# General laws of large numbers under sublinear expectations††thanks: This
work has been supported in part by the National Basic Research Program of
China (973 Program) (Grant No. 2007CB814901 ) (Financial Risk). First version:
Oct. 20, 2010. This is the third version.

Feng Hu
School of Mathematics
Shandong University
Jinan 250100, China E-mail address<EMAIL_ADDRESS>(F. Hu).

Abstract

In this paper, under some weaker conditions, we give three laws of large
numbers under sublinear expectations (capacities), which extend Peng’s law of
large numbers under sublinear expectations in [8] and Chen’s strong law of
large numbers for capacities in [1]. It turns out that these theorems are
natural extensions of the classical strong (weak) laws of large numbers to the
case where probability measures are no longer additive.

Keywords sublinear expectation, capacity, law of large numbers, maximal
distribution. 2000 MR Subject Classification 60H10, 60G48

##  1  Introduction

The classical strong (weak) laws of large numbers (strong (weak) LLN) as
fundamental limit theorems in probability theory play a fruitful role in the
development of probability theory and its applications. However, these kinds
of limit theorems have always considered additive probabilities and additive
expectations. In fact, the additivity of probabilities and expectations has
been abandoned in some areas because many uncertain phenomena can not be well
modelled by using additive probabilities and additive expectations. Motivated
by some problems in mathematical economics, statistics and quantum mechanics,
a number of papers have used non-additive probabilities (called capacities)
and nonlinear expectations (for example Choquet integral/expectation,
$g$-expectation) to describe and interpret the phenomena. Recently, motivated
by the risk measures, super-hedge pricing and model uncertainty in finance,
Peng [5-9] initiated the notion of independent and identically distributed
(IID) random variables under sublinear expectations. Furthermore, he showed
law of large numbers (LLN) and central limit theorem (CLT) under sublinear
expectations. In [1], Chen presented a strong law of large numbers for
capacities induced by sublinear expectations with the notion of IID random
variables initiated by Peng.

The purpose of this paper is to investigate one of the very important
fundamental results in the theory of Peng’s sublinear expectations: tlaw of
large numbers. All of the results on laws of large numbers in [1] and [8]
require that the sequence of random variables is independent and identically
distributed. In this paper we intend to obtain three laws of large numbers
without the requirement of identical distribution. Under some weaker
conditions, we prove three laws of large numbers under Peng’s sublinear
expectations, which extend Peng’s law of large numbers under sublinear
expectations in [8] and Chen’s strong law of large numbers for capacities in
[1].

This paper is organized as follows: in Section 2, we recall some notions and
lemmas under sublinear expectations. In Section 3, we give our main results
including the proofs.

##  2 Notions and Lemmas

In this section, we present some preliminaries in the theory of sublinear
expectations.

DEFINITION 2.1 (see [5-9]). Let $\Omega$ be a given set and let ${\cal H}$ be
a linear space of real valued functions defined on $\Omega$. We assume that
all constants are in ${\cal H}$ and that $X\in{\cal H}$ implies $|X|\in{\cal
H}$. ${\cal H}$ is considered as the space of our ”random variables”. A
nonlinear expectation $\mathbb{E}$ on ${\cal H}$ is a functional $\mathbb{E}$
: ${\cal H}\mapsto\mathbb{R}$ satisfying the following properties: for all
$X$, $Y\in{\cal H}$, we have
(a) Monotonicity: If $X\geq Y$ then $\mathbb{E}[X]\geq\mathbb{E}Y]$.
(b) Constant preserving: $\mathbb{E}[c]=c$.
The triple $(\Omega,{\cal H},\mathbb{E})$ is called a nonlinear expectation
space (compare with a probability space $(\Omega,{\cal F},P)$). We are mainly
concerned with sublinear expectation where the expectation $\mathbb{E}$
satisfies also
(c) Sub-additivity: $\mathbb{E}[X]-\mathbb{E}[Y]\leq\mathbb{E}[X-Y]$.
(d)Positive homogeneity: $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X]$,
$\forall\lambda\geq 0$.
If only (c) and (d) are satisfied, $\mathbb{E}$ is called a sublinear
functional.

The following representation theorem for sublinear expectations is very useful
(see Peng [8, 9] for the proof).

LEMMA 2.1. Let $\mathbb{E}$ be a sublinear functional defined on
$(\Omega,{\cal H})$, i.e., (c) and (d) hold for $\mathbb{E}$. Then there
exists a family $\\{E_{\theta}:\theta\in\Theta\\}$ of linear functionals on
$(\Omega,{\cal H})$ such that

$\mathbb{E}[X]=\max\limits_{\theta\in\Theta}E_{\theta}[X].$

If (a) and (b) also hold, then $E_{\theta}$ are linear expectations for
$\theta\in\Theta$. If we make furthermore the following assumption: (H) For
each sequence $\\{X_{n}\\}_{n=1}^{\infty}\subset{\cal H}$ such that
$X_{n}(\omega)\downarrow 0$ for $\omega$, we have $\hat{E}[X_{n}]\downarrow
0$. Then for each $\theta\in\Theta$, there exists a unique ($\sigma$-additive)
probability measure $P_{\theta}$ defined on $(\Omega,\sigma({\cal H}))$ such
that

$E_{\theta}[X]=\int_{\Omega}X(\omega){\rm d}P_{\theta}(\omega),\ \ X\in{\cal
H}.$

REMARK 2.1. Lemma 2.1 shows that under $(H)$, indeed, a stronger
representation holds. That is, if $\mathbb{E}$ is a sublinear expectation on
${\cal H}$ satisfying (H), then there exists a set (say $\hat{\cal P}$) of
probability measures such that

$\mathbb{E}[\xi]=\sup_{P\in\hat{\cal P}}E_{P}[\xi],\ \ \
-\mathbb{E}[-\xi]=\inf_{P\in\hat{\cal P}}E_{P}[\xi].$

Therefore, without confusion, we sometimes call supremum expectations as
sublinear expectations.

Given a sublinear expectation $\mathbb{E}$, let us denote the conjugate
expectation ${\cal E}$ of sublinear $\mathbb{E}$ by

${\cal E}[X]:=-\mathbb{E}[-X],\quad\forall X\in\mathcal{H}$

Obviously, for all $X\in\mathcal{H},$ ${\cal E}[X]\leq\mathbb{E}[X].$

Furthermore, let us denote a pair $(\mathbb{V},v)$ of capacities by

$\mathbb{V}(A):=\mathbb{E}[I_{A}],\quad v(A):={\cal E}[I_{A}],\quad\forall
A\in{\cal F}.$

It is easy to check that

$\mathbb{V}(A)+v(A^{c})=1,\quad\forall A\in{\cal F}$

where $A^{c}$ is the complement set of $A.$

DEFINITION 2.2. A set function $V$: ${\cal F}\rightarrow[0,1]$ is called upper
continuous capacity if it satisfies

(1) $V(\phi)=0,V(\Omega)=1$.

(2) $V(A)\leq V(B),$ whenever $A\subset B$ and $A,B\in{\cal F}$.

(3) $V(A_{n})\downarrow V(A)$, if $A_{n}\downarrow A$, where $A_{n},A\in{\cal
F}$.

ASSUMPTION A. Throughout this paper, we assume that $\mathbb{E}$ is a
sublinear expectation, $\mathbb{V}$ is an upper continuous capacity generated
by sublinear expectation $\mathbb{E}$.

The following is the notion of IID random variables under sublinear
expectations introduced by Peng [5-9].

DEFINITION 2.3. Independence: Suppose that $Y_{1},Y_{2},\cdots,Y_{n}$ is a
sequence of random variables such that $Y_{i}\in\mathcal{H}.$ Random variable
$Y_{n}$ is said to be independent to $X:=(Y_{1},\cdots,Y_{n-1})$ under
$\mathbb{E}$, if for each measurable function $\varphi$ on $R^{n}$ with
$\varphi(X,Y_{n})\in\mathcal{H}$ and $\varphi(x,Y_{n})\in\mathcal{H}$ for each
$x\in\mathbb{R}^{n-1},$ we have

$\mathbb{E}[\varphi(X,Y_{n})]=\mathbb{E}[\overline{\varphi}(X)],$

where $\overline{\varphi}(x):=\mathbb{E}[\varphi(x,Y_{n})]$ and
$\overline{\varphi}(X)\in\mathcal{H}$.

Identical distribution: Random variables $X$ and $Y$ are said to be
identically distributed, denoted by $X\overset{d}{=}Y$, if for each $\varphi$
such that $\varphi(X),\;\varphi(Y)\in\mathcal{H}$,

$\mathbb{E}[\varphi(X)]=\mathbb{E}[\varphi(Y)].$

Sequence of IID random variables: A sequence of IID random sequence
$\\{X_{i}\\}_{i=1}^{\infty}$ is said to be IID random variables, if
$X_{i}\overset{d}{=}X_{1}$ and $X_{i+1}$ is independent to
$Y:=(X_{1},\cdots,X_{i})$ for each $i\geq 1.$

The following lemma shows the relation between Peng’s independence and
pairwise independence in Maccheroni and Marinacci [3] and Marinacci [4].

LEMMA 2.2 (see Chen [1]). Suppose that $X,Y\in\mathcal{H}$ are two random
variables. $\mathbb{E}$ is a sub-linear expectation and $(\mathbb{V},v)$ is
the pair of capacities generated by $\mathbb{E}.$ If random variable $X$ is
independent to $Y$ under $\mathbb{E}$, then $X$ also is pairwise independent
to $Y$ under capacities $\mathbb{V},$ and $v$ e.g. for all subsets $D$ and
$G\in{\cal B}(\mathbb{R}),$

$V(X\in D,Y\in G)=V(X\in D)V(Y\in G)$

holds for both capacities $\mathbb{V}$ and $v$.

Borel-Cantelli Lemma is still true for capacity under some assumptions.

LEMMA 2.3 (see Chen [1]). Let $\\{A_{n},n\geq 1\\}$ be a sequence of events in
${\cal F}$ and $(\mathbb{V},v)$ be a pair of capacities generated by sublinear
expectation $\mathbb{E}$.

(1) If $\sum\limits_{n=1}^{\infty}\mathbb{V}(A_{n})<\infty,$ then
$\mathbb{V}\left(\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{i=n}^{\infty}A_{i}\right)=0.$

(2) Suppose that $\\{A_{n},n\geq 1\\}$ are pairwise independent with respect
to $v$, e.g.

$v\left(\bigcap\limits_{i=1}^{\infty}A_{i}^{c}\right)=\prod_{i=1}^{\infty}v\left(A_{i}^{c}\right).$

If $\sum\limits_{n=1}^{\infty}{\mathbb{V}}(A_{n})=\infty$, then

$\mathbb{V}\left(\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{i=n}^{\infty}A_{i}\right)=1.$

DEFINITION 2.4 ( Maximal distribution) (see Peng [8, 9]). Let
$C_{b,Lip}(\mathbb{R})$ denote the space of bounded and Lipschitz continuous
functions. A random variable $\eta$ on sublinear expectation space
$(\Omega,{\cal H},\mathbb{E})$ is called maximal distributed if

$\mathbb{E}[\varphi(\eta)]=\sup\limits_{\underline{\mu}\leq
y\leq\overline{\mu}}\varphi(y),\ \ \ \forall\varphi\in C_{b,Lip}(\mathbb{R}),$

where $\overline{\mu}:=\mathbb{E}[\eta]$ and $\underline{\mu}:={\cal
E}[\eta]$.

REMARK 2.2 (see Peng [8, 9]). Let $\eta$ be maximal distributed with
$\overline{\mu}=\mathbb{E}[\eta]$, $\underline{\mu}={\cal E}[\eta]$, the
distribution of $\eta$ is characterized by the following parabolic PDE:

$\partial_{t}u-g(\partial_{x}u)=0,\quad u(0,x)=\varphi(x),$

where $u(t,x):=\mathbb{E}[\varphi(x+t\eta)],$
$(t,x)\in[0,\infty)\times\mathbb{R}$,
$g(x):=\overline{\mu}x^{+}-\underline{\mu}x^{-}$ and $x^{+}:=\max\\{x,0\\}$,
$x^{-}:=(-x)^{+}$.

With the notion of IID under sublinear expectations, Peng shows a law of large
numbers under sublinear expectations (see Theorem 5.1 in Peng [8]).

LEMMA 2.4 (Law of large numbers under sublinear expectations). Let
$\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of IID random variables with finite
means $\overline{\mu}=\mathbb{E}[X_{1}],\;\;\underline{\mu}={\cal E}[X_{1}].$
Suppose $\mathbb{E}[|X_{1}|^{2}]<\infty$. Then for any continuous and linear
growth function $\varphi,$

$\mathbb{E}\left[\varphi\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\right)\right]\to\sup_{\underline{\mu}\leq
y\leq\overline{\mu}}\varphi(y),\;\hbox{ as}\;n\to\infty.$

The following lemma is a strong law of large numbers for capacities, which can
be found in Chen [1].

LEMMA 2.5 ( Strong law of large numbers for capacities). Let
$\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of IID random variables for
sublinear expectation $\mathbb{E}$. Suppose $\mathbb{E}[|X_{1}|^{2}]<\infty$.
Set $\overline{\mu}:=\mathbb{E}[X_{1}],$ $\underline{\mu}:={\cal E}[X_{1}]$
and $S_{n}:=\sum\limits_{i=1}^{n}X_{i}.$ Then

(I)

$v\left(\underline{\mu}\leq\liminf\limits_{n\to\infty}S_{n}/n\leq\limsup\limits_{n\to\infty}S_{n}/n\leq\overline{\mu}\right)=1.$

(II)

$\mathbb{V}\left(\limsup\limits_{n\to\infty}S_{n}/n=\overline{\mu}\right)=1,\quad\mathbb{V}\left(\liminf\limits_{n\to\infty}S_{n}/n=\underline{\mu}\right)=1.$

(III) $\forall b\in[\underline{\mu},\overline{\mu}]$,

$\mathbb{V}\left(\liminf\limits_{n\to\infty}|S_{n}/n-b|=0\right)=1.$

##  3 Main Results

3.1 General law of large numbers under sublinear expectations

THEOREM 3.1 (General law of large numbers under sublinear expectations). Let a
sequence $\\{X_{i}\\}_{i=1}^{\infty}$ which is in a sublinear expectation
space $(\Omega,{\cal H},\mathbb{E})$ satisfy the following conditions:

(i) each $X_{i+1}$ is independent of $(X_{1},\cdots,X_{i})$, for
$i=1,2,\cdots$;

(ii) $\mathbb{E}[X_{i}]=\overline{\mu}_{i}$, ${\cal
E}[X_{i}]=\underline{\mu_{i}}$, where
$-\infty<\underline{\mu_{i}}\leq\overline{\mu}_{i}<\infty$;

(iii) there are two constants ${\overline{\mu}}$ and ${\underline{\mu}}$ such
that

$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}|\underline{\mu_{i}}-\underline{\mu}|=0,\
\ \
\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum\limits_{i=1}^{n}|\overline{\mu_{i}}-\overline{\mu}|=0;$

(iv) $\sup\limits_{i\geq 1}\mathbb{E}[|X_{i}|^{2}]<\infty$. Then for any
continuous and linear growth function $\varphi,$

$\mathbb{E}\left[\varphi\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\right)\right]\to\sup_{\underline{\mu}\leq
y\leq\overline{\mu}}\varphi(y),\;\hbox{ as}\;n\to\infty.$

PROOF. The main idea comes from Theorem 3.1 Li and Shi [2]. First we prove the
case that $\varphi$ is a bounded and Lipschitz continuous function. For a
small but fixed $h>0$, Let $V$ be the unique viscosity solution of the
following equation

$\partial_{t}V+g(\partial_{x}V)=0,\ \ \ (t,x)\in[0,1+h]\times\mathbb{R},\ \ \
V(1+h,x)=\varphi(x),$ (1)

where $g(x):=\overline{\mu}x^{+}-\underline{\mu}x^{-}$. According to the
definition of maximal distribution, we have

$V(t,x)=\mathbb{E}[\varphi(x+(1+h-t)\eta)].$

Particularly,

$V(h,0)=\mathbb{E}[\varphi(\eta)],\ \ \ V(1+h,x)=\varphi(x).$ (2)

Since (1) is a uniformly parabolic PDE, by the interior regularity of $V$ (see
Wang [10]), we have

$||V||_{C^{1+\alpha/2,1+\alpha}([0,1]\times\mathbb{R})}<\infty,\ \ \ \hbox{for
some}\ \ \ \alpha\in(0,1).$

We set $\delta:=\frac{1}{n}$ and ${S}_{0}:=0$. Then

$\begin{array}[]{lcl}&&V(1,\delta{S}_{n})-V(0,0)=\sum\limits_{i=0}^{n-1}\\{V((i+1)\delta,\delta{S}_{i+1})-V(i\delta,\delta{S}_{i})\\}\\\
&&=\sum\limits_{i=0}^{n-1}\left\\{\left[V((i+1)\delta,\delta{S}_{i+1})-V(i\delta,\delta{S}_{i+1})\right]+\left[V(i\delta,\delta{S}_{i+1})-V(i\delta,\delta{S}_{i})\right]\right\\}\\\
&&=\sum\limits_{i=0}^{n-1}\\{I_{\delta}^{i}+J_{\delta}^{i}\\},\end{array}$

with, by Taylor’s expansion,

$J_{\delta}^{i}=\partial_{t}V(i\delta,\delta{S}_{i})\delta+\partial_{x}V(i\delta,\delta{S}_{i})X_{i+1}\delta,$

$\begin{array}[]{lcl}I_{\delta}^{i}&=&\int_{0}^{1}\left[\partial_{t}V((i+\beta)\delta,\delta{S}_{i+1})-\partial_{t}V(i\delta,\delta{S}_{i+1})\right]{\rm
d}\beta\delta\\\
&+&\left[\partial_{t}V(i\delta,\delta{S}_{i+1})-\partial_{t}V(i\delta,\delta{S}_{i})\right]\delta\\\
&+&\int_{0}^{1}\left[\partial_{x}V(i\delta,\delta{S}_{i}+\beta\delta
X_{i+1})-\partial_{x}V(i\delta,\delta{S}_{i})\right]{\rm d}\beta
X_{i+1}\delta.\end{array}$

Thus

$\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]+{\cal
E}\left[\sum\limits_{i=0}^{n-1}I_{\delta}^{i}\right]\leq\mathbb{E}[V(1,\delta{S}_{n})]-V(0,0)\leq\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]+\mathbb{E}\left[\sum\limits_{i=0}^{n-1}I_{\delta}^{i}\right].$
(3)

From (1) as well as the independence of $X_{i+1}$ to $(X_{1},\cdots,X_{i})$,
it follows that

$\begin{array}[]{lcl}&&\mathbb{E}[J_{\delta}^{i}]=\mathbb{E}\left[\partial_{t}V(i\delta,\delta{S}_{i})\delta+\partial_{x}V(i\delta,\delta{S}_{i})X_{i+1}\delta\right]\\\
&&=\mathbb{E}\left\\{\partial_{t}V(i\delta,\delta{S}_{i})\delta+\delta[(\partial_{x}V(i\delta,\delta{S}_{i}))^{+}\overline{\mu_{i+1}}-(\partial_{x}V(i\delta,\delta{S}_{i}))^{-}\underline{\mu_{i+1}}]\right\\}\\\
&&\leq\mathbb{E}\left\\{\partial_{t}V(i\delta,\delta{S}_{i})\delta+\delta[(\partial_{x}V(i\delta,\delta{S}_{i}))^{+}\overline{\mu}-(\partial_{x}V(i\delta,\delta{S}_{i}))^{-}\underline{\mu}]\right\\}\\\
&&+\delta\mathbb{E}[(\partial_{x}V(i\delta,\delta{S}_{i}))^{+}(\overline{\mu_{i+1}}-\overline{\mu})+(\partial_{x}V(i\delta,\delta{S}_{i}))^{-}(\underline{\mu_{i+1}}-\underline{\mu})]\\\
&&=\delta\mathbb{E}[(\partial_{x}V(i\delta,\delta{S}_{i}))^{+}(\overline{\mu_{i+1}}-\overline{\mu})+(\partial_{x}V(i\delta,\delta{S}_{i}))^{-}(\underline{\mu_{i+1}}-\underline{\mu})]\\\
&&\leq\delta(|\overline{\mu_{i+1}}-\overline{\mu}|+|\underline{\mu_{i+1}}-\underline{\mu}|)\mathbb{E}[|\partial_{x}V(i\delta,\delta{S}_{i})|].\end{array}$

But since both $\partial_{t}V$ and $\partial_{x}V$ are uniformly
$\alpha$-hölder continuous in $x$ and $\frac{\alpha}{2}$-hölder continuous in
$t$ on $[0,1]\times\mathbb{R}$, it follows that

$|\partial_{x}V(i\delta,\delta{S}_{i})-\partial_{x}V(0,0)|\leq
C|\delta{S}_{i}|^{\alpha}+|i\delta|^{\frac{\alpha}{2}},$

where $C$ is some positive constant. Since

$\mathbb{E}[|\delta{S}_{i}|^{\alpha}]\leq\mathbb{E}[|\delta{S}_{i}|]+1\leq\sup\limits_{i\geq
1}\mathbb{E}[|X_{i}|]+1.$

Hence, by (iv), we claim that there exists a constant $C_{1}>0$, such that

$\mathbb{E}[|\partial_{x}V(i\delta,\delta{S}_{i})|]\leq C_{1}.$

Then we obtain

$\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]\leq\sum\limits_{i=0}^{n-1}\mathbb{E}\left[J_{\delta}^{i}\right]\leq
C_{1}\frac{1}{n}\sum\limits_{i=0}^{n-1}(|\overline{\mu_{i+1}}-\overline{\mu}|+|\underline{\mu_{i+1}}-\underline{\mu}|).$

From (iii), we have
$\limsup\limits_{n\rightarrow\infty}\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]\leq
0.$

In a similar manner of the above, we also have

$\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]\geq-\sum\limits_{i=0}^{n-1}\delta(|\overline{\mu_{i+1}}-\overline{\mu}|+|\underline{\mu_{i+1}}-\underline{\mu}|)\mathbb{E}[|\partial_{x}V(i\delta,\delta{S}_{i})|].$

By (iii), it follows that
$\liminf\limits_{n\rightarrow\infty}\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]\geq
0.$ So we can claim that

$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]=0.$
(4)

For $I_{\delta}^{i}$, since both $\partial_{t}V$ and $\partial_{x}V$ are
uniformly $\alpha$-hölder continuous in $x$ and $\frac{\alpha}{2}$-hölder
continuous in $t$ on $[0,1]\times\mathbb{R}$, we then have

$|I_{\delta}^{i}|\leq
C\delta^{1+\alpha/2}(1+|X_{i+1}|^{\alpha}+|X_{i+1}|^{1+\alpha}).$

It follows that

$\mathbb{E}[|I_{\delta}^{i}|]\leq
C\delta^{1+\alpha/2}(1+\mathbb{E}[|X_{i+1}|^{\alpha}]+\mathbb{E}[|X_{i+1}|^{1+\alpha}]).$

Thus

$\begin{array}[]{lcl}&&-C(\frac{1}{n})^{\frac{\alpha}{2}}\frac{1}{n}\sum\limits_{i=0}^{n-1}\left(1+\mathbb{E}\left[|X_{i+1}|^{\alpha}\right]+\mathbb{E}\left[|X_{i+1}|^{1+\alpha}\right]\right)+\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right]\\\
&&\leq\mathbb{E}[V(1,\delta{S}_{n})]-V(0,0)\\\ &&\leq
C(\frac{1}{n})^{\frac{\alpha}{2}}\frac{1}{n}\sum\limits_{i=0}^{n-1}\left(1+\mathbb{E}\left[|X_{i+1}|^{\alpha}\right]+\mathbb{E}\left[|X_{i+1}|^{1+\alpha}\right]\right)+\mathbb{E}\left[\sum\limits_{i=0}^{n-1}J_{\delta}^{i}\right].\end{array}$
(5)

Therefore, from (iv) and (4), as $n\rightarrow\infty$, we have

$\lim\limits_{n\rightarrow\infty}\mathbb{E}[V(1,\delta S_{n})]=V(0,0).$ (6)

On the other hand, for each $t$, $t^{{}^{\prime}}\in[0,1+h]$ and
$x\in\mathbb{R}$,

$|V(t,x)-V(t^{{}^{\prime}},x)|\leq C|t-t^{{}^{\prime}}|.$

Thus

$|V(0,0)-V(h,0)|\leq Ch$ (7)

and, by (2)

$\begin{array}[]{lcl}&&|\mathbb{E}[V(1,\delta{S}_{n})]-\mathbb{E}[\varphi(\delta
S_{n})]|\\\
&&=|\mathbb{E}[V(1,{\delta}{S}_{n})]-\mathbb{E}[V(1+h,{\delta}S_{n})|\leq
Ch.\end{array}$ (8)

It follows from (6)-(8) that

$\limsup\limits_{n\rightarrow\infty}\left|\mathbb{E}\left[\varphi\left(\frac{S_{n}}{{n}}\right)\right]-\mathbb{E}[\varphi(\eta)]\right|\leq
2C{h}.$

Since $h$ can be arbitrarily small, we have

$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[\varphi\left(\frac{S_{n}}{{n}}\right)\right]=\mathbb{E}\left[\varphi\left(\eta\right)\right]=\sup_{\underline{\mu}\leq
y\leq\overline{\mu}}\varphi(y).$

The rest proof of Theorem 3.1 is very similar to that of Lemma 5.5 in peng
[8]. So we omit it.

From Theorem 3.1, we easily claim the following corollary.

COROLLARY 3.1. Let a sequence $\\{X_{i}\\}_{i=1}^{\infty}$ which is in a
sublinear expectation space $(\Omega,{\cal H},\mathbb{E})$ satisfy the
following conditions:

(i) each $X_{i+1}$ is independent of $(X_{1},\cdots,X_{i})$, for
$i=1,2,\cdots$;

(ii) $\mathbb{E}[X_{i}]=\overline{\mu}_{i}$, ${\cal
E}[X_{i}]=\underline{\mu}_{i}$, where
$-\infty<\underline{\mu}_{i}\leq\overline{\mu}_{i}<\infty$;

(iii) there are two constants ${\overline{\mu}}$ and ${\underline{\mu}}$ such
that $\lim\limits_{i\rightarrow\infty}\underline{\mu}_{i}=\underline{\mu}$,
$\lim\limits_{i\rightarrow\infty}\overline{\mu}_{i}=\overline{\mu};$

(iv) $\sup\limits_{i\geq 1}\mathbb{E}[|X_{i}|^{2}]<\infty$. Then for any
continuous and linear growth function $\varphi,$

$\mathbb{E}\left[\varphi\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\right)\right]\to\sup_{\underline{\mu}\leq
y\leq\overline{\mu}}\varphi(y),\;\hbox{ as}\;n\to\infty.$

3.2 General strong law of large numbers for capacities induced by sublinear
expectations

THEOREM 3.2 (General strong law of large numbers for capacities). Under the
conditions of Theorem 3.1, then

(I)

$v\left(\underline{\mu}\leq\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}\leq\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}\leq\overline{\mu}\right)=1.$

(II) $\forall b\in[\underline{\mu},\overline{\mu}]$,

$\mathbb{V}\left(\liminf\limits_{n\to\infty}\left|\frac{\sum_{i=1}^{n}X_{i}}{n}-b\right|=0\right)=1.$

(III)

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}=\overline{\mu}\right)=1,\quad\mathbb{V}\left(\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}=\underline{\mu}\right)=1.$

In order to prove Theorem 3.2, we need the following lemma.

LEMMA 3.1 (see Chen [1]). In a sublinear expectation space $(\Omega,{\cal
H},\mathbb{E})$, let $\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of independent
random variables such that $\sup\limits_{i\geq
1}\mathbb{E}[|X_{i}|^{2}]<\infty$. Suppose that there exists a constant $C>0$
such that

$|X_{n}-\mathbb{E}[X_{n}]|\leq C\frac{n}{\log(1+n)},\quad n=1,2,\cdots.$

Then there exists a sufficiently large number $m>1$ such that

$\sup_{n\geq
1}\mathbb{E}\left[\exp\left(\frac{m\log(1+n)}{n}S_{n}\right)\right]<\infty.$

Where $S_{n}:=\sum\limits_{i=1}^{n}[X_{i}-\mathbb{E}[X_{i}]].$

PROOF OF THEOREM 3.2. First, it is easy to show that (I) is equivalent to the
conjunction of

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}>\overline{\mu}\right)=0,$
(9)
$\mathbb{V}\left(\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}<\underline{\mu}\right)=0.$
(10)

Indeed, write
$A:=\left\\{\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}>\overline{\mu}\right\\},$
$B:=\left\\{\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}<\underline{\mu}\right\\},$
the equivalence can be proved from the inequality

$\max\\{\mathbb{V}(A),V(B)\\}\leq\mathbb{V}\left(A\bigcup
B\right)\leq\mathbb{V}(A)+\mathbb{V}(B).$

We shall turn the proofs of (9) and (10) into two steps.

Step 1 Assume that there exists a constant $C>0$ such that
$|X_{n}-\overline{\mu}_{n}|\leq\frac{Cn}{\log(1+n)}$ for $n\geq 1.$ To prove
(9), we shall show that for any $\varepsilon>0,$

$\mathbb{V}\left(\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}\left\\{\frac{\sum_{i=1}^{n}(X_{i}-\overline{\mu_{i}})}{n}\geq\varepsilon\right\\}\right)=0.$
(11)

Indeed, by Lemma 3.1, for any $\varepsilon>0,$ let us choose $m>1/\varepsilon$
such that

$\sup_{n\geq
1}\mathbb{E}\left[\exp\left(\frac{m\log(1+n)}{n}\sum_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\right)\right]<\infty.$

By Chebyshev’s inequality,

$\begin{array}[]{lcl}\mathbb{V}\left(\frac{\sum_{i=1}^{n}(X_{i}-\overline{\mu}_{i})}{n}\geq\varepsilon\right)&=&\mathbb{V}\left(\frac{m\log(1+n)}{n}\sum\limits_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\geq\varepsilon
m\log(1+n)\right)\\\ &\leq&{\rm e}^{-\varepsilon
m\log(1+n)}\mathbb{E}\left[\exp\left(\frac{m\log(1+n)}{n}\sum\limits_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\right)\right]\\\
&\leq&\frac{1}{(1+n)^{\varepsilon m}}\sup\limits_{n\geq
1}\mathbb{E}\left[\exp\left(\frac{m\log(1+n)}{n}\sum\limits_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\right)\right].\end{array}$

Since $\varepsilon m>1,$ and $\sup\limits_{n\geq
1}\mathbb{E}\left[\exp\left(\frac{m\log(1+n)}{n}\sum\limits_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\right)\right]<\infty.$
It then follows from the convergence of
$\sum\limits_{n=1}^{\infty}\frac{1}{(1+n)^{\varepsilon m}},$ we have

$\sum_{n=1}^{\infty}\mathbb{V}\left(\frac{\sum_{i=1}^{n}(X_{i}-\overline{\mu}_{i})}{n}\geq\varepsilon\right)<\infty.$

Using the first Borel-Cantelli Lemma, we have

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}(X_{i}-\overline{\mu}_{i})}{n}\geq\varepsilon\right)=0\quad\forall\varepsilon>0,$

which implies

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}>\overline{\mu}\right)=0.$

Also

$v\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}\leq\overline{\mu}\right)=1.$

Similarly, considering the sequence $\\{-X_{i}\\}_{i=1}^{\infty}.$ By step 1,
it suffices to obtain

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{-\sum_{i=1}^{n}(X_{i}-\underline{\mu_{i}})}{n}>0\right)=0.$

Hence,

$\mathbb{V}\left(\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}<\underline{\mu}\right)=0.$

Also

$v\left(\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}\geq\underline{\mu}\right)=1.$

Step 2. Write

$\overline{X}_{n}:=(X_{n}-\overline{\mu}_{n})I_{\left\\{|X_{n}-\overline{\mu}_{n}|\leq\frac{Cn}{\log(1+n)}\right\\}}-\mathbb{E}\left[(X_{n}-\overline{\mu}_{n})I_{\left\\{|X_{n}-\overline{\mu}_{n}|\leq\frac{Cn}{\log(1+n)}\right\\}}\right]+\overline{\mu}_{n}.$

It is easy to check that $\\{\overline{X}_{i}\\}_{i=1}^{\infty}$ satisfies the
assumptions in Lemma 3.1. Indeed, obviously for each $n\geq 1,$

$|\overline{X}_{n}-\overline{\mu}_{n}|\leq\frac{2Cn}{\log(1+n)}.$

On the other hand, for each $n\geq 1,$ it easy to check that

$|\overline{X}_{n}-\overline{\mu}_{n}|\leq|X_{n}-\overline{\mu}_{n}|+\mathbb{E}[|X_{n}-\overline{\mu}_{n}|].$

Then, by (iv),

$\mathbb{E}[|\overline{X}_{n}-\overline{\mu}_{n}|^{2}]\leq
4\left(\mathbb{E}[|X_{n}-\overline{\mu}_{n}|^{2}]+(\mathbb{E}[|X_{n}-\overline{\mu}_{n}|])^{2}\right)<\infty.$

Set
$\overline{S}_{n}:=\sum\limits_{i=1}^{n}(\overline{X}_{i}-\overline{\mu}_{i}),$
immediately,

$\frac{1}{n}\sum\limits_{i=1}^{n}(X_{i}-\overline{\mu}_{i})\leq\frac{1}{n}\overline{S}_{n}+\frac{1}{n}\sum_{i=1}^{n}|X_{i}-\overline{\mu}_{i}|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}+\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[|X_{i}-\overline{\mu}_{i}|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}\right].$
(12)

Since $\sup\limits_{i\geq 1}\mathbb{E}[|X_{i}|^{2}]<\infty$, we have

$\begin{array}[]{lcl}\sum\limits_{i=1}^{\infty}\frac{\mathbb{E}\left[\left|X_{i}-\overline{\mu}_{i}\right|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}\right]}{i}&\leq&\sum\limits_{i=1}^{\infty}\frac{\log(1+i)}{ci^{2}}\mathbb{E}[|X_{i}-\overline{\mu}_{i}|^{2}]\\\
&\leq&C_{1}\sum\limits_{i=1}^{\infty}\frac{\log(1+i)}{i^{2}}<\infty.\end{array}.$

By Kronecker Lemma,

$\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[|X_{i}-\overline{\mu}_{i}|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}\right]\to
0.$ (13)

Furthermore, write
$A_{i}:=\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}$ for
$i\geq 1.$ It suffices now to prove that

$\mathbb{V}\left(\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}A_{i}\right)=0.$

Indeed, by Chebyshev’s inequality,

$\mathbb{V}\left(|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right)\leq\left(\frac{\log(1+i)}{Ci}\right)^{2}\mathbb{E}[|X_{i}-\overline{\mu}_{i}|^{2}]$

Hence,

$\sum_{i=1}^{\infty}\mathbb{V}\left(|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right)<\infty$

and by the first Borel-Cantelli Lemma, we have
$\mathbb{V}\left(\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{i=n}^{\infty}A_{i}\right)=0.$
This implies that
$\omega\not\in\bigcap\limits_{n=1}^{\infty}\bigcup\limits_{i=n}A_{i},$ the
sequence

$\sum_{i=1}^{n}\frac{|X_{i}-\overline{\mu}_{i}|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}}{i}$

converges almost surely with respect to $\mathbb{V}$ as $n\rightarrow\infty$.
Applying Kronecker Lemma again,

$\frac{1}{n}\sum_{i=1}^{n}\left(|X_{i}-\overline{\mu}_{i}|I_{\left\\{|X_{i}-\overline{\mu}_{i}|>\frac{Ci}{\log(1+i)}\right\\}}\right)\to
0,\hbox{ a.s}\quad\mathbb{V}.$ (14)

Set $\lim\sup$ on both side of (12), then by (13) and (14), we have

$\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}(X_{i}-\overline{\mu}_{i})}{n}\leq\limsup\limits_{n\to\infty}\frac{\overline{S}_{n}}{n},\quad\hbox{a.s.}\quad\mathbb{V}.$

Since $\\{\overline{X}_{n}\\}_{n=1}^{\infty}$ satisfies the assumption of Step
1, by Step 1,

$\mathbb{V}\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}\overline{X}_{i}}{n}>\overline{\mu}\right)=0.$

Also

$v\left(\limsup\limits_{n\to\infty}\frac{\sum_{i=1}^{n}\overline{X}_{i}}{n}\leq\overline{\mu}\right)=1.$

In a similar manner, we can prove

$v\left(\liminf\limits_{n\to\infty}\frac{\sum_{i=1}^{n}X_{i}}{n}\geq\underline{\mu}\right)=1.$

Therefore, the proof of (I) is complete.

To prove (II). Denote $S_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i}$. If
$\overline{\mu}=\underline{\mu}$, it is trivial. Suppose
$\overline{\mu}>\underline{\mu},$ we prove that, for any
$b\in(\underline{\mu},\overline{\mu})$

$\mathbb{V}\left(\liminf_{n\to\infty}|S_{n}/{n}-b|=0\right)=1.$

To do so, we only need to prove that there exists an increasing subsequence
$\\{n_{k}\\}$ of $\\{n\\}$ such that for any
$b\in(\underline{\mu},\overline{\mu})$

$\mathbb{V}\left(\bigcap_{m=1}^{\infty}\bigcup_{k=m}^{\infty}\\{|S_{n_{k}}/{n_{k}}-b|\leq\varepsilon\\}\right)=1.$
(15)

Indeed, for any
$0<\varepsilon\leq\min\\{\overline{\mu}-b,b-\underline{\mu}\\},$ let us choose
$n_{k}=k^{k}$ for $k\geq 1.$

Set $\overline{S}_{n}:=\sum\limits_{i=1}^{n}(X_{i}-b),$ then

$\begin{array}[]{lcl}\mathbb{V}\left(\left|\frac{S_{n_{k}}-S_{n_{k-1}}}{n_{k}-n_{k-1}}-b\right|\leq\epsilon\right)&=&\mathbb{V}\left(\left|\frac{S_{n_{k}-n_{k-1}}}{n_{k}-n_{k-1}}-b\right|\leq\varepsilon\right)\\\
&=&\mathbb{V}\left(\left|\frac{S_{n_{k}-n_{k-1}}-(n_{k}-n_{k-1})b}{n_{k}-n_{k-1}}\right|\leq\varepsilon\right)\\\
&=&\mathbb{V}\left(\left|\frac{\overline{S}_{n_{k}-n_{k-1}}}{n_{k}-n_{k-1}}\right|\leq\varepsilon\right)\\\
&\geq&\mathbb{E}\left[\phi\left(\frac{\overline{S}_{n_{k}-n_{k-1}}}{n_{k}-n_{k-1}}\right)\right]\end{array}$

where $\phi(x)$ is defined by

$\phi(x)=\left\\{\begin{array}[]{l}1-e^{|x|-\varepsilon},\quad\quad|x|\leq\varepsilon;\\\
0,\quad\quad\quad\quad\quad\;\;|x|>\varepsilon.\end{array}\right.$

Considering the sequence of random variables $\\{X_{i}-b\\}_{i=1}^{\infty}.$
Obviously

$\mathbb{E}[X_{i}-b]=\overline{\mu}_{i}-b,\quad{\cal
E}[X_{i}-b]=\underline{\mu_{i}}-b.$

Applying Theorem 3.1, we have, as $k\to\infty$,

$\mathbb{E}\left[\phi\left(\frac{\overline{S}_{n_{k}-n_{k-1}}}{n_{k}-n_{k-1}}\right)\right]\to\sup_{\underline{\mu}-b\leq
y\leq\overline{\mu}-b}\phi(y)=\phi(0)=1-{\rm e}^{-\varepsilon}>0.$

Thus

$\sum_{k=1}^{\infty}\mathbb{V}\left(\left|\frac{S_{n_{k}}-S_{n_{k-1}}}{n_{k}-n_{k-1}}-b\right|\leq\varepsilon\right)\geq\sum_{k=1}^{\infty}\mathbb{E}\left[\phi\left(\frac{\overline{S}_{n_{k}-n_{k-1}}}{n_{k}-n_{k-1}}\right)\right]=\infty.$

Note the fact that the sequence of $\\{S_{n_{k}}-S_{n_{k-1}}\\}_{k\geq 1}$ is
independent for all $k\geq 1$ . Using the second Borel-Cantelli Lemma, we have

$\liminf_{k\to\infty}\left|\frac{S_{n_{k}}-S_{n_{k-1}}}{n_{k}-n_{k-1}}-b\right|\leq\varepsilon,\quad\hbox{a.s.}\;\mathbb{V}.$

But

$\left|\frac{S_{n_{k}}}{n_{k}}-b\right|\leq\left|\frac{S_{n_{k}}-S_{n_{k-1}}}{n_{k}-n_{k-1}}-b\right|\cdot\frac{n_{k}-n_{k-1}}{n_{k}}+\left[\frac{|S_{n_{k-1}}|}{n_{k-1}}+|b|\right]\frac{{n_{k-1}}}{n_{k}}.$
(16)

Noting the following fact,

$\frac{n_{k}-n_{k-1}}{n_{k}}\to 1,\quad\frac{{n_{k-1}}}{n_{k}}\to
0,\;\hbox{as}\;k\to\infty$

and

$\limsup_{n\to\infty}S_{n}/n\leq\overline{\mu},\quad\limsup_{n\to\infty}(-S_{n})/n\leq-\underline{\mu},$

which implies

$\limsup_{n\to\infty}|S_{n}|/n\leq\max\\{|\overline{\mu}|,|\underline{\mu}|\\}<\infty.$

Hence, from inequality (16), for any $\varepsilon>0,$

$\liminf_{k\to\infty}\left|\frac{S_{n_{k}}}{n_{k}}-b\right|\leq\varepsilon,\hbox{
a.s.}\;\mathbb{V}.$

That is

$\mathbb{V}\left(\liminf_{n\to\infty}|S_{n}/{n}-b|\leq\varepsilon\right)=1.$

Since $\varepsilon$ can be arbitrarily small, we have

$\mathbb{V}\left(\liminf_{n\to\infty}|S_{n}/{n}-b|=0\right)=1.$

Now we prove that

$\mathbb{V}\left(\limsup\limits_{n\to\infty}S_{n}/n=\overline{\mu}\right)=1,\quad\mathbb{V}\left(\liminf\limits_{n\to\infty}S_{n}/n=\underline{\mu}\right)=1.$

Indeed, from
$\mathbb{V}\left(\liminf\limits_{n\rightarrow\infty}\left|\frac{S_{n}}{n}-b\right|=0\right)=1$,
$\forall b\in(\underline{\mu},\overline{\mu})$, we can obtain

$\mathbb{V}\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\geq
b\right)=1.$

Then

$\mathbb{V}\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\geq\overline{\mu}\right)=1.$

On the other hand, we have, for each $P\in{\cal P}$,

$P\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}=\overline{\mu}\right)=P\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\geq\overline{\mu}\right)+P(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\leq\overline{\mu})-1.$

But

$P\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\leq\overline{\mu}\right)\geq
v\left(\underline{\mu}\leq\liminf\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\leq\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}\leq\overline{\mu}\right)=1.$

Thus

$\mathbb{V}\left(\limsup\limits_{n\rightarrow\infty}\frac{S_{n}}{n}=\overline{\mu}\right)=1.$

In a similar manner, we can obtain

$\mathbb{V}\left(\liminf\limits_{n\rightarrow\infty}\frac{S_{n}}{n}=\underline{\mu}\right)=1.$

The proof of Theorem 3.2 is complete.

3.2 General weak law of large numbers for capacities induced by sublinear
expectations

THEOREM 3.3 (General weak law of large numbers for capacities). Under the
conditions of Theorem 3.1, then for any $\varepsilon>0$,

$\lim\limits_{n\rightarrow\infty}v\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\in(\underline{\mu}-\varepsilon,\overline{\mu}+\varepsilon)\right)=1.$
(17)

PROOF. To prove (17), we only need to prove that

$\lim\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq\underline{\mu}-\varepsilon\right)=0$

and

$\lim\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\geq\underline{\mu}+\varepsilon\right)=0.$

Now we prove

$\lim\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq\underline{\mu}-\varepsilon\right)=0.$

For any $\delta<\varepsilon$, construct two functions $f$, $g$ such that

$f(x)=1\ \ \hbox{for}\ \ x\leq\underline{\mu}-\varepsilon-\delta,$ $f(x)=0\ \
\hbox{for}\ \ x\geq\underline{\mu}-\varepsilon,$
$f(x)=\frac{1}{\delta}(\underline{\mu}-\varepsilon-x)\ \ \hbox{for}\ \
\underline{\mu}-\varepsilon-\delta<x<\underline{\mu}-\varepsilon$

and

$g(x)=1\ \ \hbox{for}\ \ x\leq\underline{\mu}-\varepsilon,$ $g(x)=0\ \
\hbox{for}\ \ x\geq\underline{\mu}-\varepsilon+\delta,$
$g(x)=\frac{1}{\delta}(\underline{\mu}-\varepsilon-x)+1\ \ \hbox{for}\ \
\underline{\mu}-\varepsilon<x<\underline{\mu}-\varepsilon+\delta.$

Obviously, $f$ and $g\in C_{b,Lip}(\mathbb{R})$. By Theorem 3.1, we have

$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[f\left(\frac{1}{{n}}\sum\limits_{i=1}^{n}X_{i}\right)\right]=\sup\limits_{\underline{\mu}\leq
y\leq\overline{\mu}}f(y)=0,$
$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[g\left(\frac{1}{{n}}\sum\limits_{i=1}^{n}X_{i}\right)\right]=\sup\limits_{\underline{\mu}\leq
y\leq\overline{\mu}}g(y)=0.$

So

$0=\sup\limits_{\underline{\mu}\leq
y\leq\overline{\mu}}f(y)\leq\liminf\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq\underline{\mu}-\varepsilon\right)\leq\limsup\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq\underline{\mu}-\varepsilon\right)\leq\sup\limits_{\underline{\mu}\leq
y\leq\overline{\mu}}g(y)=0.$ (18)

Hence

$\lim\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\leq\underline{\mu}-\varepsilon\right)=0.$

In a similar manner, we can obtain

$\lim\limits_{n\rightarrow\infty}\mathbb{V}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\geq\underline{\mu}+\varepsilon\right)=0.$

The proof of Theorem 3.3 is complete.

## References

  * [1] Chen, Z. J., Strong laws of large numbers for capacities, arXiv:1006.0749v1 [math.PR] 3 Jun 2010.
  * [2] Li, M. and Shi, Y. F., A general central limit theorem under sublinear expectations, Science China Mathematics, 53, 2010, 1989-1994.
  * [3] Maccheroni, F. and Marinacci, M., A strong law of large number for capacities, Ann. Proba., 33, 2005, 1171-1178.
  * [4] Marinacci, M., Limit laws for non-additive probabilities and their frequentist interpretation, J. Econom. Theory, 84, 1999, 145-195.
  * [5] Peng, S. G., G-expectation, G-Brownian motion and related stochastic calculus of Ito type, in: F.E. Benth, et al. (Eds.), Proceedings of the Second Abel Symposium, 2005, Springer-Verlag, 2006, 541-567.
  * [6] Peng, S. G., Law of large number and central limit theorem under nonlinear expectations, arXiv: math. PR/0702358vl 13 Feb 2007.
  * [7] Peng, S. G., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 118(12), 2008, 2223-2253.
  * [8] Peng, S. G., A new central limit theorem under sublinear expectations, arXiv:0803.2656v1 [math.PR] 18 Mar. 2008.
  * [9] Peng, S. G., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A-Mathematics, 52(7), 2009, 1391-1411.
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# Evolution of QPOs in GX 339$-$4 and EXO 1846$-$031 with Insight-HXMT and
NICER

Zuobin Zhang11affiliation: Center for Field Theory and Particle Physics and
Department of Physics, Fudan University, 200438 Shanghai, China. †E-mail:
<EMAIL_ADDRESS>, Honghui Liu11affiliation: Center for Field Theory and
Particle Physics and Department of Physics, Fudan University, 200438 Shanghai,
China. †E-mail<EMAIL_ADDRESS>, Divya Rawat22affiliation: Inter-
University Center for Astronomy and Astrophysics, Ganeshkhind, Pune 411007,
India , Cosimo Bambi11affiliation: Center for Field Theory and Particle
Physics and Department of Physics, Fudan University, 200438 Shanghai, China.
†E-mail<EMAIL_ADDRESS>††affiliationmark:  , Ranjeev Misra22affiliation:
Inter-University Center for Astronomy and Astrophysics, Ganeshkhind, Pune
411007, India , Pengju Wang33affiliation: University of Chinese Academy of
Sciences, Chinese Academy of Sciences, Beijing 100049, China 44affiliation:
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China , Long Ji55affiliation:
School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China
, Shu Zhang33affiliation: University of Chinese Academy of Sciences, Chinese
Academy of Sciences, Beijing 100049, China 44affiliation: Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China and Shuangnan Zhang33affiliation: University
of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049,
China 44affiliation: Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China [

###### Abstract

We conduct a spectral and timing analysis of GX 339$-$4 and EXO 1846$-$031
with the aim of studying the evolution of Type-C QPOs with spectral
parameters. The high cadence data from Insight-HXMT and NICER allow us to
track them. Type-C QPOs appear at the end of low-hard state and/or hard-
intermediate state. The results reveal that the QPO frequency is closely
related to the inner disk radius and mass accretion rate in the two sources.
Such a correlation is nicely consistent with the dynamic frequency model.

###### Subject headings:

High energy astrophysics; X-ray astronomy; Low mass X-ray Binary; Stellar mass
black holes

##  1\. Introduction

Quasi-periodic oscillations (QPOs) refers to narrow peaks structure in the
power density spectrum (PDS) that are commonly observed in X-ray binaries
(XRBs) (van der Klis, 2005). In black hole systems, QPOs are mainly split into
low-frequency QPOs (LFQPOs, centroid frequency $0.1-30$ Hz), and high
frequency QPOs (HFQPOs, centroid frequency $\geq 60$ Hz) (Belloni, 2010).
Samimi et al. (1979) reported the ‘sporadic quasi-periodic behaviour’ in the
light curve of GX 339$-$4, and Motch et al. (1983) reported the first rigorous
detection of QPOs for the same source. It was immediately recognized that the
QPOs would have been a powerful tool to study the accretion process around
black holes.

Over the last forty years, especially after the launch of RXTE, we have
accumulated a lot of knowledge about QPOs. Using a truncated power-law to fit
the broadband noise in PDS, and a Lorentz function with the center frequency
of $\nu_{\rm QPO}$ to fit the low-frequency QPOs, Wijnands et al. (1999) found
that there is an obvious positive correlation between the truncation frequency
$\nu_{\rm b}$ and the frequency of the low-frequency QPOs $\nu_{\rm LF}$.
Psaltis et al. (1999) reported that there is also a good positive correlation
between the frequency of low-frequency QPOs and the frequency of the broadband
noise (or high-frequency QPOs) in low mass XRBs, including black holes and
neutron stars.

We have observed QPOs in most black hole XRBs, and realized that low frequency
QPOs can be divided into three types: Type-A, -B, and -C QPOs, based on
quality factor, noise type, fractional rms, and phase delay (e.g., Wijnands et
al. 1999; Sobczak et al. 2000; Casella et al. 2005; Motta et al. 2011).
Different types QPOs occupy different regions on the hardness-intensity
diagram, as well as obviously distribute in different areas on the center
frequency and rms plots (e.g., Motta et al. 2011). The phenomenon of rapid
transition between different types of QPOs has been found in some sources, and
the time scale of this phenomenon can be very short (10 s) (e.g. Homan et al.
2020). In this work, we only focus on the Type-C QPOs.

Type-C QPOs appear in the early stage of the outburst, particularly in the
hard-intermediate state and the end of low-hard state. The centroid frequency
varies from a few mHz to $\sim 10$ Hz, and is tightly correlated with the
spectral state. Vignarca et al. (2003) reported a positive correlation between
the centroid frequency and the photon index $\Gamma$. Motta et al. (2011)
found Type-C QPOs trace out a well-defined track, and the centroid frequency
obviously correlate with the corona flux and disk flux. The dependence of the
QPO frequency and photon energy was illustrated by Qu et al. (2010).

In addition to the phenomenological study of QPOs, many studies has been done
on the theoretical explanation behind it. Most theoretical models explain QPO
phenomenon through the following two different mechanisms: instabilities of
the corona-disk system (e.g., Titarchuk & Fiorito 2004; Mastichiadis et al.
2022; Varnière et al. 2012) or the geometrical effects of general relativity
(e.g., Stella & Vietri 1998; Ingram et al. 2009). Titarchuk & Fiorito (2004)
introduced a transition layer in corona-disk system that can explain the QPO
phenomenon in XRBs. The disk-corona natural frequency model was proposed by
Mastichiadis et al. (2022), and they argued that type-C QPOs arise from the
interaction of the hot corona with the cold accretion disk. Varnière et al.
(2012) suggested that LFQPOs could result from the relativistic accretion-
ejection instability (AEI). The geometrical effects model mainly refers to the
precession of the corona region. This model interprets the QPOs as a
Lense–Thirring precession of the innermost region of the accretion disk (e.g.,
Stella & Vietri 1998; Ingram et al. 2009). In recent years, more and more
observations have been analyzed to test these models. However, a unified model
that can explain all QPO behaviors has not been found yet.

Recently, Misra et al. (2020) identified the QPO frequency of GRS 1915+105 as
the relativistic dynamic frequency of a truncated accretion disk with AstroSat
observations of that source. The authors found a strong correlation between
the QPO frequency divided by the accretion rate and the inner disk radius. The
correlation is actually consistent with the prediction of dynamic frequency
under the assumption of a standard relativistic accretion model (Novikov &
Thorne, 1973). Liu et al. (2021a) extended the relation to cover a wider range
of variations, and confirmed the high spin nature of the black hole in GRS
1915+105 with the data of Insight-HXMT (dubbed HXMT; Zhang et al. 2014). We
note that GRS 1915+105 is a persistent source with particular properties
(Belloni et al., 2000). We would like to test if this relation holds for other
sources different from GRS 1915+105, and we notice that there are two
appropriate sources, GX 339$-$4 and EXO 1846$-$031, in the archive.

The XRB transient GX 339$-$4 is a typical low mass X-ray binary (LMXB)
discovered in 1973 (Markert et al., 1973). It goes into bright outburst every
a few years and all four X-ray states typically seen in XRBs have been
detected in this system (e.g., Miyamoto et al. 1995; Homan & Belloni 2005;
Plant et al. 2014). GX 339$-$4 is located at $8–12$ kpc with a black hole mass
of $4–11$ M⨀ (Zdziarski et al., 2019). Strong relativistic reflection
signatures have been found in this source in the hard and soft states (e.g.,
García et al. 2015; Miller et al. 2004; Liu et al. 2022). Previous studies
have found that the black hole in GX 339$-$4 has a very high spin ($a_{*}\sim
0.95$, García et al. 2015; Parker et al. 2016). The inclination angle of the
accretion disk should have an intermediate value (Fürst et al. 2015; Parker et
al. 2016). Motta et al. (2011) systematically studied the properties and the
behaviour of QPOs, as a function of the integrated broad-band variability and
the spectral parameters. The authors suggested that the frequencies of all
QPOs (including Type-C QPOs) correlate with the disk flux.

EXO 1846$-$031 was discovered by the European X-ray Observatory Satellite
(EXOSAT) when it went into outburst in April 1985 and is considered a LMXB
(Parmar et al. 1993; Draghis et al. 2020). CGRO/BATSE detected a second
outburst in 1994 (Zhang et al., 1994). Since then, the source was in a
quiescent state for 25 years. EXO 1846$-$031 had a new outburst in 2019, which
was monitored by X-ray missions (e.g., MAXI/GSC; HXMT; NuSTAR) and radio
missions (e.g., MeerKAT; AMI-LA). NuSTAR conducted a high-quality observation
on August 3, 2019 with a 22.2 ks exposure time. Draghis et al. (2020) reported
strong relativistic reflection features with the extremely sensitive NuSTAR
spectra, and argued that the source is a black hole with a nearly maximal spin
parameter ($a_{*}=0.997$) at disk inclination of $\theta=73^{\circ}$. EXO
1846$-$031 is located at $2.4–7.5$ kpc according to the previous studies on
X-ray and radio data (Parmar et al. 1993; Williams et al. 2022), with a black
hole mass of $\sim 9$ M⨀ (Draghis et al. 2020; Williams et al. 2022). Liu et
al. (2021b) reported the observational results from a detailed timing analysis
of EXO 1846$-$031 2019 outburst with the observations of HXMT and NICER.

In this work, we focus on the latest HXMT and NICER observations of GX 339$-$4
and EXO 1846$-$031, and present a detailed temporal and spectral analysis. The
paper is organized as follows. Sec. 2 presents the observational data
reduction. The spectral timing analysis is reported in Sec. 3. We discuss the
results and report our conclusions in Sec. 4 and Sec. 5, respectively.

Figure 1.— MAXI GSC light curve and corresponding hardness of GX 339$-$4
starting from 2021 February 5 (MJD = 59250). The hardness is defined as the
ratio between the count rates at $4–10$ keV and $2–4$ keV bands. The vertical
black lines mark the HXMT observations analyzed in this work and the vertical
red lines mark the NICER observations.
Figure 2.— MAXI GSC light curve and corresponding hardness of EXO 1846$-$031
starting from 2019 June 16 (MJD = 58650). The hardness is defined as the ratio
between the count rates at $4–10$ keV and $2–4$ keV bands. The vertical black
lines mark the HXMT observations analyzed in this work and the vertical red
lines mark the NICER observations.

##  2\. Observations and data reduction

###  2.1. Data selection

Starting from February 2021, GX 339$-$4 went into a new outburst that lasted
for a few months. Fig. 1 shows the long-term light curve in the $2–20$ keV
band and the corresponding hardness ratio observed with MAXI GSC. The hardness
is defined as the ratio between the count rates at $4–10$ keV and $2–4$ keV.
HXMT and NICER extensively observed the 2021 outburst of the source. We went
through all available HXMT and NICER data and picked out those observations
that show Type–C QPO signatures. The selected observations analyzed in this
work are marked in the light curve of GX 339$-$4 in Fig. 1. Information about
these observations is listed in Tab. A1.

The 2019 outburst of EXO 1846$-$031 was first detected by MAXI/GSC on 2019
July 23 (Negoro et al., 2019), and it lasted about 3 months. The MAXI X-ray
Hardness–Intensity diagram (HID) of the outburst shows a characteristic
q-shaped hysteresis of X-ray binaries in outburst (Williams et al. 2022; Liu
et al. 2021a). The long-term MAXI light curve and the corresponding hardness
ratio are shown in Fig. 2. HXMT and NICER conducted high–cadence pointing
observations of EXO 1846$-$031\. Type–C QPOs appear during the transition from
hard to soft state (Liu et al., 2021a). We selected observations showing
Type–C QPO signatures. The selected observations are marked in the light curve
in Fig. 2 and listed in Tab. A2.

###  2.2. Data reduction

HXMT covers the broadband energy range of 1–250 keV with low-energy, medium-
energy, and high-energy detectors (Zhang et al. 2020; Chen et al. 2020; Cao et
al. 2020; Liu et al. 2020). The light curves and spectra are extracted with
the HXMT data analysis software (HXMTDAS) version 2.05 and CALDB version 2.06,
following the official user guide. The background is estimated by the
standalone scripts hebkgmap, mebkgmap, and lebkgmap (Guo et al. 2020; Liao et
al. 2020a; Liao et al. 2020b). The data are screened following the recommended
criteria, i.e., an elevation angle $\textgreater 10^{\circ}$, a geomagnetic
cutoff rigidity $\textgreater 10$ GeV, a pointing offset angle $\textless
0.1$, and at least 300 s away from the South Atlantic Anomaly.

The NICER data are processed with the NICER data analysis software (NICERDAS)
version 2021-04-01_V008 and CALDB version 20210707. We use the standard
filtering criteria: the pointing offset is less than $54^{\prime\prime}$, and
the pointing direction is more than $40^{\circ}$ away from the bright Earth
limb, more than $30^{\circ}$ away from the dark Earth limb, and outside the
South Atlantic Anomaly. In addition, we remove the data of detectors # 14 and
# 34 which are affected by episodes of increased electronic noise, and we
select events that are not flagged as “overshoot” or “undershoot” resets
(EVENT_FLAGS = bxxxx00) or forced triggers (EVENT_FLAGS=bx1x000). The standard
NICER reduction routine nicerl2 is used to process the data. The cleaned
events are barycenter-corrected using the FTOOL barycorr. We extract the
energy spectra of the background in each observation using the nibackgen3C50
tool (Remillard et al., 2022). The Redistribution Matrix File (RMF) and
Ancillary Response File (ARF) are created by using the tasks nicerrmf and
nicerarf, respectively.

Figure 3.— Light curves of GX 339$-$4 by NICER in the 0.5–10 keV band (top
panel) and corresponding hardness (bottom panel) (ObsID: 4133010103,
4133010104). The hardness is defined as the ratio between the count rates in
the $4–10$ keV and $0.5–4$ keV bands. The source undergoes a process of flux
increase and hardness decrease during this period. The intervals with exposure
$>200$ s are marked with yellow shadow. The selected sub-observations are
numbered 4133010103-1 through 4133010103-9, and 4133010104-1 through
4133010104-13.

##  3\. Data analysis

###  3.1. Timing analysis

We extract HXMT LE and NICER XTI light curves with a time resolution of 1 ms
from the full energy band (1-10 keV in HXMT; 0.5-10 keV in NICER) for each
HXMT observation and NICER observation. In order to calculate hardness ratios,
we also produce LE light curves from the 1-5 keV and 5–10 keV bands, and
produce XTI light curves from the 0.5-4 keV and 4-10 keV bands.

We carefully check the extracted light curves from all observations of GX
339$-$4 and find there are two NICER observations (ObsID: 4133010103,
4133010104) that show a relatively strong variability of count rate and
hardness. Fig. 3 shows the light curves of these two NICER observations. The
gaps in the light curves are due to the low Earth orbit of the telescope or
the SAA. We can clearly see that the source went through a period of
luminosity increase and hardness decrease. Comparing with the location of
these two observations in Fig. 1 (the last red dotted line; in fact, since the
two lines are quite close, they look like one line), we conclude that the two
observations are during the hard-to-soft transition so the hardness keeps
getting lower. Then we divide the observations according to the light curve,
counting each snapshot as a sub-observation. We check over all sub-observation
data and pick out those with exposure $>200$ s. The selected sub-observations
are numbered 4133010103-1 through 4133010103-9, and 4133010104-1 through
4133010104-13, as shown in Fig. 3. The other light curves do not show strong
variability in the count rate i.e., no distinctive evidence of flares, dips,
or state transitions, making it safe for timing and spectral analysis to
characterize the source properties.

For EXO 1846$-$031, the count rate of the source remain fairly stable during
each HXMT and NICER interval, and the hardness does not change dramatically.
Therefore, we conclude that we can carry out timing and spectral analysis in
the unit of one observation.

To measure the QPO frequency of GX 339$-$4 and EXO 1846$-$031, we employ the
Python package Stingray (Huppenkothen et al., 2019) to create PDS for each
observation. The light curve is splited into 64 s segments, and then the final
PDS is generated by averaging all 64 s segments. The PDS is normalized
according to the ”rms” method (Belloni & Hasinger, 1990), and logarithmically
rebinned so that each bin size is 1.02 times larger than the previous bin.
Note that we focus on the HXMT LE 1-10 keV light curve and the NICER XTI
0.5-10 keV light curve to extract PDS and search for QPO signal. The 8–30 keV
HXMT ME light curves have been analyzed in the same way and return consistent
measurements of the QPO frequencies. So we report only the results from LE
data in this work.

We use XSPEC v12.12.1 (Arnaud, 1996) to analyze the PDS. The typical PDS of
XRBs manifests broad components and one or two narrow peaks at different
frequencies, corresponding to broad band noise, the possible QPO fundamental
and (sub) harmonics, respectively. We need at least one narrow Lorentzian for
the QPO to fit the Poisson-extracted PDS (Belloni et al., 2002). More narrow
Lorentzians are sometimes included to model harmonic peaks. All QPOs we detect
have a quality factor (Q) greater than 4 and detection significance greater
than $3\sigma$. Fig. A1 and Fig. A2 show a typical PDS and the fit results
with several Lorentzian models for GX 339$-$4\. Fig. A3 and Fig. A4 show the
counterpart for EXO 1846$-$031\. The QPO frequencies for each observation are
listed in Tab. A3 for GX 339$-$4, and Tab. A4 for EXO 1846$-$031.

###  3.2. Spectral analysis

For spectral analysis of the HXMT data, we consider the LE data in the 2–10
keV band and ME data in the 8–20 keV band. ME data above 20 keV and HE data
are ignored because of the very high background. Note that we ignore the data
below 2 keV of the LE instrument in spectral analysis (instead of the 1 keV
for timing analysis) because of calibration uncertainties in the low energy
band. For NICER data, we consider the 1-10 keV band in this section, ignoring
the data below 1 keV because of calibration issues.

The HXMT and NICER spectra are fitted with the XSPEC (v12.12.1) package, using
the recommended photoelectric cross sections of Verner et al. (1996) and
element abundances of Wilms et al. (2000). The $\chi^{2}$ statistics is
employed and all parameter uncertainties are estimated at 90% confidence
level, corresponding to $\Delta\chi^{2}=2.71$. All spectra are grouped to
ensure a minimum counts of 20 per bin. A systematic error of 1% is added in
the NICER spectra.

The HXMT and NICER spectra of GX 339$-$4 are fitted with the model combination
Tbabs $\times$ (simpl $\times$ kerrd + relxill). Tbabs is included to account
for absorption by the interstellar medium. We set its column density ($n_{\rm
H}$) to be a free parameter for NICER spectra. While with HXMT spectra, we can
not constrain its column density ($n_{\rm H}$), so we fix it at best-fit
value, $0.55\times 10^{22}$ cm-2, which is consistent with the result of the
NICER data and the value in literature (e.g., Wang et al. 2020; Liu et al.
2022). kerrd accounts for the thermal emission from the geometrically thin and
optically thick accretion disk (Ebisawa et al., 2003), in which the black hole
distance, mass, and inclination angle of the accretion disk are set to 8.4
kpc, 9.0 M⨀, and 30∘ (Parker et al., 2016), respectively. The spectral
hardening factor of kerrd is set to 1.7 (Shimura & Takahara, 1995). simpl
(Steiner et al., 2009) is used to take into account for the Comptonization of
disk photons by the corona. The source has been found to have strong
reflection features (Liu et al., 2022), and we use the full reflection model
relxill (García et al., 2014) to fit them. The spin parameter ($a_{*}$) is
fixed at 0.95 (Parker et al., 2016), and the index of the emissivity profile
is fixed at 3 because it cannot be constrained by the fit. The best-fit values
and uncertainties of GX 339$-$4 are shown in Tab. A3. Fig. A1 and Fig. A2 show
typical spectra and fit results of HXMT data and NICER data, respectively.

In the case of EXO 1846$-$031, the best-fit model combination is Tbabs
$\times$ (simpl $\times$ kerrd + relxill) for HXMT spectra. The black hole
distance, mass, and inclination angle of the accretion disk are set to 4.5 kpc
(Williams et al., 2022), 10.0 M⨀ (Williams et al. 2022, Draghis et al. 2020)
and 73∘ (Draghis et al., 2020), respectively. The spin parameter ($a_{*}$) in
relxill is fixed at 0.998 (Draghis et al., 2020). We use a simple power-law to
model the emissivity profile ($q_{\rm in}=q_{\rm out}$ free). The other
parameters are set exactly as in the case of GX 339$-$4\. For NICER spectra,
we notice that there are still some large residuals in the soft X-ray band
with the same model, including a Gaussian-like emission near 1.1 keV and edge-
like shapes near 1.8 keV. These energies correspond to features in the
effective area of NICER versus energy (e.g., Wang et al. 2020), where 1.1 and
1.8 keV are attributed to sodium and silicon, respectively. Therefore, we
adopt the following model for the NICER spectra: Tbabs $\times$ (simpl
$\times$ kerrd + relxill + gaussian) $\times$ edge. This calibration issue
arises in EXO 1846$-$031 because the source has a high interstellar
absorption, which makes the photon count rate in the lower energy band
relatively low, making the calibration issue prominent. Typical spectra and
fit results of HXMT and NICER are shown in Fig. A3 and Fig. A4. In Tab. A4, we
summarize the best-fit values and errors of EXO 1846$-$031.

Figure 4.— Evolution of disk inner radius $R_{\rm in}$ (left panel) and QPO
frequency ($f_{\rm QPO}$) along with MJD in GX 339$-$4\. The black points
indicate HXMT data, and red dots indicate NICER data.  Figure 5.— Evolution of
disk inner radius $R_{\rm in}$ (left panel) and QPO frequency ($f_{\rm QPO}$)
along with MJD in EXO 1846$-$031\. The black points and red points represent
the results of HXMT data and NICER data, respectively.  Figure 6.— Correlation
between the parameters involved in the temporal and spectral analysis in the
case of GX 339$-$4\. Correlation of the QPO frequency vs. inner disk radius
and the QPO frequency vs. accretion rate are shown in the upper left and upper
right panels. The two central panels illustrate the accretion rate vs. inner
disk radius and inner disk radius vs. photon index. The photon index vs. QPO
frequency is depicted in the bottom panel. The black and red crosses denote
the results of HXMT data and NICER data, respectively. In the left top panel,
the dashed gray lines represent the correlation of the frequency and inner
radius predicted by Lense–Thirring precession model. The lines from left to
right depict $a_{*}=0.3$, 0.5, 0.7, 0.9 and 0.998, respectively.  Figure 7.—
Correlation between the parameters involved in temporal and spectral analysis
in the case of EXO 1846$-$031\. It is organized as in Fig. 6.  Figure 8.—
Variation of QPO frequency divided by the accretion rate vs. inner disk radius
in the case of GX 339$-$4\. The results of HXMT data and NICER data are
denoted with black and red crosses, respectively. The orange curve represents
the best fit ( $a_{*}=0.603$, $N=1.02$), and the blue curve corresponds to
$a_{*}=0.900$, $N=0.52$.  Figure 9.— Variation of QPO frequency divided by the
accretion rate vs. inner disk radius in the case of EXO 1846$-$031\. The black
and red crosses denote the results of HXMT data and NICER data, respectively.
The blue curve represents the best fit ( $a_{*}=0.9978$, $N=0.281$). The
orange curve represents the best fit ( $a_{*}=0.754$, $N=0.853$) when we only
include the data with $R_{\rm in}>5$ $R_{\rm g}$. It proves that the data with
small $R_{\rm in}$ are important to fit the spin parameter.

##  4\. Results and discussion

Fig. 4 and Fig. 5 show the evolution of inner radius ($R_{in}$) and QPO
frequency ($f_{\rm QPO}$) with time for GX 339$-$4 and EXO 1846$-$031,
respectively. Generally speaking, we clearly see that the value of $f_{\rm
QPO}$ monotonically increases with time. The behaviour is consistent with that
reported in Motta et al. (2011) and Liu et al. (2021b). It has also been
observed in other XRBs, for example, XTE J1859+226 (Casella et al., 2004). In
addition, a notable feature for both sources is the decrease of $R_{in}$. For
GX 339$-$4, the inner disk moves toward the ISCO (Innermost Stable Circular
Orbit), from $>50R_{\rm g}$ to $\sim 7R_{\rm g}$ ($R_{\rm g}$, gravitional
radius), which coincide with the result in the previous study (e.g. Wang-Ji et
al. 2018; Wang et al. 2020). Although there is some variable feature, EXO
1846$-$031 shows a similar trend.

Correlation between the parameters involved in temporal and spectral analysis
are shown in Fig. 6 and Fig. 7. An interesting result is the relationship
between the photon index ($\Gamma$) and the QPO frequency ($f_{\rm QPO}$). The
results we get from both sources share the same tendency, as shown in the
bottom panels of Fig. 6 and Fig. 7. There is a strong positive correlation
between $f_{\rm QPO}$ and $\Gamma$ of the power-law in the beginning which
flattens or starts reversing at the highest values of the $f_{\rm QPO}$. The
turnoff in the correlation is not apparent in GX 339$-$4, while it is evident
in EXO 1846$-$031 (around $\Gamma\sim 2.7$). A similar kind of correlation
have been reported in a number of other LMXBs (e.g., Vignarca et al. 2003;
Titarchuk & Fiorito 2004; Titarchuk & Seifina 2009; Fürst et al. 2016).
Titarchuk & Fiorito (2004) introduced the transition layer (TL) model to
explain the observed correlations. The TL model depicts how the QPOs related
to the corona properties (e.g., the size, optical depth, temperature and
spectral index), and predicts the correlation between photon index and QPO
frequency. The results we get are in good agreement with the model’s
predictions, except for the observations of EXO 1846$-$031 with $f_{\rm
QPO}>5.18$, where a negative correlation between $f_{\rm QPO}$ and $\Gamma$
appears. A universal explanation of this correlation between $\Gamma$ and
$f_{\rm QPO}$ is still missing.

The upper left panel of Fig. 6 shows a broad anti-correlation between the QPO
frequency ($f_{\rm QPO}$) and inner radius ($R_{in}$) in GX 339$-$4\. This
anti-correlation is not particularly significant in EXO 1846$-$031, and we can
only see a general tendency of larger $R_{in}$ corresponding to a smaller
frequency. The same correlation between the QPO frequency and the disk inner
radius was reported in other sources (e.g., GRS 1915+105; Rodriguez et al.
2002). The Lense–Thirring precession model would predict anti-correlation
between $f_{\rm QPO}$ and $R_{in}$, and a direct dependence of the QPO
frequency on the inner radius (Ingram et al. 2009; Ingram & Done 2010). To
check the possibility of modeling the results with the relativistic precession
model, we use equation 2 in Ingram et al. (2009) to fit data points. The model
cannot explain the results we obtained, both in the case of GX 339$-$4 and EXO
1846$-$031, as shown in the plot.

The variation of the frequency with the accretion rate is shown in the upper
right panels of Fig. 6 and Fig. 7. Liu et al. (2021a) reported a strongest
correlation between the QPO frequency ($f_{\rm QPO}$) and mass accretion rate
($\dot{M}$) in GRS 1915+105 with HXMT data. We do not find any significant
correlation between them in GX 339$-$4, while there is a weak anti-correlation
in EXO 1846$-$031\. In fact, a positive correlation between $f_{\rm QPO}$ and
$\dot{M}$ is proposed in the TL model by Titarchuk & Fiorito (2004), and in
the disk-corona natural frequency model by Mastichiadis et al. (2022). Fig. 3
of Titarchuk & Fiorito (2004) depicts the positive correlation between $f_{\rm
QPO}$ and the $\gamma$-parameter (which is proportional to mass accretion
rate), which is opposite to what we find. Besides, Mastichiadis et al. (2022)
argue that type-C QPOs could arise from the interaction of the hot corona with
the cold accretion disk, and predict a formula $f_{\rm 0}\propto\dot{M}^{1/2}$
below a certain mass accretion rate (Fig. 5 of Mastichiadis et al. (2022)).
The results we get do not fit well with the predictions of that model. These
discrepancies may suggest that the “transition layer model” and the disk-
corona natural frequency model are not favored in our case.

Misra et al. (2020) identified QPOs as the dynamic frequency of a truncated
relativistic accretion disk in the case of GRS 1915+105. The dynamic frequency
is defined as the ratio of the sound propagation velocity of the inner disk to
the truncation radius, i.e., the inverse of the sound crossing time (Misra et
al., 2020). Based on the assumption that the accretion disk is a standard
relativistic accretion disk (Novikov & Thorne, 1973), the dynamic frequency is
a function of the inner radius ($R_{\rm in}$), black hole spin ($a_{*}$), mass
accretion rate ($\dot{M}$) and a normalization factor ($N$). Liu et al.
(2021a) extended the results in Misra et al. (2020) to a larger range of
accretion rates with HXMT data of GRS 1915+105, and confirmed the high spin
nature of the source. Following the work of Misra et al. (2020) and Liu et al.
(2021a), we illustrate the relation between QPO frequency ($f_{\rm QPO}$)
divided by accretion rate ($\dot{M}$) and disk inner radius ($R_{\rm in}$) in
Fig. 8 and Fig. 9. Both sources show negative correlation between $f_{\rm
QPO}/\dot{M}$ and $R_{\rm in}$, Moreover, the correlation follows the
prediction of the dynamic frequency model.

We fit the relation between $f_{\rm QPO}/\dot{M}$ and $R_{\rm in}$ using
Equation (3) in Misra et al. (2020). The fit returns $a_{*}=0.9978\pm 0.0009$
and $N=0.281\pm 0.025$ for EXO 1846$-$031, indicating a rapidly spinning black
hole. This result is consistent with what has been reported by analyzing the
blurred reflection spectra (e.g., Draghis et al. 2020; Abdikamalov et al.
2021). The best-fit curve is shown in Fig. 9. In the case of GX 339$-$4, the
fit returns $a_{*}=0.603\pm 0.026$ and $N=1.02\pm 0.05$. Such a low spin
result is somewhat different from the result obtained by analyzing the blurred
reflection spectra or thermal spectra (e.g. Reis et al. 2008; Ludlam et al.
2015; García et al. 2015; Parker et al. 2016; Wang et al. 2020). We note that
for this source we do not have data below 6 $R_{\rm g}$. The relativistic
effects are more evident at lower $R_{\rm in}$ (3 $\sim$ 5 $R_{\rm g}$).
Hence, data points at lower $R_{\rm g}$ plays a crucial role in the estimation
of spin parameter value. In Fig. 8, we simultaneously show two curves,
$a_{*}=0.603$ and $a_{*}=0.900$. It is worth noting that the most important
difference between the two curves is reflected in the region with low $R_{\rm
in}$. This also proves our view that a reasonable fitting value cannot be
obtained because of the lack of data with relatively small $R_{\rm in}$.

The middle right panels of Fig. 6 and Fig. 7 show that the inner disk radius
tends to decrease when the photon index ($\Gamma$) increases. The behaviour is
consistent with that expected during a hard-to-soft transition. A noteworthy
positive correlation between the mass accretion rate ($\dot{M}$) and the inner
radius ($R_{in}$) in EXO 1846$-$031 is described in the middle left panel of
Fig. 7. A similar relationship was reported in GRS 1915+105 (Misra et al.
2020; Liu et al. 2021a; Rawat et al. 2022) and MAXI J1535-571 (Garg et al.,
2022). The correlation is beyond the expectation of truncated disk model (Done
et al., 2007). But Dullemond & Spruit (2005) predicted a positive correlation
between $\dot{M}$ and $R_{\rm in}$ (see their Fig. 8), calculating the
evaporation of the cool accretion disk on account of the ion-bombardment. An
alternative explanation is discussed in Abramowicz et al. (1978), where the
authors suggested a larger inner edge is required when the mass accretion rate
increases to dissipate the angular momentum of accretion material.

##  5\. Conclusion

We investigate the temporal and spectral properties from the latest HXMT and
NICER observing campaign of GX 339$-$4 and EXO 1846$-$031\. Temporal and
spectral analysis show that the evolution of the QPO frequency is closely
related to mass accretion rate and inner disk radius in both sources.
Comparing the relativistic precession model, the natural frequency model and
the dynamic frequency model, we find the dynamic frequency model gives the
best fit to our results. We extend the application of the model from GRS
1915+105, a persistent source, to these two transient sources, and confirm the
high spin nature of the black hole in EXO 1846$-$031.

## Acknowledgements

This work was supported by the Natural Science Foundation of Shanghai, Grant
No. 22ZR1403400, the Shanghai Municipal Education Commission, Grant No.
2019-01-07-00-07-E00035, and the National Natural Science Foundation of China
(NSFC), Grant No. 12250610185, 11973019, and 12261131497.

##  Appendix A Appendix A

Mission |  Obs. ID |  Start data |  Exposure (s)
---|---|---|---
HXMT | P0304024026 | 2021-03-12 | 2274
P0304024028 | 2021-03-14 | 1401
P0304024032 | 2021-03-18 | 1597
P0304024035 | 2021-03-22 | 1669
P0304024036 | 2021-03-24 | 1193
P0304024038 | 2021-03-26 | 2088
NICER | 3558011402 | 2021-03-17 | 1595
3558011501 | 2021-03-19 | 7560
4133010101 | 2021-03-19 | 2030
4133010102 | 2021-03-20 | 1860
4133010103 | 2021-03-26 | 6111
4133010104 | 2021-03-27 | 8709
Table A1HXMT and NICER observations of GX 339$-$4 analyzed in this work. For HXMT, the listed exposure time is for the LE instrument.  Mission |  Obs. ID |  Start data |  Exposure (s)
---|---|---|---
HXMT | P021405000101 | 2019-08-02 | 718
P021405000102 | 2019-08-02 | 1436
P021405000103 | 2019-08-02 | 762
P021405000104 | 2019-08-02 | 718
P021405000105 | 2019-08-02 | 1715
P021405000106 | 2019-08-03 | 563
P021405000107 | 2019-08-03 | 656
P021405000301 | 2019-08-05 | 700
P021405000302 | 2019-08-05 | 1102
P021405000303 | 2019-08-05 | 678
P021405000401 | 2019-08-06 | 718
P021405000502 | 2019-08-07 | 691
P021405000503 | 2019-08-07 | 539
P021405000601 | 2019-08-08 | 1130
P021405000701 | 2019-08-08 | 1163
P021405000702 | 2019-08-09 | 1795
NICER | 2200760101 | 2019-07-31 | 5658
2200760102 | 2019-08-01 | 1165
2200760103 | 2019-08-02 | 2562
2200760104 | 2019-08-03 | 1488
2200760105 | 2019-08-04 | 1130
2200760106 | 2019-08-05 | 3564
2200760107 | 2019-08-06 | 912
2200760108 | 2019-08-07 | 927
2200760109 | 2019-08-08 | 3293
2200760110 | 2019-08-09 | 4629
2200760112 | 2019-08-11 | 2749
2200760113 | 2019-08-12 | 3341
2200760114 | 2019-08-13 | 7154
2200760115 | 2019-08-13 | 8181
2200760116 | 2019-08-15 | 4703
2200760117 | 2019-08-16 | 8739
2200760118 | 2019-08-17 | 4875
2200760119 | 2019-08-17 | 3341
2200760120 | 2019-08-19 | 3894
Table A2HXMT and NICER observations of EXO 1846$-$031 analyzed in this work.
As in the case of GX 339$-$4, the listed exposure time is for the LE
instrument of HXMT.

Figure A1.— Left: a typical PDS of GX 339$-$4 from HXMT data (Obs ID:
P0304024028). Right: the HXMT spectrum and residuals to the best-fit model for
the same observation. Data from the LE and ME detector are denoted in black
and red, respectively.

Figure A2.— Left: a typical PDS of GX 339$-$4 from NICER data (Obs ID:
4133010103-7). Right: the NICER spectrum and residuals to the best-fit model
for the same observation.

Figure A3.— Left: a typical PDS of EXO 1846$-$031 from HXMT data (Obs ID:
P021405000105). Right: the HXMT spectrum and residuals to the best-fit model
for the same observation. Same as Fig. A1, data from the LE and ME detector
are denoted in black and red, respectively.

Figure A4.— Left: a typical PDS of EXO 1846$-$031 from NICER data (Obs ID: 2200760106). Right: the NICER spectrum and residuals to the best-fit model for the same observation.  Obs. ID |  $N_{\rm H}$ [$10^{22}$ cm-2] | QPO frequency | $\Gamma$ | $f_{\rm sc}$ |  $\dot{M}$ [$10^{15}$ g s-1] |  $R_{\rm in}$ [$r_{\rm g}$] | $\chi^{2}/\nu$
---|---|---|---|---|---|---|---
P0304024026 | $0.55^{*}$ | $0.276_{-0.025}^{+0.022}$ | $1.748_{-0.025}^{+0.022}$ | $0.40_{-0.21}^{+P}$ | $0.39_{-0.13}^{+1.7}$ | $40_{-13}^{+57}$ | $893.93/1057$
P0304024028 | $0.55^{*}$ | $0.370_{-0.012}^{+0.016}$ | $1.793_{-0.005}^{+0.005}$ | $0.8_{-0.5}^{+P}$ | $0.174_{-0.007}^{+0.007}$ | $25.3_{-1.0}^{+2.2}$ | $857.35/976$
P0304024032 | $0.55^{*}$ | $0.428_{-0.019}^{+0.013}$ | $1.812_{-0.014}^{+0.025}$ | $0.44_{-0.11}^{+P}$ | $0.8_{-0.4}^{+2.9}$ | $55.6_{-7}^{+44}$ | $911.31/1013$
P0304024035 | $0.55^{*}$ | $0.66_{-0.04}^{+0.04}$ | $1.851_{-0.024}^{+0.028}$ | $0.56_{-0.11}^{+0.09}$ | $0.31_{-0.08}^{+0.3}$ | $18.2_{-2.4}^{+13}$ | $930.44/1033$
P0304024036 | $0.55^{*}$ | $0.93_{-0.05}^{+0.05}$ | $1.831_{-0.04}^{+0.022}$ | $0.50_{-0.04}^{+0.04}$ | $0.33_{-0.05}^{+0.05}$ | $15.5_{-2.7}^{+3.0}$ | $948.06/1021$
P0304024038 | $0.55^{*}$ | $2.53_{-0.06}^{+0.06}$ | $2.338_{-0.023}^{+0.021}$ | $0.432_{-0.017}^{+0.016}$ | $0.84_{-0.04}^{+0.03}$ | $15.4_{-0.7}^{+0.6}$ | $959.94/977$
3558011402 | $0.62_{-0.04}^{+0.04}$ | $0.407_{-0.019}^{+0.019}$ | $1.703_{-0.024}^{+0.023}$ | $0.5_{-0.4}^{+0.2}$ | $0.35_{-0.18}^{+0.2}$ | $36.3_{-12}^{+8}$ | $684.37/852$
3558011501 | $0.557_{-0.004}^{+0.004}$ | $0.492_{-0.015}^{+0.018}$ | $1.7186_{-0.0024}^{+0.0022}$ | $0.69_{-0.1}^{+0.11}$ | $0.207_{-0.005}^{+0.005}$ | $21.7_{-1.1}^{+1.1}$ | $587.78/899$
4133010101 | $0.617_{-0.025}^{+0.12}$ | $0.491_{-0.017}^{+0.03}$ | $1.711_{-0.029}^{+0.02}$ | $0.48_{-0.22}^{+0.21}$ | $0.30_{-0.09}^{+0.16}$ | $31.1_{-7}^{+10}$ | $725.14/889$
4133010102 | $0.580_{-0.018}^{+0.04}$ | $0.520_{-0.028}^{+0.026}$ | $1.715_{-0.024}^{+0.019}$ | $0.6_{-0.5}^{+0.2}$ | $0.23_{-0.06}^{+0.04}$ | $23.1_{-1.9}^{+4}$ | $691.04/886$
4133010103-1 | $0.457_{-0.03}^{+0.018}$ | $1.97_{-0.06}^{+0.05}$ | $2.247_{-0.018}^{+0.03}$ | $0.517_{-0.04}^{+0.005}$ | $0.64_{-0.11}^{+0.16}$ | $20_{-3}^{+3}$ | $581.61/709$
4133010103-2 | $0.487_{-0.029}^{+0.021}$ | $1.98_{-0.06}^{+0.04}$ | $2.12_{-0.06}^{+0.05}$ | $0.32_{-0.3}^{+0.11}$ | $0.57_{-0.25}^{+0.25}$ | $18.2_{-5}^{+5}$ | $604.52/719$
4133010103-3 | $0.46_{-0.03}^{+0.03}$ | $2.20_{-0.04}^{+0.04}$ | $2.16_{-0.06}^{+0.05}$ | $0.43_{-0.19}^{+0.07}$ | $0.41_{-0.11}^{+0.08}$ | $13.9_{-2.2}^{+2.1}$ | $594.45/683$
4133010103-4 | $0.489_{-0.027}^{+0.027}$ | $2.54_{-0.04}^{+0.05}$ | $2.26_{-0.07}^{+0.08}$ | $0.38_{-0.25}^{+0.1}$ | $0.39_{-0.13}^{+0.07}$ | $12.3_{-3}^{+3}$ | $609.25/715$
4133010103-5 | $0.481_{-0.023}^{+0.027}$ | $2.69_{-0.05}^{+0.04}$ | $2.30_{-0.12}^{+0.05}$ | $0.45_{-0.15}^{+0.07}$ | $0.46_{-0.13}^{+0.11}$ | $12.9_{-2.7}^{+2.6}$ | $605.14/718$
4133010103-6 | $0.473_{-0.018}^{+0.021}$ | $2.55_{-0.04}^{+0.03}$ | $2.20_{-0.07}^{+0.06}$ | $0.37_{-0.13}^{+0.07}$ | $0.46_{-0.12}^{+0.11}$ | $12.7_{-2.5}^{+2.5}$ | $550.91/763$
4133010103-7 | $0.488_{-0.022}^{+0.023}$ | $2.64_{-0.02}^{+0.03}$ | $2.28_{-0.09}^{+0.06}$ | $0.39_{-0.13}^{+0.08}$ | $0.47_{-0.11}^{+0.13}$ | $13.2_{-2.6}^{+2.8}$ | $641.36/781$
4133010103-8 | $0.495_{-0.021}^{+0.018}$ | $2.91_{-0.04}^{+0.03}$ | $2.35_{-0.08}^{+0.06}$ | $0.44_{-0.11}^{+0.08}$ | $0.51_{-0.11}^{+0.13}$ | $12.7_{-2.4}^{+2.7}$ | $609.30/776$
4133010103-9 | $0.504_{-0.02}^{+0.015}$ | $3.06_{-0.02}^{+0.03}$ | $2.35_{-0.11}^{+0.06}$ | $0.40_{-0.14}^{+0.09}$ | $0.44_{-0.09}^{+0.11}$ | $11.0_{-2.3}^{+2.2}$ | $620.33/788$
4133010104-1 | $0.500_{-0.019}^{+0.020}$ | $3.28_{-0.05}^{+0.05}$ | $2.35_{-0.16}^{+0.09}$ | $0.37_{-0.22}^{+0.16}$ | $0.42_{-0.13}^{+0.14}$ | $10.1_{-2.6}^{+2.7}$ | $645.09/762$
4133010104-2 | $0.502_{-0.021}^{+0.023}$ | $3.20_{-0.04}^{+0.03}$ | $2.39_{-0.09}^{+0.05}$ | $0.45_{-0.10}^{+0.06}$ | $0.51_{-0.12}^{+0.08}$ | $11.9_{-2.5}^{+1.5}$ | $618.63/763$
4133010104-3 | $0.500_{-0.029}^{+0.027}$ | $3.04_{-0.26}^{+0.14}$ | $2.46_{-0.22}^{+0.07}$ | $0.50_{-0.29}^{+0.08}$ | $0.69_{-0.17}^{+0.19}$ | $13.2_{-4}^{+2.8}$ | $547.62/656$
4133010104-4 | $0.507_{-0.029}^{+0.028}$ | $3.04_{-0.26}^{+0.14}$ | $2.34_{-0.22}^{+0.14}$ | $0.36_{-P}^{+0.21}$ | $0.6_{-0.3}^{+0.3}$ | $12_{-5}^{+5}$ | $565.97/656$
4133010104-5 | $0.48_{-0.03}^{+0.03}$ | $3.41_{-0.06}^{+0.07}$ | $2.4_{-0.3}^{+0.1}$ | $0.41_{-P}^{+0.19}$ | $0.34_{-0.17}^{+0.08}$ | $8.5_{-2.8}^{+2.9}$ | $502.43/616$
4133010104-6 | $0.50_{-0.02}^{+0.02}$ | $3.63_{-0.06}^{+0.06}$ | $2.38_{-0.12}^{+0.09}$ | $0.40_{-0.14}^{+0.09}$ | $0.50_{-0.11}^{+0.12}$ | $10.6_{-2.2}^{+2.2}$ | $559.10/776$
4133010104-7 | $0.500_{-0.022}^{+0.019}$ | $3.97_{-0.08}^{+0.10}$ | $2.39_{-0.18}^{+0.11}$ | $0.37_{-0.17}^{+0.1}$ | $0.36_{-0.08}^{+0.09}$ | $8.2_{-1.9}^{+1.8}$ | $559.05/700$
4133010104-8 | $0.534_{-0.024}^{+0.03}$ | $3.99_{-0.06}^{+0.05}$ | $2.52_{-0.28}^{+0.17}$ | $0.40_{-0.4}^{+0.23}$ | $0.34_{-0.14}^{+0.11}$ | $7.7_{-2.9}^{+2.8}$ | $515.87/692$
4133010104-9 | $0.512_{-0.026}^{+0.023}$ | $3.99_{-0.06}^{+0.05}$ | $2.50_{-0.19}^{+0.13}$ | $0.37_{-0.18}^{+0.1}$ | $0.34_{-0.08}^{+0.07}$ | $7.1_{-1.3}^{+1.4}$ | $611.50/685$
4133010104-10 | $0.528_{-0.018}^{+0.019}$ | $4.55_{-0.08}^{+0.08}$ | $2.57_{-0.15}^{+0.11}$ | $0.36_{-0.13}^{+0.1}$ | $0.48_{-0.09}^{+0.09}$ | $7.4_{-1.1}^{+1.5}$ | $591.18/774$
4133010104-11 | $0.500_{-0.018}^{+0.022}$ | $4.34_{-0.04}^{+0.04}$ | $2.42_{-0.21}^{+0.14}$ | $0.30_{-0.17}^{+0.12}$ | $0.38_{-0.07}^{+0.06}$ | $6.8_{-1.3}^{+1.4}$ | $544.57/701$
4133010104-12 | $0.526_{-0.022}^{+0.025}$ | $4.81_{-0.07}^{+0.07}$ | $2.62_{-0.2}^{+0.13}$ | $0.37_{-0.13}^{+0.11}$ | $0.42_{-0.09}^{+0.08}$ | $6.8_{-1.4}^{+1.4}$ | $536.68/698$
4133010104-13 | $0.528_{-0.024}^{+0.016}$ | $5.37_{-0.19}^{+0.23}$ | $2.70_{-0.29}^{+0.14}$ | $0.39_{-0.2}^{+0.12}$ | $0.54_{-0.1}^{+0.09}$ | $7.7_{-1.7}^{+1.7}$ | $564.02/718$
Table A3Best-fit Parameters of GX 339$-$4\. The symbol $*$ means that the parameter is fixed at its best-fit value because it cannot be constrained. All uncertainties are quoted at the 90% confidence level.  Obs. ID |  $N_{\rm H}$ [$10^{22}$ cm-2] | QPO frequency | $\Gamma$ | $f_{\rm sc}$ |  $\dot{M}$ [$10^{15}$ g s-1] |  $R_{\rm in}$ [$r_{\rm g}$] | $\chi^{2}/\nu$
---|---|---|---|---|---|---|---
P021405000101 | $6.73_{-0.19}^{+0.24}$ | $0.69_{-0.01}^{+0.01}$ | $1.81_{-0.04}^{+0.07}$ | $0.09_{-0.04}^{+0.08}$ | $8.0_{-2.2}^{+1.1}$ | $99_{-58}^{+P}$ | $1017.35/1051$
P021405000102 | $7.4_{-0.7}^{+0.8}$ | $0.85_{-0.01}^{+0.01}$ | $1.82_{-0.04}^{+0.04}$ | $0.32_{-0.11}^{+0.21}$ | $0.9_{-0.5}^{+0.9}$ | $25_{-10}^{+20}$ | $988.95/1022$
P021405000103 | $8.7_{-1.1}^{+1.5}$ | $0.99_{-0.01}^{+0.01}$ | $1.95_{-0.05}^{+0.04}$ | $0.13_{-0.03}^{+0.05}$ | $5.2_{-1.8}^{+2.1}$ | $55_{-16}^{+41}$ | $1026.72/1022$
P021405000104 | $7.6_{-0.5}^{+0.6}$ | $1.13_{-0.02}^{+0.02}$ | $2.02_{-0.06}^{+0.06}$ | $0.17_{-0.05}^{+0.16}$ | $6.4_{-4}^{+2.5}$ | $83_{-50}^{+17}$ | $1030.04/1019$
P021405000105 | $8.0_{-0.5}^{+P}$ | $1.25_{-0.01}^{+0.01}$ | $1.964_{-0.024}^{+0.05}$ | $0.20_{-0.03}^{+0.1}$ | $2.8_{-1.3}^{+2.0}$ | $41_{-12}^{+15}$ | $1073.18/1139$
P021405000106 | $6.13_{-0.17}^{+0.3}$ | $1.34_{-0.01}^{+0.01}$ | $2.029_{-0.017}^{+0.06}$ | $0.989_{0.021}^{+P}$ | $0.29_{-0.04}^{+0.09}$ | $9.94_{-0.16}^{+4}$ | $994.17/922$
P021405000107 | $6.59_{-0.5}^{+0.27}$ | $1.60_{-0.05}^{+0.05}$ | $2.05_{-0.07}^{+0.07}$ | $0.51_{-0.29}^{+P}$ | $1.0_{-0.3}^{+1}$ | $29_{-8}^{+20}$ | $954.56/967$
P021405000201 | $6.35_{-0.23}^{+0.5}$ | $2.26_{-0.06}^{+0.06}$ | $2.245_{-0.028}^{+0.05}$ | $0.28_{-0.08}^{+0.4}$ | $8.2_{-6}^{+8}$ | $100_{-56}^{+P}$ | $947.89/1006$
P021405000301 | $5.5_{-0.5}^{+0.4}$ | $3.14_{-0.03}^{+0.03}$ | $2.41_{-0.07}^{+0.09}$ | $0.55_{-0.06}^{+0.12}$ | $0.39_{-0.15}^{+0.3}$ | $6.0_{-2.3}^{+6}$ | $1043.07/1056$
P021405000302 | $6.12_{-0.26}^{+0.4}$ | $3.18_{-0.07}^{+0.07}$ | $2.43_{-0.07}^{+0.07}$ | $0.50_{-0.05}^{+0.05}$ | $0.36_{-0.1}^{+0.15}$ | $5.5_{-1.2}^{+3}$ | $1007.96/1113$
P021405000303 | $6.2_{-0.7}^{+1.2}$ | $3.73_{-0.05}^{+0.05}$ | $2.11_{-0.11}^{+0.18}$ | $0.004_{-P}^{+0.5}$ | $0.09_{-0.06}^{+0.9}$ | $3.5_{-P}^{+14.1}$ | $889.98/912$
P021405000401 | $5.1_{-0.6}^{+0.6}$ | $3.26_{-0.12}^{+0.12}$ | $2.22_{-0.06}^{+0.10}$ | $0.47_{-0.06}^{+0.09}$ | $0.29_{-0.09}^{+0.19}$ | $4.4_{-1.4}^{+2.5}$ | $835.45/909$
P021405000502 | $5.6_{-0.6}^{+0.3}$ | $4.19_{-0.15}^{+0.15}$ | $2.43_{-0.04}^{+0.09}$ | $0.420_{-0.05}^{+0.018}$ | $0.50_{-0.17}^{+0.18}$ | $6.0_{-1.6}^{+0.7}$ | $993.15/989$
P021405000503 | $6.4_{-0.5}^{+0.7}$ | $3.90_{-0.09}^{+0.09}$ | $2.54_{-0.11}^{+0.1}$ | $0.35_{-0.05}^{+0.04}$ | $0.62_{-0.24}^{+0.17}$ | $6.8_{-2.2}^{+1.1}$ | $937.30/953$
P021405000601 | $6.01_{-0.4}^{+0.27}$ | $4.48_{-0.05}^{+0.05}$ | $2.57_{-0.08}^{+0.06}$ | $0.403_{-0.03}^{+0.027}$ | $0.75_{-0.19}^{+0.12}$ | $7.4_{-1.6}^{+0.9}$ | $1047.54/1021$
P021405000701 | $6.9_{-0.5}^{+0.3}$ | $6.69_{-0.33}^{+0.33}$ | $2.44_{-0.09}^{+0.05}$ | $0.123_{--0.067}^{+0.07}$ | $0.84_{-0.25}^{+0.4}$ | $7.1_{-1.1}^{+1}$ | $1059.74/1136$
P021405000702 | $6.24_{-0.29}^{+0.3}$ | $6.31_{-0.14}^{+0.14}$ | $2.42_{-0.06}^{+0.1}$ | $0.209_{-0.022}^{+0.025}$ | $1.16_{-0.13}^{+0.14}$ | $7.4_{-0.7}^{+0.7}$ | $1062.41/1101$
2200760101 | $6.60_{-0.24}^{+0.23}$ | $0.262_{-0.012}^{+0.008}$ | $1.609_{-0.04}^{+0.017}$ | $0.16_{-0.06}^{+0.15}$ | $0.57_{-0.08}^{+0.27}$ | $33_{-14}^{+25}$ | $799.14/899$
2200760102 | $7.2_{-0.3}^{+0.4}$ | $0.412_{-0.015}^{+0.017}$ | $1.81_{-0.06}^{+0.11}$ | $0.053_{-0.007}^{+0.016}$ | $3.1_{-1}^{+1.0}$ | $90_{-28}^{+6}$ | $704.88/672$
2200760103 | $6.83_{-0.2}^{+0.12}$ | $1.007_{-0.028}^{+0.029}$ | $1.972_{-0.026}^{+0.04}$ | $0.257_{-0.04}^{+0.026}$ | $0.71_{-0.11}^{+0.2}$ | $33_{-3}^{+4}$ | $861.35/853$
2200760104 | $6.9_{-0.4}^{+0.3}$ | $1.37_{-0.06}^{+0.05}$ | $1.877_{-0.05}^{+0.027}$ | $0.10_{-P}^{+0.23}$ | $0.9_{-0.6}^{+0.7}$ | $32_{-9}^{+11}$ | $841.57/849$
2200760105 | $5.66_{-0.11}^{+0.11}$ | $2.43_{-0.06}^{+0.06}$ | $1.85_{-0.11}^{+0.12}$ | $0.39_{-0.07}^{+0.05}$ | $0.020_{-0.004}^{+0.004}$ | $1.9_{-P}^{+0.4}$ | $728.60/759$
2200760106 | $5.90_{-0.08}^{+0.09}$ | $3.25_{-0.04}^{+0.04}$ | $2.15_{-0.11}^{+0.16}$ | $0.46_{-0.13}^{+0.17}$ | $0.04_{-0.02}^{+0.03}$ | $2.8_{-1.3}^{+1.2}$ | $780.47/863$
2200760107 | $6.20_{-0.1}^{+0.11}$ | $3.31_{-0.05}^{+0.05}$ | $2.367_{-0.028}^{+0.03}$ | $0.6^{*}$ | $0.07_{-0.03}^{+0.07}$ | $4.9_{-1.6}^{+2.9}$ | $755.51/798$
2200760108 | $5.99_{-0.12}^{+0.17}$ | $3.69_{-0.06}^{+0.06}$ | $2.50_{-0.04}^{+0.07}$ | $0.6^{*}$ | $0.179_{-0.07}^{+0.026}$ | $6.3_{-1.7}^{+0.7}$ | $709.60/798$
2200760109 | $6.32_{-0.11}^{+0.09}$ | $6.93_{-0.3}^{+0.28}$ | $2.20_{-0.6}^{+0.18}$ | $0.12_{-P}^{+0.11}$ | $0.87_{-0.11}^{+0.07}$ | $6.3_{-0.6}^{+0.6}$ | $755.68/898$
2200760110 | $6.19_{-0.04}^{+0.09}$ | $6.46_{-0.28}^{+0.5}$ | $1.80_{-0.28}^{+0.6}$ | $0.04_{-0.04}^{+0.11}$ | $0.95_{-0.1}^{+0.06}$ | $6.3_{-0.4}^{+0.6}$ | $750.97/899$
2200760112 | $6.35_{-0.06}^{+0.08}$ | $6.16_{-0.34}^{+0.5}$ | $2.40_{-0.23}^{+0.12}$ | $0.22_{-0.06}^{+0.05}$ | $0.86_{-0.1}^{+0.08}$ | $7.3_{-0.4}^{+0.6}$ | $784.13/889$
2200760113 | $6.56_{-0.15}^{+0.11}$ | $6.6_{-0.5}^{+1.2}$ | $2.50_{-0.24}^{+0.14}$ | $0.17_{-P}^{+0.07}$ | $0.12_{-0.05}^{+0.04}$ | $3.9_{-0.7}^{+0.5}$ | $782.20/858$
2200760114 | $6.28_{-0.03}^{+0.15}$ | $5.93_{-0.20}^{+0.5}$ | $2.17_{-0.17}^{+0.27}$ | $0.08_{-P}^{+0.11}$ | $0.40_{-0.08}^{+0.06}$ | $5.6_{-0.5}^{+0.6}$ | $766.57/899$
2200760115 | $6.35_{-0.14}^{+0.11}$ | $5.76_{-0.23}^{+0.26}$ | $2.33_{-0.19}^{+0.4}$ | $0.12_{-P}^{+0.25}$ | $0.13_{-0.06}^{+0.04}$ | $4.0_{-0.5}^{+1.1}$ | $715.46/899$
2200760116 | $6.19_{-0.1}^{+0.12}$ | $5.35_{-0.18}^{+0.16}$ | $2.27_{-0.22}^{+0.4}$ | $0.18_{-0.11}^{+0.3}$ | $0.23_{-0.06}^{+0.05}$ | $5.2_{-0.5}^{+2.8}$ | $812.60/899$
2200760117 | $6.31_{-0.14}^{+0.12}$ | $5.27_{-0.24}^{+0.24}$ | $2.25_{-0.23}^{+0.5}$ | $0.09_{-P}^{+0.3}$ | $0.21_{-0.06}^{+0.07}$ | $5.0_{-0.5}^{+2.5}$ | $723.59/899$
2200760118 | $6.54_{-0.09}^{+0.11}$ | $5.77_{-0.23}^{+0.26}$ | $2.81_{-0.13}^{+0.11}$ | $0.50_{-0.1}^{+0.15}$ | $0.40_{-0.07}^{+0.12}$ | $7.8_{-1.8}^{+2.4}$ | $766.21/899$
2200760119 | $6.13_{-0.08}^{+0.05}$ | $5.18_{-0.12}^{+0.15}$ | $2.70_{-0.18}^{+0.4}$ | $0.13_{-0.06}^{+0.11}$ | $0.281_{-0.029}^{+0.04}$ | $4.9_{-0.4}^{+0.7}$ | $755.06/856$
2200760120 | $6.41_{-0.3}^{+0.26}$ | $5.34_{-0.15}^{+0.22}$ | $2.96_{-0.4}^{+0.28}$ | $0.22_{-0.07}^{+0.14}$ | $0.148_{-0.029}^{+0.027}$ | $4.2_{-0.7}^{+0.5}$ | $736.47/837$
Table A4Best-fit Parameters of EXO 1846$-$031\. The symbol $*$ means that the
parameter is fixed at its best-fit value because it cannot be constrained. All
uncertainties are quoted at the 90% confidence level.

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