# Localized State-Induced Enhanced Emission in Perovskite Nanocrystals

Feilong Wang  Qiongrong Ou  Shuyu Zhang<EMAIL_ADDRESS>Institute for
Electric Light Sources, School of Information Science and Technology, Fudan
University, Shanghai 200433, China.

###### Abstract

The luminescent mechanism of nano-sized materials with indirect bandgap and
parity-forbidden transition has always been a critical issue in breaking
through bottlenecks in traditional luminescent materials. The lack of
understanding has resulted in great disputes regarding the origin of
fluorescence in weak-transition nanocrystals. This paper proposes a new
physical luminescence model, named as localized state-induced enhanced
emission, i.e., localized processes induced by size-limitation or partial
doping, to explain the anomalous luminescence from non-luminescent state to
luminescent state in various perovskite nanocrystals with indirect/direct
bandgap or parity-forbidden transition. These findings provide a theoretical
viewpoint to design efficient lead-free perovskite NCs and promote the
development of fast optical transitions in luminescent materials.

††preprint: APS/123-QED

##  I  INTRODUCTION

Nanocrystals (NCs) have emerged as a promising luminescent material, playing a
significant role in the fields of luminous displays[1, 2], laser devices[3,
4], and biochemical sensing[5, 6]. Consequently, they have greatly advanced
the development of luminescence physics. Although basic phenomenological
physical luminescence models have been established over the past few decades,
including emission from quantum confinement[7, 8], luminescence due to defect
or doping[9, 10], self-trapped exciton luminescence[11, 12], and aggregation-
induced emission[13, 14], disputes still exist regarding the phase transition
in indirect bandgap semiconductors from a non-luminescent to a luminescent
state and in direct bandgap semiconductors from a forbidden transition to an
allowable one.

Effective luminescence with high photoluminescence quantum yield (PLQY) has
been observed in nanocrystalline systems corresponding bulk materials that
typically exhibit difficulty in emitting light, such as silicon NCs[15] and
lead-free double perovskite Cs2AgBiX6 (X=Cl, Br) NCs[16]. The origin of
luminescence in silicon NCs has been elucidated recently with a new physical
luminescence model called Localized State-Induced Enhanced Emission
(LIEE)[17], which successfully explained the luminescence in silicon NCs and
the fast optical transition resulting from size reduction. However, research
has yet to demonstrate the universality of this model, and the fluorescence
mechanism in indirect-bandgap Cs2AgBiX6 NCs remains elusive[16, 18].

Despite the potential of NCs as luminescent materials, there remains a lack of
precise understanding of the anomalous luminescence mechanisms and how to
improve their luminous performance. Luo et al. suggested that the change in
electron wavefunction and increased electron-hole (e-h) pair overlap,
resulting from Jahn-Teller distortion of the AgCl6 octahedron in sodium-doped
Cs2AgInCl6, led to an enlarged transition dipole moment (TDM)[19]. However,
the impact of the surface effects on e-h pairs was not taken into account in
(sodium-doped) Cs2AgInCl6 NCs. In a related study, the yellow emission in
silver-doped Cs2NaInCl6 was ascribed to bright self-trapped excitons[20]
without consideration of the surface scattering (electron scattering at
nanocrystalline surface). To date, few reports have been able to pinpoint the
origins of luminescence in double perovskite NCs or clarify the enhanced
emission caused by partial doping. In short, one of the primary obstacles in
the development of lead-free double perovskites is the absence of a
comprehensive understanding of the luminescence phenomena in systems with
intrinsically low luminescent properties.

In this manuscript, we presents a new approach to explore the luminescence
properties of perovskite NCs with poor optical transitions using LIEE due to
the scattering of electron at nanocrystalline surface, a method previously
applied to silicon NCs. We began by calculating the wavefunction distribution
of direct and indirect bandgap perovskites in real space. Then reconstructed
the e-h pairs in real space under the single particle excitation approximation
and demonstrated that LIEE can produce anomalous luminescence in perovskite
NCs without the need for electron-phonon coupling or Jahn–Teller distortion
effects. Our results provide a theoretical foundation for designing efficient
lead-free double perovskites NCs and may advance the development of fast
optical transitions in luminescent materials.

##  II COMPUTATIONAL DETAILS

All-electron full-potential linearised augmented-plane wave (LAPW) method was
employed with ELK code[21] in bulk-materials calculation. All structures were
optimized with Perdew-Burke-Ernzerhof (PBE)[22] functional and a
6$\times$6$\times$6 $k$-points grid was adopted.
$r\textsuperscript{2}$SCAN[23] was employed to obtain energy band and Kohn-
Sham (KS) wavefunction.

All ground-state calculations of clusters were performed in CP2K.[24] Gaussian
plane waves method (GPW) in the QUICKSTEP[25] module was employed with a
double-zeta DZVP-MOLOPT-SR-GTH basis set[26] and norm-conserving GTH-PBE[27,
28] pseudopotential. A 500-Ry cutoff energy was applied for planewave
expansion, and each Gaussian was mapped onto a 55-Ry cutoff grid. Dispersion
with D3[29] was considered in all calculations. All structures were placed in
larger non-periodic boxes (thus in cluster model) and then optimized for
minimal energy with Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm in
PBE[30] level.

All excited state calculation were performed in TDA-TDDFT ( Tamm-Dancoff
approximated time-dependent density functional theory). Configuration
coefficients greater than 0.001 were printed out for e-h pairs analysis.
Multiwfn[31] was employed to conduct e-h pairs analysis.

In TDDFT framework, all KS orbital are involved in electron excited state. The
excited state wavefunction is

$\Psi_{TD}=\sum_{i\rightarrow a}\omega_{i}^{a}\Phi_{i}^{a}+\sum_{a\rightarrow
i}\omega_{i}^{{}^{\prime}a}\Phi_{i}^{a}$ (1)

$\Phi_{i}^{a}$ is the configuration state wavefunction corresponding to moving
an electron from originally occupied $i$-th MO to virtual $a$-th MO.
$\omega_{i}^{a}$ and $\omega_{i}^{{}^{\prime}a}$ are configuration coefficient
of excitation and de-excitation, respectively, limited by normalizing
conditions. In TDA-TDDFT, the de-excitation term disappears and the hole and
electron density can be perfectly defined as[32]

$\rho_{h}(\mathbf{r})=\sum_{i\rightarrow
a}{(\omega_{i}^{a})}^{2}\varphi_{i}(\mathbf{r})\varphi_{i}(\mathbf{r})+\sum_{i\rightarrow
a}\sum_{j\neq i\rightarrow
a}\omega_{i}^{a}\omega_{j}^{a}\varphi_{i}(\mathbf{r})\varphi_{j}(\mathbf{r})$
(2)

$\rho_{e}(\mathbf{r})=\sum_{i\rightarrow
a}{(\omega_{i}^{a})}^{2}\varphi_{a}(\mathbf{r})\varphi_{a}(\mathbf{r})+\sum_{i\rightarrow
a}\sum_{i\rightarrow b\neq
a}\omega_{i}^{a}\omega_{i}^{b}\varphi_{a}(\mathbf{r})\varphi_{b}(\mathbf{r})$
(3)

$\varphi_{i}$ and $\varphi_{j}$ are occupied molecular orbitals, while
$\varphi_{a}$ and $\varphi_{b}$ are virtual orbitals.

The transition density in real space can be writen as

$\mathbf{T}_{\rho}(\mathbf{r})=\sum_{i\rightarrow
a}\omega_{i}^{a}\psi_{i}(\mathbf{r})\psi_{a}(\mathbf{r})$ (4)

The integral in the whole space after multiplying it with the coordinate
$\mathbf{r}$

$\mathbf{d}=-\mathbf{r}\cdot\mathbf{T}_{\rho}(\mathbf{r})$ (5)

is the transition dipole moment (TDM) $\mathbf{d}$.

Figure 1:  Energy band diagram and isosurface of electron-hole in (a) and (e)
CsPbBr3, (b) and (f) Cs2AgInBr6, (c) and (g) Cs2NaInBr6 and (d) and (h)
Cs2AgInBr6. Red arrows in (g) show two antipodal transition dipole sub-vector
due to crystal symmetry, .ie. , the electron and hole possess the same parity.
The red dotted lines facilitate observation.  Table 1: A brief summary of
transition condition in bulk perovskite. Orbital symmetry dictates the atomic
optical transition (the partial TDM “$\uparrow$” or “$\downarrow$”). The
symmetry of these partial TDMs, in turn, defines the overall TDM for the
system. The orientation of the orbitals then governs the degree of overlap.

kind | overlap(a. u.) | symmetry | orientation | $f_{\mathrm{osc}}$ (a.u.) | emit
---|---|---|---|---|---
orbital | partial TDM
CsPbBr3 | 7.295 | ✓✓✓ | ✓✓✓ | ✓✓✓ | $2.7$ | excellent
Cs2AgInBr6 | 3.307 | ✓✓✓ | ✗ | ✓✓ | $3.6\times 10^{-22}$ | no
Cs2NaInBr6 | 2.976 | ✓ | ✓ | ✗ | $9.1\times 10^{-17}$ | no
Cs2AgBiBr6 | 3.119 | ✓ | ✓ | ✗ |  $1.3\times 10^{-5}$ (regardless of momentum) | no

  * ✓ favorable degree of luminescence, ✗ not conducive to luminescence.

Figure 2:  A brief illustration in the formation of localized states. (a) In
periodic bulk, e-h wavefunctions are described as periodic Bloch functions.
The “even-to-even” transition due to the cancellation of the two partial TDM
“$\downarrow\uparrow$” and the weak “even-to-odd” transition with little
overlap are both shown. (b) In a good nano-structured “cavity”, the e-h are
highly concentrated in some areas. This situation is extremely beneficial to
the enhancement of emission. However, (c) surface-to-core localized states and
(d) surface-to-surface localized states could be arised limited by the surface
and overlap of the e-h is decreased sharply. Therefore, their enhancement in
luminescence is much limited. Nonetheless, emission in nanomaterials is more
efficient than bulk for media difficult to emit. It should be noted that the
TDM strongly dependents on the size of nanomaterials in the latter two cases.
Figure 3:  Optical transition in perovskite nanocrystals with (a)-(d)
absorbing oscillator strength. (e)-(h) e-h distribution of the lowest
transition state (blue isosurface for hole, green isosurface for electron).
And (i)-(l) the YZ-plane projection of transition density with TDM above the
figure. The arrows represent transition dipole sub-vectors which the opposite
direction would partially cancel out.

##  III RESULTS AND DISCUSSION

###  III.1 E-h pair distribution in bulk perovskite

Firstly, we have calculated energy band and obtained Kohn-Sham wavefunction
for both direct and indirect bulk-perovskite structures (Fig. 1). In CsPbBr3,
the hole is made up of the _p_ -orbital of bromine and the 6 _s_ -orbital of
lead, while the electron is mainly located on the 6 _p_ -orbital of lead. This
transition satisfies the quantum transition selection rule and possesses an
excellent luminous rate, as electrons and holes in the same real space greatly
increase the transition probability according to the Dirac-Fermi’s golden
rule. The TDM is maximal due to opposing orbital symmetry, as shown in TAB. 1.
Unfortunately, the lead-free perovskite structures Cs2AgInBr6 Cs2NaInBr6 with
direct bandgaps exhibit poor luminescence. The holes are provided by the _p_
-orbital of bromine (and the _$d_{z^{2}}$_ -orbital of silver in Cs2AgInBr6),
while indium (with silver) contributes _s_ -orbital to the electrons (Fig. 1g
and h). From the perspective of the quantum transition selection rule, the
silver-to-indium (silver) transition is strictly prohibited, but the bromine-
to-indium (silver) (odd-to-even parity) transition can occur. In Cs2AgInBr6,
atoms along the In-Br-Ag-Br-In axis (the red line in Fig. 1b) contributes to
the TDM presenting an extremely symmetrical transition density “$+--+$” or
“$-++-$”. That means if the “$+$” represents In($s$)$\rightarrow$Br($p$),
“$-$” represents Br($p$)$\rightarrow$Ag($d$) and the “$+\rightarrow-$” is the
partial TDM. The TDM should add the two partial TDM,
“$+\rightarrow-$”($\downarrow$) and “$-\rightarrow+$”($\uparrow$) in Fig. 1
(two reds arrows). Thus, opposite partial TDM in Fig. 1 (two reds arrows) are
simultaneously contributing to the TDM. If the system only have one partial
TDM mentioned above, The optical transition clearly occurrs. The cancellation
of opposite direction of the two partial TDM results in a zero TDM, known as a
crystal symmetry-forbidden transition. Another interesting forbidden mechanism
is presented in Cs2NaInBr6. The absence of coupling interaction between sodium
and bromine induces a nodal plane of bromine _p_ -orbital that is parallel to
the indium-bromide bond, meaning that the hole and electron cannot effectively
overlap (Cs2NaInBr6 vs. Cs2AgInBr6, $2.976$ vs. $3.307$ in TAB. 1). Thus, an
extremely limited transition probability arises in Cs2NaInBr6.

In another lead-free perovskite, Cs2AgBiBr6 is considered poorly luminescent
due to its indirect bandgap. The silver 4 _$d_{z^{2}}$_ -orbital, bismuth 6
_p_ -orbital, bromine 4 _p_ -orbital, and silver 5 _s_ -orbital can satisfy
the necessary symmetry requirements. However, the perpendicular orientation of
orbitals and mismatching momentum results in repulsion between holes and
electrons, hindering luminescence.

Indeed, to determine luminescence probability, it is necessary to consider the
electric dipole transition, $\bra{j}\mathbf{r}\ket{i}$, which involves the
vector sum of partial TDMs, crystal orbital, or molecular orbital symmetry and
effective electron-hole overlap. Additionally, the transition must satisfy the
selection rule ($\Delta{l}=\pm 1$).

###  III.2 Origin of the anomalous emission in NCs

Obviously, the primary approach to obtain emission in poor-luminescence
materials is to change the e-h spatial distribution, so as to overcome the
luminescent restrictions as list in TAB. 1. Localized states provide an
effective means to change the non-luminescent situation. In the past few
decades, despite unclear description to abnormal emission in space-limited
nano-system, researchers have observed luminescence in silicon through porous
silicon [33], quantum well [34, 35], and quantum dot [36]. A recent study
suggests that localized states can modify the spatial orientation of electron-
hole pairs and produce anomalous emission. In nanomaterials, the electron-hole
wavefunction does not satisfy the perfect Bloch function and oscillates in a
”cavity” (Fig. 2). Influenced by relaxation of surface atoms, which is related
to the surface dielectric layer such as ligands and solutions, the
wavefunction evolves into two forms of localized states - core states and
surface states - that are no longer symmetrical and highly localized.
Therefore, global symmetry is not the principal consideration for efficient
emission. Instead, the orientation of the electron-hole pairs overlap and the
symmetry of partial TDM are more crucial.

Similar to silicon, can space-limited double perovskite emit? In order to
further elucidate this problem, we calculated the optical transitions of all
nanocrystals (Fig. 3). All double perovskite NCs present the first exciton
absorption peak with medium oscillator strength (10-2-10-4) that is
significantly greater than that observed in bulk materials.This observation
suggests that double perovskite NCs possess the capacity to emit light
similarly to silicon NCs. To gain a more intuitive understanding of the
localized states, we first generated an isosurface graph of electron-hole
pairs, and then projected transition density onto the YZ-plane, as shown in
Fig. 3. Nearly all transitions were localized and capable of changing the
spatial distribution of electron-hole pairs, thereby causing the transition
from a non-luminescent (or weakly luminescent) state to a strong optical
transition in all nanocrystals.

In some nanocrystals, the excited state contributed by surface atoms is lower
than that contributed by atoms inside nanocrystals, which causes electrons to
form surface-localized states that are unsuitable for emission. In CsPbBr3
NCs, the oscillator strength of the first excitated peak is only 0.03 (Fig.
3a), which is significantly smaller than the value of 2.7 observed in bulk
materials (TAB. 1). Observing the e-h wavefunctions, it is evident that the
overlap of e-h pairs is decrescent (Fig. 3e), and surface-localized electrons
and core-localized holes could cause this reduction. A more conservative case
can be seen in Cs2NaInBr6 NCs (Fig. 3g), where the e-h pairs tend to be
completely separate along the two sides of the surface, resulting in a much
smaller oscillator strength of ($10^{-4}$). Nevertheless, it has been greatly
improved by approximately $10^{12}$ (Fig. 3c) compared to Cs2NaInBr6 bulk
(TAB. 1).

In Cs2AgInBr6 and Cs2AgBiBr6, core-to-core states (shown in Fig. 2b) are
generated, as depicted in Fig. 3f and h. In the former material, localized
states have broken the transition symmetry, which means the cancellation of
partial TDM (vector addition of “$\uparrow$” and “$\downarrow$”) is avoided,
resulting in a $10^{19}$-fold (Fig. 3b vs. TAB. 1) improvement in oscillator
strength. This material inherits the direct bandgap and high overlap
characteristics of the bulk material. On the other hand, for the indirect-
bandgap Cs2AgBiBr6, the main factor limiting the oscillator strength is the
effective partial TDM. The optical transition between the $p$-orbital of
bismuth and the $p$-orbital of bromine is prevented, and there is only one
transition channel—the $s$-orbital of silver and the $p$-orbital of bromine
(“$\downarrow$”). However, induced by the localized state, the partial TDM
vector between the silver and the surrounding four bromine atoms is not
completely symmetrical. Therefore, the vector sum of partial TDM is a non-zero
value, albeit an extremely limited one.

###  III.3 To precisely control the localized state for efficient luminescence

Localized states arising from quantum confinement exhibit limited robustness,
as depicted in Fig. 3. Influenced by the atoms relaxation on surface, the
elimination of atomic degeneracy, and the presence of surrounding ligands, the
localized state tends to move in the opposite direction of the desired. To
ensure that localized states in NCs contribute positively to luminescence,
more accurate methods beyond intrinsic quantum confinement are necessary. It
meets two essential criteria: 1) creating a scattering surface for
localization, and 2) positioning the localized energy level in the lowest
unoccupied orbital. Proper doping serves as a simple and effective strategy.
Using CsPbBr3 NCs as an illustration, it is observed that localized states
have a negative impact, as shown in Fig. 3e.

Figure 4:  Sb3+-doped CsPbBr3 with (a) enhanced absorption sepectrum and (b)
e-h pairs spatial distribution of the lowest excited states (the TDM vector is
marked).

In Fig. 4, Sb3+-doped CsPbBr3 NCs present three distinct features, consistent
with recent experimental results [10]: 1) A spectral redshift attributed to
the lower Sb3+ $5p$ orbital energy level compared to the Pb2+ $6p$ orbital,
enabling the formation of the lowest localized excited state around the Sb3+
atom. 2) Increased radiative recombination rate achieved through a
redistribution of electron distribution from Fig. 2c to Fig. 2b. Notably, Sb3+
does not directly involved in luminescence and help to scatter electron to
form the Fig. 2b model. 3) The redshift narrower emission spectrum results
from a trapped state, which widens the energy gap between the several lowest
excited states and makes it difficult for thermal transition from the lowest
excited states (emitting center) to higher excited states, thereby preventing
spectral broadening.

##  IV CONCLUSION

In summary, we have investigated localized state-enhanced emisssion in
different types of perovskite NCs. The results indicate that LIEE can control
the distribution of the e-h wavefunction and lead to an anomalous luminescence
in different kinds of poor luminescent perovskite NCs, via destroying the
lattice symmetry, .ie., surface scattering, surface doping, etc. These
findings provide a theoretical viewpoint to design efficient lead-free double
perovskite NCs and promote the development of fast optical transitions in
luminescent materials.

##  V AUTHOR CONTRIBUTIONS

Conceptualization, Data curation, Investigation, Methodology, Software,
Visualization, Writing – original draft: F.W.; Funding acquisition, Project
administration, Supervision: S.Z.; Resources: S.Z. and Q.O.; Writing – review
& editing: F.W. and S.Z..

###### Acknowledgements.

This work is supported by Science and Technology commission of Shanghai
Municipality (21ZR1408800), and National Natural Science Foundation of China
(11975081).

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# A closed-form expression for the variance of truncated distribution and its
uses

Roberto Vila<EMAIL_ADDRESS>Department of Statistics, University of
Brasília, Brasília, Brazil  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada  Narayanaswamy Balakrishnan
<EMAIL_ADDRESS>Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada  Raul Matsushita<EMAIL_ADDRESS>Department of Statistics, University of Brasília, Brasília, Brazil

###### Abstract

This work sheds some light on the relationship between a distribution’s
standard deviation and its range, a topic that has been discussed extensively
in the literature. While many previous studies have proposed inequalities or
relationships that depend on the shape of the population distribution, the
approach here is built on a family of bounded probability distributions based
on skewing functions. We offer closed-form expressions for its moments and the
asymptotic behavior as the support’s semi-range tends to zero and $\infty$. We
also establish an inequality in which the well-known Popoviciu’s one is a
special case. Finally, we provide an example using US dollar prices in four
different currencies traded on foreign exchange markets to illustrate the
results developed here.

Keywords. Truncated distribution $\cdot$ Skewing function $\cdot$ Popoviciu’s
inequality $\cdot$ Skew-symmetric distribution $\cdot$ Econophysics.
Mathematics Subject Classification (2010). MSC 60E05 $\cdot$ MSC 62Exx $\cdot$
MSC 62Fxx.

##  1 Introduction

Relating the standard deviation ($\sigma$) to the range is a well-studied
topic (_see_ Tippett,, 1925; Popoviciu,, 1935; Shone,, 1949; David et al.,,
1954). However, most of the results found in the literature in this regard
propose inequalities or relationships that depend on the shape of the
population distribution. Matsushita et al., (2020, 2023) recently suggested a
power law between $\sigma$ and the semi-range $\ell$ without knowing the
population distribution, but assuming symmetric truncated forms restricted to
$\ell\ll 1$. They argued that truncation is a phenomenon naturally generated
by the sampling process. In their approach, conditional distribution
properties can link the usual unbounded distribution for describing unobserved
data.

Here, we establish a closed-form expression for the variance of truncated
distributions valid for all $\ell>0$. We start with the general case (Section
2.1), by considering a truncated variable $X$ over the interval $[a,b]$ based
on a cumulative distribution function $G$ with unbounded support as a skewing
function. For $c\in(a,b)$ and $p>0$, we present a general form for
$\mathbb{E}[(X-c)^{p}]$ as well as its asymptotic behavior as the support’s
semi-range tends to zero and $\infty$. We also present some inequalities that
included Popoviciu’s inequality as a special case.

Section 2.2 presents some properties and examples regarding the symmetrically
truncated distribution as a particular case. Importantly, we deduce the form
of the ratio $\sigma/\ell$ as a function of $G$ and $\ell$. For illustrative
purposes, we have presented an example using actual financial data (Section
3). They consist of 16 million tick-by-tick returns of four currencies against
the US dollar transacted on foreign exchange markets. Finally, Section 4 makes
some brief concluding remarks.

##  2 Main results

###  2.1 General case

Suppose we have a random variable with cumulative distribution function (CDF)
$G(x)$ and with infinite support. Based on $G$, given two real numbers $a$ and
$b$, such that $a<b$, we have

$\displaystyle F_{X}(x)={G(x)-G(a)\over G(b)-G(a)},\quad a<x<b,$ (2.1)

to be the truncated CDF of a random variable $X$ with support $(a,b)$. Then,
we have the following result concerning its moments.

######  Theorem 2.1.

Let $c$ and $p$ be real numbers such that $a<c<b$ and $p>0$. Then, the $p$-th
moment about $c$ is given by

$\displaystyle\mathbb{E}[(X-c)^{p}]=\dfrac{1}{G(b)-G(a)}\,\big{[}(b-c)^{p}G(b)-(a-c)^{p}G(a)-pI_{G}(c;a-c,b-c,p)\big{]},$

where $I_{G}(c;s,t,p)$ is defined as

$\displaystyle I_{G}(c;s,t,p)=\int_{s}^{t}y^{p-1}G(y+c)\,{\rm d}y,\quad s<t.$

###### Proof.

From (2.1), we have

$\displaystyle\mathbb{E}[(X-c)^{p}]$ $\displaystyle={1\over
G(b)-G(a)}\int_{a}^{b}(x-c)^{p}\,{\rm d}G(x)$ $\displaystyle={p\over
G(b)-G(a)}\left\\{\int_{a}^{c}\left[\,\int_{0}^{x-c}y^{p-1}\,{\rm
d}y\right]{\rm d}G(x)+\int_{c}^{b}\left[\,\int_{0}^{x-c}y^{p-1}\,{\rm
d}y\right]{\rm d}G(x)\right\\}.$

Change the order of integration, the above expression can be written as

$\displaystyle={p\over
G(b)-G(a)}\left\\{\int_{0}^{a-c}y^{p-1}\left[\,\int_{a}^{y+c}{\rm
d}G(x)\right]{\rm d}y+\int_{0}^{b-c}y^{p-1}\left[\,\int_{y+c}^{b}{\rm
d}G(x)\right]{\rm d}y\right\\}$ $\displaystyle={1\over
G(b)-G(a)}\left[{(b-c)^{p}G(b)}-{(a-c)^{p}G(a)}\right]$ $\displaystyle-{p\over
G(b)-G(a)}\int_{a-c}^{0}y^{p-1}G(y+c)\,{\rm d}y-{p\over
G(b)-G(a)}\int_{0}^{b-c}y^{p-1}G(y+c)\,{\rm d}y,$

which completes the proof of the theorem. ∎

######  Corollary 2.2.

Under the conditions of Theorem 2.1, we have

$\displaystyle\mathbb{E}[(X-c)^{p}]\leqslant\begin{cases}(b-c)^{p}[1-F_{X}(c)]+(a-c)^{p}F_{X}(c),&\text{if}\
p\ \text{is even},\\\\[5.69046pt] (b-c)^{p}[1-F_{X}(c)],&\text{if}\ p\
\text{is odd},\end{cases}$

where $F_{X}$ is as given in (2.1).

###### Proof.

It is evident that

$\displaystyle I_{G}(c;a-c,b-c,p)=I_{G}(c;a-c,0,p)+I_{G}(c;0,b-c,p).$ (2.2)

Assuming $p$ to be even, for $a-c<y<0$, we have
$y^{p-1}G(c)<y^{p-1}G(y+c)<y^{p-1}G(a)$, and for $0<y<b-c$, we have
$y^{p-1}G(c)<y^{p-1}G(y+c)<y^{p-1}G(b)$. Consequently, we get

$\displaystyle I_{G}(c;a-c,0,p)\geqslant-{G(c)(c-a)^{p}\over
p}\quad\text{and}\quad I_{G}(c;0,b-c,p)\geqslant{G(c)(b-c)^{p}\over p}.$

Now, upon using these inequalities in (2.2), we obtain

$\displaystyle I_{G}(c;a-c,b-c,p)\geqslant{G(c)\over
p}\,\big{[}(b-c)^{p}-(c-a)^{p}\big{]}.$

Using the above inequality in the expression in Theorem 2.1, we get

$\displaystyle\mathbb{E}[(X-c)^{p}]\leqslant\dfrac{1}{G(b)-G(a)}\,\big{\\{}(b-c)^{p}G(b)-(a-c)^{p}G(a)-G(c)\big{[}(b-c)^{p}-(c-a)^{p}\big{]}\big{\\}}.$

From (2.1), the right-hand side of the above inequality can be rewritten as

$\displaystyle=(b-c)^{p}[1-F_{X}(c)]+(a-c)^{p}F_{X}(c).$

This proves the inequality for the case when $p$ is even.

The inequality for the case when $p$ is odd can be established in an analogous
manner. ∎

When $p$ is even, a lower bound for $\mathbb{E}[(X-c)^{p}]$ can be established
as below.

######  Proposition 2.3.

Under the conditions of Theorem 2.1, for $p$ even, we have

$\displaystyle\mathbb{E}[(X-c)^{p}]\geqslant(t-c)^{p}\left[F_{X}(b)-F_{X}(t)\right],\quad\text{if}\
c<t<b.$

###### Proof.

Suppose $p$ is even. Then, as $(X-c)^{p}\geqslant 0$ and
$(X-c)^{p}\geqslant(t-c)^{p}$ for $X>t>c$, it is clear that

$\displaystyle\mathbb{E}[(X-c)^{p}]\geqslant\mathbb{E}\big{[}(X-c)^{p}\mathds{1}_{\\{X>t\\}}\big{]}\geqslant(t-c)^{p}\,\mathbb{E}\big{[}\mathds{1}_{\\{X>t\\}}\big{]},$

which yields the required result. ∎

######  Proposition 2.4.

Under the conditions of Theorem 2.1, we have

$\displaystyle\min_{a<c<b}\mathbb{E}[(X-c)^{p}]\leqslant\begin{dcases}\left(\dfrac{b-a}{2}\right)^{p},&\text{if}\
p\ \text{is even},\\\\[5.69046pt] 0,&\text{if}\ p\geqslant 1\ \text{is
odd}.\end{dcases}$

###### Proof.

Suppose $p$ is even. In this case, from Corollary 2.2, we have

$\displaystyle\mathbb{E}[(X-c)^{p}]\leqslant(b-c)^{p}[1-F_{X}(c)]+(a-c)^{p}F_{X}(c)\leqslant
S(F(c))T(c),$ (2.3)

where $S(F(c))=\max\left\\{1-F_{X}(c),F_{X}(c)\right\\}$ and
$T(c)=(b-c)^{p}+(a-c)^{p}$. As $F_{X}(c)=1/2$ is a minimum point of $M(F(c))$,
we have

$\displaystyle\min_{0<F(c)<1}S(F(c))={1\over 2}.$

Taking the minimum over $0<F(c)<1$ in (2.3), we get

$\displaystyle\mathbb{E}[(X-c)^{p}]\leqslant{1\over 2}\,T(c).$

Now, taking the minimum over $a<c<b$ in the above inequality and using the
fact that the function $T(c)$ reaches a minimum value at the point
$c=(a+b)/2$, we get

$\displaystyle\min_{a<c<b}\mathbb{E}[(X-c)^{p}]\leqslant{1\over
2}\,T\left({a+b\over 2}\right)=\left(\dfrac{b-a}{2}\right)^{p}.$

This proves the inequality for the case when $p$ is even. Further, the
inequality for the case when $p$ is odd trivially follows from Corollary 2.2.
∎

######  Proposition 2.5.

The variance $\sigma^{2}$ can be expressed as

$\displaystyle\sigma^{2}={\rm
Var}(X)=\dfrac{1}{G(b)-G(a)}\,\big{[}(b-\mu)^{2}G(b)-(a-\mu)^{2}G(a)-2I_{G}(\mu;a-\mu,b-\mu)\big{]},$

where $\mu=\mathbb{E}(X)$ and

$\displaystyle I_{G}(\mu;s,t)=\int_{s}^{t}yG(y+\mu)\,{\rm d}y,\quad s<t.$

###### Proof.

By taking $c=\mu$ and $p=2$ in Theorem 2.1, we obtain the required result. ∎

######  Proposition 2.6.

The Popoviciu inequality (Popoviciu,, 1935) on variances given by

$\displaystyle\sigma^{2}\leqslant\left(\dfrac{b-a}{2}\right)^{2},$

follows from Proposition 2.4.

###### Proof.

The proof follows immediately by setting $p=2$ in Proposition 2.4. ∎

A reverse form of Popoviciu’s inequality can be obtained upon taking
$p=2,c=\mu=\mathbb{E}(X)$ and $t=(\mu+b)/2$ in Proposition 2.3.

######  Proposition 2.7.

We have

$\displaystyle\sigma^{2}\geqslant\left(\dfrac{b-\mu}{2}\right)^{2}\left[F_{X}(b)-F_{X}\left({\mu+b\over
2}\right)\right].$

#### Asymptotic behavior

In the following theorem, we establish the asymptotic behaviour of the $p$-th
moment.

######  Theorem 2.8.

If $X$ is distributed as in (2.1), then we have

$\displaystyle\lim_{{b-a\over 2}\longrightarrow
0^{+}}\mathbb{E}\left[\left(\dfrac{X-a}{b-a}\right)^{p}\,\right]={1\over
p+1},\quad p>-1,$

and

$\displaystyle\lim_{{b-a\over
2}\longrightarrow\infty}\mathbb{E}\left[\left(\dfrac{X-a}{b-a}\right)^{p}\,\right]={1\over
2^{p}}.$

###### Proof.

From (2.1), we find

$\displaystyle F_{\frac{X-a}{b-a}}(z)={G\big{(}z(b-a)+a\big{)}-G(a)\over
G(b)-G(a)}=\dfrac{G\big{(}{b-a\over 2}\,(2z-1)+{a+b\over
2}\big{)}-G\big{(}{a+b\over 2}-{b-a\over 2}\big{)}}{G\big{(}{b-a\over
2}+{a+b\over 2}\big{)}-G\big{(}{a+b\over 2}-{b-a\over 2}\big{)}},\ 0<z<1.$

Now, it is a simple task to verify that

$\displaystyle\lim_{{b-a\over 2}\longrightarrow
0}F_{\frac{X-a}{b-a}}(z)=F_{U}(z),\quad\forall z\in\mathbb{R},$

and

$\displaystyle\lim_{{b-a\over
2}\longrightarrow\infty}F_{\frac{X-a}{b-a}}(z)=F_{Y}(z),\quad\forall
z\neq{1\over 2},$

where $U\sim U(0,1)$ and $Y$ is a discrete random variable such that
$\mathbb{P}(Y=1/2)=1$.

Moreover, we note that $[(X-a)/(b-a)]^{p}$ is uniformly integrable because
$0<(X-a)/(b-a)<1$. As convergence in distribution along with uniform
integrability imply convergence in mean (cf. Billingsley,, 1968, Theorem 5.4),
we have

$\displaystyle\lim_{{b-a\over 2}\longrightarrow
0^{+}}\mathbb{E}\left[\left(\dfrac{X-a}{b-a}\right)^{p}\,\right]=\mathbb{E}(U^{p}),\quad
p>-1,$

and

$\displaystyle\lim_{{b-a\over
2}\longrightarrow\infty}\mathbb{E}\left[\left(\dfrac{X-a}{b-a}\right)^{p}\,\right]=\mathbb{E}(Y^{p}),$

which completes the proof of the theorem. ∎

######  Corollary 2.9.

We further have

$\displaystyle\lim_{{b-a\over 2}\longrightarrow
0^{+}}{\sigma^{2}+(\mu-a)^{2}\over(b-a)^{2}}={1\over
3}\quad\text{and}\quad\lim_{{b-a\over
2}\longrightarrow\infty}{\sigma^{2}\over(b-a)^{2}}=0.$

where $\mu=\mathbb{E}(X)$ and $\sigma^{2}={\rm Var}(X)$.

###### Proof.

By taking $p=2$ in Theorem 2.8, the above results follow readily. ∎

###  2.2 Symmetric case

In this section, we assume that $G$ is a skewing function, i.e., it is such
that $G(x)\geqslant 0$ and $G(-x)=1-G(x)$, and $X$ distributed as in (2.1)
with $a=-\ell$ and $b=\ell>0$. That is, $X$ has its CDF as

$\displaystyle F_{X}(x)={G(x)+G(\ell)-1\over 2G(\ell)-1},\quad-\ell<x<\ell.$
(2.4)

######  Proposition 2.10.

The variance $\sigma^{2}$ can be expressed as

$\displaystyle\sigma^{2}=\ell^{2}H(\ell),$

where

$\displaystyle H(\ell)=1-\dfrac{2C(\ell)-1}{2G(\ell)-1},$

with $C(\ell)=C(\ell,G)$ defined as

$\displaystyle C(\ell)={2\over\ell^{2}}\int_{0}^{\ell}yG(y)\,{\rm d}y.$

Moreover, $1/2\leqslant C(\ell)\leqslant G(\ell)$.

###### Proof.

As $G$ is a skewing function, we have $\mu=\mathbb{E}(X)=0$. Moreover,
$I_{G}(\mu;a-\mu,b-\mu)$ in Proposition 2.5 satisfies the identity

$\displaystyle I_{G}(\mu;a-\mu,b-\mu)=\ell^{2}\left[C(\ell)-{1\over
2}\right].$

Upon substituting this in Proposition 2.5 and carrying out some simple
algebraic steps, the required result follows. ∎

######  Remark 2.11.

It is useful to observe that, knowing $C(\ell)$ (see Table 1 for some explicit
examples of these constants), Proposition 2.10 gives a more informative result
than Popoviciu’s inequality and present in particular a method for the exact
calculation of the variance of truncated distributions of the form in (2.4).

Table 1: Some examples of constants $C(\ell)$, for use in Proposition 2.10.

Distribution | $G(x)$ | $C(\ell)$
---|---|---
Normal | $\Phi(x)$ | ${1\over 2\ell^{2}}\left\\{\ell\left[\ell+\exp\big{(}-{\ell^{2}\over 2}\big{)}\sqrt[\ ]{{2\over\pi}}\,\right]+(\ell^{2}-1){\rm erf}\big{(}{\ell\over\sqrt[\ ]{2}}\big{)}\right\\}$
Student-$t$ ($\nu=2$) | ${1\over 2}\left(1+\sqrt[\ ]{x^{2}\over x^{2}+1}\,\right)$ | ${1\over 2\ell^{2}}\left\\{\ell^{2}+{1\over\ell}\sqrt[\ ]{\ell^{2}\over 2+\ell^{2}}\left[2\ell+\ell^{3}-2\sqrt[\ ]{2+\ell^{2}}{\rm arcsinh}({\ell\over\sqrt[\ ]{2}})\right]\right\\}$
Cauchy | ${1\over\pi}\arctan(x)+{1\over 2}$ | ${1\over 2\pi\ell^{2}}\big{[}\ell(\ell\pi-2)+2(1+\ell^{2})\arctan(\ell)\big{]}$
Laplace | ${1\over 2}+{1\over 2}{\rm sgn}(x)[1-\exp(-|x|)]$ | ${1\over\ell^{2}}\exp(-\ell)\big{[}1+\ell+\exp(\ell)(\ell^{2}-1)\big{]}$
Logistic | ${1\over 1+\exp(-x)}$ | ${2\over\ell^{2}}\left\\{{\pi^{2}\over 12}+\ell\log[1+\exp(\ell)]+{\rm Li}_{2}[-\exp(\ell)]\right\\}$

In Table 1, $\Phi$ is the CDF of a standard normal distribution, ${\rm
Li}_{2}[z]=-\int_{0}^{z}\log(1-x)/x{\rm d}x$ is the polylogarithm function of
order $2$ and ${\rm erf}(z)=(2/\sqrt[\ ]{\pi})\int_{0}^{z}\exp(-x^{2}){\rm
d}x$ is the error function.

######  Proposition 2.12.

We further have

$\displaystyle\lim_{\ell\longrightarrow
0^{+}}H(\ell)=\lim_{\ell\longrightarrow
0^{+}}{\sigma^{2}\over\ell^{2}}={1\over
3}\quad\text{and}\quad\lim_{\ell\longrightarrow\infty}H(\ell)=\lim_{\ell\longrightarrow\infty}{\sigma^{2}\over\ell^{2}}=0.$

###### Proof.

Upon taking $a=-\ell$, $b=\ell$ and $\mu=0$ in Corollary 2.9, the required
result follows. ∎

Next, we present two further examples in addition to Table 1.

######  Example 2.1.

Let $X$ be a truncated symmetric standard Cauchy distribution with density
function $f(x)=\left(2\arctan{\ell}\right)^{-1}\cdot\left(1+x^{2}\right)^{-1}$
if $|x|<\ell$, with $\ell\leqslant 1$, and $f(x)=0$, if $|y|>\ell$. As its
variance is $\sigma^{2}=\ell/\arctan\ell-1$ (see Johnson, Kotz and
Balakrishnan,, 1994), we obtain

$\displaystyle H(\ell)=\frac{1}{\ell\arctan\ell}-\frac{1}{\ell^{2}}.$

######  Example 2.2.

Let $X$ be a symmetrically truncated standard Gaussian distribution with
density function $f(x)=\\{\sqrt[\
]{2\pi}[1-2\Phi(-\ell)]\\}^{-1}\exp({-x^{2}/2})$, if $|x|<\ell$, where
$\ell\leqslant 1$ and $\Phi$ is the standard Gaussian cumulative distribution
function, and $f(x)=0$, if $|x|>\ell$ (see Johnson, Kotz and Balakrishnan,,
1994). As its variance can be expressed as $\sigma^{2}=1-2\ell f(\ell)$, we
find

$\displaystyle H(\ell)=\frac{1}{\ell^{2}}-\frac{2}{\ell^{3}f(\ell)}.$

In both these examples, we observe $H(\ell)\rightarrow 1/3$ as $\ell\downarrow
0$ and $H(\ell)\uparrow 0$ as $\ell\uparrow\infty$, as stated in Proposition
2.12. However,

$\displaystyle\ell H(\ell)=\frac{\sigma^{2}}{\ell}$

behaves differently as $\uparrow\infty$. In the first example, $\ell
H(\ell)\rightarrow 2/\pi$, but in the second example, we find $\ell
H(\ell)\rightarrow 0$.

##  3 Illustration with financial data

We illustrate the results developed here with intraday spot exchange rate data
of four currencies against the US dollar transacted on the foreign exchange
(Forex) market. There are 16 million tick-by-tick returns of bid prices
provided by Tick Data, LLC (Table 2). Following the discussion regarding the
truncated nature of the past (Matsushita et al.,, 2020, 2023), we consider the
symmetric case here.

For each currency, let $\\{X_{ij}\\}$ be the $j$th return observed on day
$i=1,\ldots,d$ (Table 2). Taking the daily sample standard deviation $s_{i}$
and the maximum daily absolute return $\ell_{i}=\max_{1\leq j\leq
n_{i}}\\{X_{ij}\\}$, where $n_{i}$ denotes the sample size on day $i$ with
$n=\sum_{i=1}^{d}n_{i}$, Figure 1 depicts the daily sample ratios
$\\{s_{i}/\ell_{i}\\}$ in the form of dots.

Now, consider the general sequence ignoring days as $\\{X_{t}\\}$, where
$t=1,\ldots,n$. Letting $\ell^{*}=\max_{1\leq t\leq n}\\{X_{t}\\}$, we
generated a grid of 1,000 truncation points,
$\\{\ell:\ell=m\ell^{*}/1000,\text{where }m=1,\ldots,1000\\}$. For each $\ell$
over this grid, we obtained the sample standard deviation of the conditional
(truncated) data $\\{X_{t}||X_{t}|\leq\ell\\}$. In this way, we empirically
find the form of the ratio $\sigma/\ell$ for the returns of each currency
(Figure 1). Then, Proposition 2.10 provides a feasible and practical way of
describing the relationship between the variance and the cutoff $\ell$. For
small $\ell$, Matsushita et al., (2020, 2023) proposed the power law
$\sigma^{\beta}/\ell\approx\zeta$ from a second-order approximation, where
$\beta>0$ and $\zeta$ are real constants. So, we may approximate $\sigma/\ell$
as

$\displaystyle\frac{\sigma}{\ell}\approx\zeta^{1/\beta}\ell^{-1+1/\beta}.$

Figure 2 depicts the log-log plots of this approximated result, and shows the
validity of such a power law approximation for $\sigma/\ell$.

Table 2: Intraday spot exchange rate data description.

Country | Currency | Code | Period | Number of days ($d$) | Data points ($n$)
---|---|---|---|---|---
Britain | British pound | GBP | 31 Aug 08 $-$ 12 Jun 15 | 2,116 | 2,754,615
Canada | Canadian dollar | CAD | 12 Jun 00 $-$ 12 Jun 15 | 4,419 | 3,931,202
Japan | Japanese yen | JPY | 30 May 00 $-$ 12 Jun 15 | 4,598 | 4,804,463
Switzerland | Swiss franc | CHF | 30 May 00 $-$ 12 Jun 15 | 4,587 | 4,838,100
|  |  | Total | 15,720 | 16,328,380
Source: Tick Data, LLC. |  |  |  |  |

Figure 1: Display of $\ell$ _versus_ the sample ratio $\sigma/\ell$ (lines)
from data described in Table 2, where dots represent daily values.

Figure 2: Display of $\ln\ell$ _versus_ $\ln\tfrac{\sigma}{\ell}$ (dashed
lines) from data described in Table 2, where dots represent daily values.

##  4 Concluding remarks

In this work, we have presented a general approach for understanding the
relationship between the variance and the range of a general family of
truncated distributions based on skewing functions. We have established a
closed-form expression for its moments and their asymptotic behavior as the
support’s semi-range tends to zero and $\infty$.

As discussed previously by Matsushita et al., (2020, 2023), if the truncated
nature arises naturally from the past, the function relating truncation length
and standard deviation may assist in connecting the bounded past and unbounded
future data. For this reason, we expect our results to be useful in many
practical situations.

##### Acknowledgements

Roberto Vila gratefully acknowledges financial support from CNPq, CAPES, and
FAP-DF, Brazil. Raul Matsushita acknowledges financial support from CNPq,
CAPES, FAP-DF, and DPI/DPG/UnB, Brazil.

##### Disclosure statement

There are no conflicts of interest to disclose.

## References

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