)=\Big{\\{}p\in\mathcal{P}:\inf_{q\in\mathcal{P}_{0}}\sup_{A\subseteq[\ell_{1}]\times[\ell_{2}]\times[n]}\big{|}p(A)-q(A)\big{|}\geq\varepsilon\Big{\\}}.$

Given these sets of distributions, the hypotheses of interest can be
formulated as

$\displaystyle H_{0}:p_{X,Y,Z}\in\mathcal{P}_{0}\quad\text{versus}\quad
H_{1}:p_{X,Y,Z}\in\mathcal{P}_{1}(\varepsilon).$

For the above testing problem, Theorem 1.3 of Canonne et al., (2018) proves
that there exists a test with sample complexity

$\displaystyle
O\Biggl{(}\max\Biggl{\\{}\min\Biggl{\\{}\frac{n^{7/8}\ell_{1}^{1/4}\ell_{2}^{1/4}}{\varepsilon},\
\frac{n^{6/7}\ell_{1}^{2/7}\ell_{2}^{2/7}}{\varepsilon^{8/7}}\Biggr{\\}},\
\frac{n^{3/4}\ell_{1}^{1/2}\ell_{2}^{1/2}}{\varepsilon},\
\frac{n^{2/3}\ell_{1}^{2/3}\ell_{2}^{1/3}}{\varepsilon^{4/3}},\
\frac{n^{1/2}\ell_{1}^{1/2}\ell_{2}^{1/2}}{\varepsilon^{2}}\Biggr{\\}}\Biggr{)}.$
(1)

This result is achieved under Poissonization where the sample size $M$ follows
a Poisson random variable with parameter $m$. The proof of Theorem 1.3 relies
on several innovative lemmas including Claim 2.2:

Claim 2.2 of Canonne et al., (2018). _There exists an absolute constant $C>0$
such that, for $X\sim\mathrm{Poisson}(\lambda)$ and $a,b\geq 0$,_

$\displaystyle\mathrm{Var}\bigl{[}X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq
4)\bigr{]}\leq C\mathbb{E}\bigl{[}X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq
4)\bigr{]}.$

Unfortunately, the upper bound in the claim turns out to be incorrect and
therefore the proof of their Theorem 1.3 remains incomplete (confirmed via
personal communication with one of the authors). We fix this error in Lemma 1
below, which is the main technical contribution of this work. To illustrate
the error in Claim 2.2, suppose that the parameter $\lambda$ is sufficiently
large such that $X$ is approximately $N(\lambda,\lambda)$ by the central limit
theorem. Moreover, by taking $a$ and $b$ relatively small in comparison to
$\lambda$, one can have an approximation

$\displaystyle X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq 4)\approx
N(\lambda,\lambda)\sqrt{ab}.$

Hence the variance and the expectation in Claim 2.2 approximately become
$\lambda ab$ and $\lambda\sqrt{ab}$, respectively, which contradicts the given
inequality when $a,b\geq 1$.

##  2 Main Result

We now show that the upper bound in Claim 2.2 holds if we add an additional
factor of $\max\\{ab,\sqrt{a}b,\sqrt{ab}\\}$ for $b\geq a\geq 0$ in the upper
bound.

######  Lemma 1 (Correction of Claim 2.2).

There exists an absolute constant $C>0$ such that, for
$X\sim\mathrm{Poisson}(\lambda)$ and $b\geq a\geq 0$,

$\displaystyle\mathrm{Var}\bigl{[}X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq
4)\bigr{]}$ $\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
C\max\\{ab,\sqrt{a}b,\sqrt{ab}\\}\mathbb{E}\bigl{[}X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq
4)\bigr{]}.$

It is worth pointing out that one can improve the factor of
$\max\\{ab,\sqrt{a}b,\sqrt{ab}\\}$ in the upper bound with more effort.
Nevertheless, the current form is sharp enough to reprove the sample
complexity (1). We also note that in its application, $a$ and $b$ correspond
to the domain sizes $\ell_{1}\geq 2$ and $\ell_{2}\geq 2$, respectively, in
which case the additional factor becomes
$\max\\{ab,\sqrt{a}b,\sqrt{ab}\\}=ab$. Due to the non-linearity of
$X\sqrt{\min(X,a)\min(X,b)}\mathds{1}(X\geq 4)$ in $X$, the proof of Lemma 1
turns out to be non-trivial, requiring a delicate case-by-case analysis. The
details can be found in Section 3.

Having provided a correction of Claim 2.2, we next reprove Theorem 1.3 of
Canonne et al., (2018). There is essentially one place where Claim 2.2 is used
in Canonne et al., (2018), more specifically in their Lemma 5.3. Once we
establish Lemma 5.3, the rest of the proof remains the same. Before stating
the result, we need to introduce some notation. For simplicity, we denote the
conditional distribution of $X$ and $Y$ given $Z=z$ by $p_{z}$ and the product
distribution of their marginals by $q_{z}$. Define
$\varepsilon_{z}=\sup_{A\subseteq[n]}|p_{z}(A)-q_{z}(A)|$ and write
$\varepsilon_{z}^{\prime}=\varepsilon_{z}/\sqrt{4\ell_{1}\ell_{2}}$. We
further let
$\omega_{z}=\sqrt{\min(\sigma_{z},\ell_{1})\min(\sigma_{z},\ell_{2})}$. With
this notation in place, we reprove Lemma 5.3 of Canonne et al., (2018) as
follows.

######  Lemma 2 (Lemma 5.3 of Canonne et al., (2018)).

Consider $D=\sum_{z\in[n]}\sigma_{z}\omega_{z}\varepsilon_{z}^{\prime
2}\mathds{1}(\sigma_{z}\geq 4)$ where $\sigma_{1},\ldots,\sigma_{n}$ are
independent Poisson random variables with the corresponding parameters
$m\mathbb{P}(Z=1),\ldots,m\mathbb{P}(Z=n)$. Then there exists a constant $C>0$
such that $\mathrm{Var}[D]\leq C\mathbb{E}[D]$.

###### Proof.

Using the independence of $\sigma_{1},\ldots,\sigma_{n}$, we have the identity

$\displaystyle\mathrm{Var}[D]\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{z\in[n]}\varepsilon_{z}^{\prime
4}\mathrm{Var}[\sigma_{z}\omega_{z}\mathds{1}(\sigma_{z}\geq 4)].$

Moreover, Lemma 1 along with a trivial bound $\varepsilon_{z}^{\prime}\leq
1/\sqrt{4\ell_{1}\ell_{2}}$ yields

$\displaystyle\sum_{z\in[n]}\varepsilon_{z}^{\prime
4}\mathrm{Var}[\sigma_{z}\omega_{z}\mathds{1}(\sigma_{z}\geq
4)]\leavevmode\nobreak\ \leq\leavevmode\nobreak\ $ $\displaystyle
C\sum_{z\in[n]}\varepsilon_{z}^{\prime
4}\ell_{1}\ell_{2}\mathbb{E}[\sigma_{z}\omega_{z}\mathds{1}(\sigma_{z}\geq
4)]$ $\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
C\sum_{z\in[n]}\varepsilon_{z}^{\prime
2}\mathbb{E}[\sigma_{z}\omega_{z}\mathds{1}(\sigma_{z}\geq
4)]=C\mathbb{E}[D].$

This completes the proof of Lemma 2, which in turn proves Theorem 1.3 of
Canonne et al., (2018). ∎

##  3 Proof of Lemma 1

It now remains to prove Lemma 1. We begin with the notation used in the proof.

Notation. We use $C,C^{\prime}$ to denote positive absolute constants whose
values may vary from line to line. For constants $x$ and $y$, we write
$x\lesssim y$ (resp. $x\gtrsim y$) if there exists an absolute positive
constant $C$ (resp. $C^{\prime}$) such that $x\leq Cy$ (resp. $x\geq
C^{\prime}y$). $a\wedge b$ indicates $\min(a,b)$. For simplicity, we often
write

$\displaystyle X_{a,b}:=X\sqrt{(X\wedge a)(X\wedge b)}\mathds{1}(X\geq 4).$

Our proof relies on two other claims in Canonne et al., (2018). We recall them
here for completeness.

Claim 2.1 of Canonne et al., (2018). _For $X\sim\mathrm{Poisson}(\lambda)$,
the following inequality holds_

$\displaystyle\mathrm{Var}[X\mathds{1}(X\geq 4)]\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ \mathbb{E}[X\mathds{1}(X\geq 4)].$ (2)

Claim 2.3 of Canonne et al., (2018) _For $X\sim\mathrm{Poisson}(\lambda)$ and
integers $a,b\geq 2$, the following inequality holds_

$\displaystyle\mathbb{E}\bigl{[}X\sqrt{(X\wedge a)(X\wedge b)}\mathds{1}(X\geq
4)\bigr{]}\leavevmode\nobreak\ \gtrsim\leavevmode\nobreak\
\bigl{(}\lambda\sqrt{(\lambda\wedge a)(\lambda\wedge
b)}\wedge\lambda^{4}\bigr{)}.$ (3)

We split the proof into two parts depending on whether $a\geq 2$ or $0\leq
a<2$.

###  3.1 Case where $a\geq 2$

We start with the first case where $a\geq 2$. In our analysis, it is more
convenient to work with an alternative form of the variance of $X_{a,b}$.
First notice that for any random variable $W$ and its i.i.d. copy
$W^{\prime}$, the variance of $W$ is equal to

$\displaystyle\mathrm{Var}[W]=\frac{\mathbb{E}\bigl{[}(W-W^{\prime})^{2}\bigr{]}}{2}.$

Let $Y$ be an i.i.d. copy of $X\sim\mathrm{Poisson}(\lambda)$. Using the
alternative formula of the variance, we have the identity given as

$\displaystyle
2\mathrm{Var}[X_{a,b}]=\mathbb{E}\bigl{[}\bigl{(}X\sqrt{(X\wedge a)(X\wedge
b)}\mathds{1}(X\geq 4)-Y\sqrt{(Y\wedge a)(Y\wedge b)}\mathds{1}(Y\geq
4)\bigr{)}^{2}\bigr{]}.$

Equivalently, by letting $p$ be the probability mass function (pmf) of
$\mathrm{Poisson}(\lambda)$,

$\displaystyle 2\mathrm{Var}[X_{a,b}]\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\underbrace{\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge
b)}\Bigr{)}^{2}p(x)p(y)}_{:=(\mathrm{I})}$ $\displaystyle+\leavevmode\nobreak\
$
$\displaystyle\underbrace{\sum_{x=4}^{\infty}\sum_{y=0}^{3}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}\Bigr{)}^{2}p(x)p(y)}_{:=(\mathrm{II})}$
$\displaystyle+\leavevmode\nobreak\ $
$\displaystyle\underbrace{\sum_{x=0}^{3}\sum_{y=4}^{\infty}\Bigl{(}y\sqrt{(y\wedge
a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)}_{:=(\mathrm{III})},$

where we use the fact that both $X_{a,b}$ and $Y_{a,b}$ are equal to zero when
$x\leq 3$ and $y\leq 3$.

Analysis of (II) and (III). Let us start bounding the second term (II) as it
is simpler to analyze than the first term (I). The third term can be handled
similarly by symmetry. We proceed by considering the two cases (i)
$\lambda\geq 1$ and (ii) $\lambda<1$, separately.

  * •

Suppose that $\lambda\geq 1$. Using the explicit form of the Poisson pmf and
its fourth moment expression, we have

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=0}^{3}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}\Bigr{)}^{2}p(x)p(y)\leavevmode\nobreak\
\leq\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{\infty}x^{4}p(x)\sum_{y=0}^{3}p(y)$
$\displaystyle\leq\leavevmode\nobreak\ $
$\displaystyle\mathbb{E}[X^{4}]\biggl{(}1+\lambda+\frac{\lambda^{2}}{2}+\frac{\lambda^{3}}{6}\biggr{)}e^{-\lambda}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ $
$\displaystyle(\lambda^{4}+6\lambda^{3}+7\lambda^{2}+\lambda)\biggl{(}1+\lambda+\frac{\lambda^{2}}{2}+\frac{\lambda^{3}}{6}\biggr{)}e^{-\lambda}.$

We would like to verify that the above bound is less than or equal to
$\mathbb{E}[X_{a,b}]$, up to a constant. To this end, recall Claim 2.3 of
Canonne et al., (2018) in (3):

$\displaystyle\mathbb{E}[X_{a,b}]\leavevmode\nobreak\
\gtrsim\leavevmode\nobreak\ \bigl{(}\lambda\sqrt{(\lambda\wedge
a)(\lambda\wedge b)}\wedge\lambda^{4}\bigr{)}.$

Since we assume $a,b\geq 2$, we have $\sqrt{(\lambda\wedge a)(\lambda\wedge
b)}\geq 1$ and so

$\displaystyle\mathbb{E}[X_{a,b}]\leavevmode\nobreak\
\gtrsim\leavevmode\nobreak\ $
$\displaystyle\bigl{(}\lambda\wedge\lambda^{4}\bigr{)}$
$\displaystyle\gtrsim\leavevmode\nobreak\ $
$\displaystyle(\lambda^{4}+6\lambda^{3}+7\lambda^{2}+\lambda)\biggl{(}1+\lambda+\frac{\lambda^{2}}{2}+\frac{\lambda^{3}}{6}\biggr{)}e^{-\lambda},$

where the last inequality uses the fact that the following function

$\displaystyle
h(\lambda):=\frac{\lambda\wedge\lambda^{4}}{(\lambda^{4}+6\lambda^{3}+7\lambda^{2}+\lambda)\Bigl{(}1+\lambda+\frac{\lambda^{2}}{2}+\frac{\lambda^{3}}{6}\Bigr{)}e^{-\lambda}}$

is bounded below by some positive constant (numerically $\approx 0.0119$) for
all $\lambda\geq 1$. Therefore when $\lambda\geq 1$,

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=0}^{3}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}\Bigr{)}^{2}p(x)p(y)\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ \mathbb{E}[X_{a,b}].$

  * •

Now consider the case where $\lambda<1$. In this case, we observe
$\mathbb{E}[X_{a,b}]\gtrsim\lambda^{4}$. Also note that

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=0}^{3}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}\Bigr{)}^{2}p(x)p(y)$ $\displaystyle\leavevmode\nobreak\
\overset{\mathrm{(i)}}{\leq}\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{\infty}\Bigl{(}x\sqrt{(x\wedge a)(x\wedge
b)}\Bigr{)}^{2}p(x)$
$\displaystyle\overset{\mathrm{(ii)}}{\leq}\leavevmode\nobreak\ $
$\displaystyle\sqrt{ab}\sum_{x=4}^{\infty}x^{2}\sqrt{(x\wedge a)(x\wedge
b)}\frac{\lambda^{x}e^{-\lambda}}{x!}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sqrt{ab}\lambda\sum_{x=4}^{\infty}x\sqrt{(x\wedge a)(x\wedge
b)}\frac{\lambda^{x-1}e^{-\lambda}}{(x-1)!}$
$\displaystyle\overset{\mathrm{(iii)}}{=}\leavevmode\nobreak\ $
$\displaystyle\sqrt{ab}\lambda\sum_{x=3}^{\infty}(x+1)\sqrt{((x+1)\wedge
a)((x+1)\wedge b)}\frac{\lambda^{x}e^{-\lambda}}{x!}$
$\displaystyle\overset{\mathrm{(iv)}}{\leq}\leavevmode\nobreak\ $
$\displaystyle 4\sqrt{ab}\lambda\sum_{x=3}^{\infty}x\sqrt{(x\wedge a)(x\wedge
b)}\frac{\lambda^{x}e^{-\lambda}}{x!}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle 4\sqrt{ab}\lambda\times 3\sqrt{(3\wedge a)(3\wedge
b)}\frac{\lambda^{3}e^{-\lambda}}{3!}+4\sqrt{ab}\lambda\sum_{x=4}^{\infty}x\sqrt{(x\wedge
a)(x\wedge b)}\frac{\lambda^{x}e^{-\lambda}}{x!}$
$\displaystyle\lesssim\leavevmode\nobreak\ $
$\displaystyle\sqrt{ab}\lambda^{4}+\sqrt{ab}\mathbb{E}[X_{a,b}],$

where step (i) uses the fact $p$ is a probability mass function, step (ii)
uses the basic inequality $\sqrt{(x\wedge a)(y\wedge b)}\leq\sqrt{ab}$, step
(iii) follows by change of variables and step (iv) follows since $x+1\leq 2x$
for all $x\geq 3$. Therefore we can conclude that

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=0}^{3}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}\Bigr{)}^{2}p(x)p(y)\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ \sqrt{ab}\mathbb{E}[X_{a,b}].$

Similarly,

$\displaystyle\sum_{x=0}^{3}\sum_{y=4}^{\infty}\Bigl{(}y\sqrt{(y\wedge
a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ \sqrt{ab}\mathbb{E}[X_{a,b}].$

Analysis of (I). Now, we handle the first term (I) in the variance expansion:

$\displaystyle(\mathrm{I})\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}\underbrace{\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge
b)}\Bigr{)}^{2}p(x)p(y)}_{:=u_{x,y,a,b}}.$

We separate the cases into three: (i) $b\geq a\geq 4$, (ii) $2\leq a,b\leq 4$
and (iii) $2\leq a<4\leq b$ and proceed our analysis.

  * •

We first assume that (i) $b\geq a\geq 4$ and decompose the above double
summations into 9 terms:

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=4}^{a}u_{x,y,a,b}+\sum_{x=4}^{a}\sum_{y=a+1}^{b}u_{x,y,a,b}+\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}u_{x,y,a,b}$
$\displaystyle+\leavevmode\nobreak\ $
$\displaystyle\sum_{x=a+1}^{b}\sum_{y=4}^{a}u_{x,y,a,b}+\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}u_{x,y,a,b}+\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}u_{x,y,a,b}$
$\displaystyle+\leavevmode\nobreak\ $
$\displaystyle\sum_{x=b+1}^{\infty}\sum_{y=4}^{a}u_{x,y,a,b}+\sum_{x=b+1}^{\infty}\sum_{y=a+1}^{b}u_{x,y,a,b}+\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}u_{x,y,a,b}$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\mathrm{(I)}_{1}+\cdots+\mathrm{(I)}_{9}.$

Starting with the first term $\mathrm{(I)}_{1}$,

$\displaystyle\sum_{x=4}^{a}\sum_{y=4}^{a}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=4}^{a}\Bigl{(}x\sqrt{(x\wedge a)(x\wedge
b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=4}^{a}(x^{2}-y^{2})^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=4}^{a}(x-y)^{2}(x+y)^{2}p(x)p(y)$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
4a^{2}\sum_{x=4}^{a}\sum_{y=4}^{a}(x-y)^{2}p(x)p(y)$
$\displaystyle\lesssim\leavevmode\nobreak\ $ $\displaystyle
a^{2}\mathrm{Var}[X\mathds{1}(X\geq 4)],$

where the last inequality follows by the fact that for i.i.d. random variables
$X$ and $Y$,

$\displaystyle\mathrm{Var}[X\mathds{1}(X\geq 4)]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ $
$\displaystyle\frac{1}{2}\mathbb{E}\bigl{[}\bigl{(}X\mathds{1}(X\geq
4)-Y\mathds{1}(Y\geq 4)\bigr{)}^{2}\bigr{]}$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\frac{1}{2}\sum_{x=4}^{a}\sum_{y=4}^{a}(x-y)^{2}p(x)p(y)+\frac{1}{2}\sum_{x=4}^{a}\sum_{y=0}^{3}x^{2}p(x)p(y)+\frac{1}{2}\sum_{x=0}^{3}\sum_{y=4}^{a}y^{2}p(x)p(y).$

For the second term $\mathrm{(I)}_{2}$,

$\displaystyle\sum_{x=4}^{a}\sum_{y=a+1}^{b}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=a+1}^{b}\Bigl{(}x\sqrt{(x\wedge a)(x\wedge
b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=a+1}^{b}\bigl{(}x^{2}-y\sqrt{ay}\bigr{)}^{2}p(x)p(y).$

Since $x\leq a<y\leq b$, we have $y\sqrt{y}\sqrt{a}-x^{2}\geq 0$ and

$\displaystyle y\sqrt{y}\sqrt{a}-x^{2}\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle
x\sqrt{y}\sqrt{a}-x^{2}-x\sqrt{y}\sqrt{a}+y\sqrt{y}\sqrt{a}$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
x(\sqrt{y}\sqrt{a}-x)+\sqrt{y}\sqrt{a}(y-x)$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
x(y-x)+\sqrt{y}\sqrt{a}(y-x)$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle(x+\sqrt{y}\sqrt{a})(y-x),$

which implies

$\displaystyle(x^{2}-y\sqrt{y}\sqrt{a})^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ (x+\sqrt{y}\sqrt{a})^{2}(y-x)^{2}\leq 4ab(y-x)^{2}.$

Using this inequality, we have

$\displaystyle\sum_{x=4}^{a}\sum_{y=a+1}^{b}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathrm{Var}[X\mathds{1}(X\geq 4)].$

For the third term $\mathrm{(I)}_{3}$, observe that

$\displaystyle\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}\bigl{(}x^{2}-y\sqrt{ab}\bigr{)}^{2}p(x)p(y).$

In this case, we have $x\leq a\leq b<y$ and

$\displaystyle 0\leavevmode\nobreak\ \leq\leavevmode\nobreak\ $ $\displaystyle
y\sqrt{ab}-x^{2}$ $\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
x\sqrt{ab}-x^{2}+y\sqrt{ab}-x\sqrt{ab}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle x(\sqrt{ab}-x)+\sqrt{ab}(y-x)$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle(x+\sqrt{ab})(y-x),$

where the last inequality holds since $\sqrt{ab}<y$, which in turn implies
that

$\displaystyle\sum_{x=4}^{a}\sum_{y=b+1}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathrm{Var}[X\mathds{1}(X\geq 4)].$

By symmetry, the fourth term $\mathrm{(I)}_{4}$ can be similarly analyzed as
the second term. For the fifth term $\mathrm{(I)}_{5}$,

$\displaystyle\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}\bigl{(}x\sqrt{ax}-y\sqrt{ay}\bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}\bigl{(}x\sqrt{x}-y\sqrt{y}\bigr{)}^{2}p(x)p(y).$

Without loss of generality, assume $a<x\leq y\leq b$ and note that

$\displaystyle 0\leavevmode\nobreak\ \leq\leavevmode\nobreak\ $ $\displaystyle
y\sqrt{y}-x\sqrt{x}$ $\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
x\sqrt{b}-x\sqrt{x}+y\sqrt{b}-x\sqrt{b}+y\sqrt{y}-y\sqrt{b}$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
x(\sqrt{b}-\sqrt{x})+\sqrt{b}(y-x)+y(\sqrt{y}-\sqrt{b})$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
y(\sqrt{b}-\sqrt{x})+\sqrt{b}(y-x)+y(\sqrt{y}-\sqrt{b})$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
y(\sqrt{y}-\sqrt{x})+\sqrt{b}(y-x)$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle y\frac{y-x}{\sqrt{y}+\sqrt{x}}+\sqrt{b}(y-x).$

Therefore we have

$\displaystyle(x\sqrt{x}-y\sqrt{y})^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak$
$\displaystyle\frac{2y^{2}}{(\sqrt{x}+\sqrt{y})^{2}}(x-y)^{2}+2b(x-y)^{2}$ (4)
$\displaystyle\leq\leavevmode\nobreak$ $\displaystyle
2y(x-y)^{2}+2b(x-y)^{2},$

which implies along with condition $y\leq b$ that

$\displaystyle\sum_{x=a+1}^{b}\sum_{y=a+1}^{b}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathrm{Var}[X\mathds{1}(X\geq 4)].$

For the sixth term $\mathrm{(I)}_{6}$, it can be seen that

$\displaystyle\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}\bigl{(}x\sqrt{ax}-y\sqrt{ab}\bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}\bigl{(}x\sqrt{x}-y\sqrt{b}\bigr{)}^{2}p(x)p(y).$

Since $a<x\leq b<y$, we have

$\displaystyle 0\leavevmode\nobreak\ <\leavevmode\nobreak\ $ $\displaystyle
y\sqrt{b}-x\sqrt{x}\leavevmode\nobreak\
=\sqrt{x}(\sqrt{x}\sqrt{b}-x)+\sqrt{b}(y-x)$
$\displaystyle\leq\leavevmode\nobreak\ $
$\displaystyle\sqrt{b}(\sqrt{x}\sqrt{b}-x)+\sqrt{b}(y-x)\leavevmode\nobreak\
\leq\leavevmode\nobreak\ 2\sqrt{b}(y-x),$

which implies

$\displaystyle\sum_{x=a+1}^{b}\sum_{y=b+1}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathrm{Var}[X\mathds{1}(X\geq 4)].$

The seventh term $\mathrm{(I)}_{7}$ and the eighth term $\mathrm{(I)}_{8}$ can
be analyzed as the third term $\mathrm{(I)}_{3}$ and the sixth term
$\mathrm{(I)}_{6}$, respectively. The final term $\mathrm{(I)}_{9}$ is

$\displaystyle\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}\bigl{(}x\sqrt{ab}-y\sqrt{ab}\bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
ab\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}(x-y)^{2}p(x)p(y)$
$\displaystyle\lesssim\leavevmode\nobreak\ $ $\displaystyle
ab\mathrm{Var}[X\mathds{1}(X\geq 4)].$

In summary, we have established that

$\displaystyle(\mathrm{I})\leavevmode\nobreak\ =\leavevmode\nobreak\
\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ $ $\displaystyle ab\mathrm{Var}[X\mathds{1}(X\geq
4)]\leavevmode\nobreak\ \overset{\mathrm{(i)}}{\lesssim}\leavevmode\nobreak\
ab\mathbb{E}[X\mathds{1}(X\geq 4)]$ $\displaystyle\lesssim\leavevmode\nobreak\
$ $\displaystyle ab\mathbb{E}[X\sqrt{(X\wedge a)}\sqrt{(X\wedge
b)}\mathds{1}(X\geq 4)],$

where step (i) uses Claim 2.1 of Canonne et al., (2018) in (2). Combining the
pieces yields

$\displaystyle\mathrm{Var}[X_{a,b}]\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathbb{E}[X_{a,b}],$

as desired.

  * •

We are only left with the cases where (ii) $2\leq a,b<4$ and (iii) $2\leq
a<4\leq b$, and only need to re-analyze the term (I). Since both $x,y\geq 4$,
it holds that $\sqrt{(x\wedge a)(x\wedge b)}=\sqrt{ab}$ under case (i) $2\leq
a,b<4$, and therefore we have

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}\Bigl{(}x\sqrt{(x\wedge
a)(x\wedge b)}-y\sqrt{(y\wedge a)(y\wedge b)}\Bigr{)}^{2}p(x)p(y)$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
ab\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}(x-y)^{2}p(x)p(y)$
$\displaystyle\lesssim\leavevmode\nobreak\ $ $\displaystyle
ab\mathrm{Var}[X\mathds{1}(X\geq 4)]\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathbb{E}[X_{a,b}].$

Similarly, when case (ii) $2\leq a<4\leq b$ holds,

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}\underbrace{\Bigl{(}x\sqrt{x\wedge
b}-y\sqrt{y\wedge b}\Bigr{)}^{2}p(x)p(y)}_{:=v_{x,y,b}}$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=4}^{b}\sum_{y=4}^{b}v_{x,y,b}+a\sum_{x=b+1}^{\infty}\sum_{y=4}^{b}v_{x,y,b}$
$\displaystyle+\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=4}^{b}\sum_{y=b+1}^{\infty}v_{x,y,b}+a\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}v_{x,y,b}$
$\displaystyle=\leavevmode\nobreak\ $
$\displaystyle(\mathrm{I})_{1^{\prime}}+\cdots+(\mathrm{I})_{4^{\prime}}.$

Following a similar analysis in (4), the first term
$(\mathrm{I})_{1^{\prime}}$ in the decomposition satisfies

$\displaystyle a\sum_{x=4}^{b}\sum_{y=4}^{b}v_{x,y,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\
ab\sum_{x=4}^{b}\sum_{y=4}^{b}(x-y)^{2}p(x)p(y)\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathbb{E}[X_{a,b}].$

The second term $(\mathrm{I})_{2^{\prime}}$ (and also the third term
$(\mathrm{I})_{3^{\prime}}$ by symmetry) satisfies

$\displaystyle
a\sum_{x=b+1}^{\infty}\sum_{y=4}^{b}v_{x,y,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle
a\sum_{x=b+1}^{\infty}\sum_{y=4}^{b}(x\sqrt{b}-y\sqrt{y})^{2}p(x)p(y)$
$\displaystyle\overset{(\star)}{\lesssim}\leavevmode\nobreak\ $ $\displaystyle
ab\sum_{x=b+1}^{\infty}\sum_{y=4}^{b}(x-y)^{2}p(x)p(y)\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathbb{E}[X_{a,b}],$

where inequality ($\star$) follows since it holds

$\displaystyle 0\leq
x\sqrt{b}-y\sqrt{y}=(x-y)\sqrt{b}+\sqrt{y}(\sqrt{yb}-y)\leq
2\sqrt{b}(x-y)\quad\text{when $4\leq y\leq b<x$.}$

For the fourth term $(\mathrm{I})_{4^{\prime}}$, we have

$\displaystyle
a\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}v_{x,y,b}\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle
ab\sum_{x=b+1}^{\infty}\sum_{y=b+1}^{\infty}(x-y)^{2}p(x)p(y)$
$\displaystyle\lesssim\leavevmode\nobreak\ $ $\displaystyle
ab\mathbb{E}[X_{a,b}].$

Combining the pieces yields

$\displaystyle\sum_{x=4}^{\infty}\sum_{y=4}^{\infty}u_{x,y,a,b}\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\ ab\mathbb{E}[X_{a,b}].$

This completes the proof of Lemma 1 when $a\geq 2$.

###  3.2 Case where $0\leq a<2$

Next we consider the remaining case where $0\leq a<2$. In this case along with
$b\geq 2$, observe that $X_{a,b}=X\sqrt{a(X\wedge b)}\mathds{1}(X\geq 4)$.
Consequently it follows that

$\displaystyle\mathrm{Var}[X_{a,b}]\leavevmode\nobreak\ =\leavevmode\nobreak\
$ $\displaystyle a\mathrm{Var}\bigl{[}X\sqrt{(X\wedge b)}\mathds{1}(X\geq
4)\bigr{]}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\frac{a}{2}\mathrm{Var}\bigl{[}X\sqrt{(X\wedge 2)(X\wedge
b)}\mathds{1}(X\geq 4)\bigr{]}$ $\displaystyle\lesssim\leavevmode\nobreak\ $
$\displaystyle ab\mathbb{E}\bigl{[}X\sqrt{(X\wedge 2)(X\wedge
b)}\mathds{1}(X\geq 4)\bigr{]},$

where the last inequality uses the previously established result in Section
3.1 for $a=2$ and $b\geq 2$. The desired result then follows by noting that

$\displaystyle ab\mathbb{E}\bigl{[}X\sqrt{(X\wedge 2)(X\wedge
b)}\mathds{1}(X\geq 4)\bigr{]}\leavevmode\nobreak\ =\leavevmode\nobreak\
\sqrt{a}b\mathbb{E}\bigl{[}X\sqrt{(X\wedge a)(X\wedge b)}\mathds{1}(X\geq
4)\bigr{]}.$

When $0\leq a,b<2$, Claim 2.1 of Canonne et al., (2018) in (2) yields

$\displaystyle\mathrm{Var}[X_{a,b}]\leavevmode\nobreak\ =\leavevmode\nobreak\
$ $\displaystyle ab\mathrm{Var}\bigl{[}X\mathds{1}(X\geq 4)\bigr{]}$
$\displaystyle\lesssim\leavevmode\nobreak\ $ $\displaystyle
ab\mathbb{E}\bigl{[}X\mathds{1}(X\geq 4)\bigr{]}$
$\displaystyle\lesssim\leavevmode\nobreak\ $
$\displaystyle\sqrt{ab}\mathbb{E}\bigl{[}X\sqrt{(X\wedge a)(X\wedge
b)}\mathds{1}(X\geq 4)\bigr{]}.$

Therefore the statement of Lemma 1 holds when $0\leq a<2$. Combining this with
the previous result in Section 3.1 completes the proof of Lemma 1.

#### Acknowledgements.

The author would like to thank Clément L. Canonne for his careful proofreading
and helpful comments. The author also thanks Matey Neykov, Sivaraman
Balakrishnan and Larry Wasserman for their helpful discussion. Finally, the
author acknowledges support from the Yonsei University Research Fund of
2021-22-0332 as well as support from the Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry
of Education (2022R1A4A1033384).

## References

  * Canonne et al., (2018) Canonne, C. L., Diakonikolas, I., Kane, D. M., and Stewart, A. (2018).  Testing Conditional Independence of Discrete Distributions.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 735–748, New York, NY, USA. Association for Computing Machinery.  https://arxiv.org/pdf/1711.11560v2.pdf.