#  Conditional uncorrelation equals independence

Dawid Tarłowski<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID 0000-0002-6824-4568 Faculty of Mathematics and Computer Science,
Jagiellonian University, Łojasiewicza 6, Kraków, 30-348, Poland

###### Abstract

It is well known that the independent random variables $X$ and $Y$ are
uncorrelated in the sense $E[XY]=E[X]\cdot E[Y]$ and that the implication may
be reversed in very specific cases only. This paper proves that under general
assumptions the conditional uncorrelation of random variables, where the
conditioning takes place over the suitable class of test sets, is equivalent
to the independence. It is also shown that the mutual independence of
$X_{1},\dots,X_{n}$ is equivalent to the fact that any conditional correlation
matrix equals to the identity matrix.

###### keywords:

correlation, independence, linear dependence, conditional moments

##  1 Introduction

It is well known that if $X$ and $Y$ are two independent integrable real-
valued random variables then they are uncorrelated in the sense
$E[XY]=E[X]\cdot E[Y]$ and that this implication cannot be reversed unless
very specific assumptions on $X$ and $Y$ are satisfied, David (2009). More,
even the uncorrelation of all the higher moments of $X$ and $Y$ is not enough
to force the independence, De Paula (2008). This paper proves that under
general assumptions the conditional uncorrelation of random variables, where
the conditioning takes place over the class $\mathcal{K}=\\{[a,b]\colon
a<b\\}$ or $\mathcal{K}=\\{[t,\infty)\colon t\in\mathbb{R}\\}$, is equivalent
to the independence. This solves Problem 6327 from Chernoff, Móri et al.
(1981) which, according to my knowledge, is solved only in the case where the
vector $(X,Y)$ has a joint probability density
$f\colon\mathbb{R}^{2}\to\mathbb{R}^{+}$, see Jaworski, Jelito, Pitera (2024)
which applies copula functions to the problem (under some additional technical
assumptions on $f$). Theorem 1 from the next section shows that the
independence of $X$ and $Y$ is equivalent to the conditional uncorrelation for
all types of random variables $X$ and $Y$ and any dependencies between them.
The direct conclusion is that if $X$ and $Y$ are not independent then there is
some rectangle $[a,b]\times[c,d]$ conditioning on which we have the nonzero
Pearson correlation coefficient. The motivation for such study comes from the
developing field of conditional statistics which have proved to provide useful
applications and nontrivial theoretical problems, see for instance Hebda-
Sobkowicz et al.(2020), Jaworski-Pitera (2016), Jaworski-Pitera (2020),
Jelito-Pitera (2021), Jaworski et al. (2024), Pitera et al. (2022), Navarro et
al. (1998). For the previous papers on the conditional correlations one can
see, for instance, Akeman et al. (1984), Baba et al. (2004), Lawrance (1976).
The general methodology we will present here is, roughly speaking, about using
the properties of the Radon-Nikodym derivative to reduce the problem of
conditional uncorrelation to the one- dimensional conditioning which is well
understood, Navarro et al. (1998). The presented approach allows to prove
other general statements about the independence and Theorem 2 shows that the
mutual independence of the marginals of the random vector
$(X_{1},\dots,X_{n})$ is equivalent to the fact that any conditional
correlation matrix of $(X_{1},\dots,X_{n})$ equals to the identity matrix.
Section 2 introduces the basic definitions and presents the main results.
Section 3 presents the necessary theory and proves Theorems 1 and 2.

##  2 Conditional uncorrelation and the main results

Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space. The space
$\mathbb{R}^{n}$ is naturally equipped with the sigma-algebra of Borel sets
$\mathcal{B}(\mathbb{R}^{n})$ and $Z\colon\Omega\to\mathbb{R}^{n}$ is a random
vector iff $Z^{-1}(C)\in\Sigma$ for any $C\in\mathcal{B}(\mathbb{R}^{n})$. The
probability distribution of the random vector $Z$ is denoted by
$\mathbb{P}_{Z}$ and given by $\mathbb{P}_{Z}(C)=\mathbb{P}[Z\in C]$,
$C\in\mathcal{B}(\mathbb{R}^{n})$. For any $A\in\Sigma$ with $\mathbb{P}[A]>0$
the $\mathbb{P}[\cdot|A]$ will denote the conditional probability measure:

$\mathbb{P}[C|A]=\frac{\mathbb{P}[C\cap A]}{\mathbb{P}[A]},\ C\in\Sigma.$

Let $X\colon\Omega\to\mathbb{R}$ be a random variable and let $1_{A}$ denote
the characteristic function of $A$, i.e. $1_{A}(x)=1$ for $x\in A$ and
$1_{A}(x)=0$ for $x\notin A$. If the expectation
$E[X1_{A}]=\int\limits_{A}Xd\mathbb{P}$ exists, the conditional expectation
$E[X|A]$ is defined by $E[X|A]=\int\limits_{\Omega}Xd\mathbb{P}{(\cdot|A)}$
and satisfies $E[X|A]=\frac{E[X1_{A}]}{\mathbb{P}[A]}.$ If $A=\\{X\in C\\}$,
where $C\subset\mathbb{R}$ is a Borel set with
$\mathbb{P}_{X}(C)=\mathbb{P}[X\in C]>0$, then

$E[X|A]=E[X|X\in C]=\frac{E[X1_{C}(X)]}{\mathbb{P}_{X}(C)}.$

If $X$ and $Y$ are non-constant random variables with $E[X^{2}]<\infty$ and
$E[Y^{2}]<\infty$ then both the covariance and the (Pearson’s) correlation are
well defined:

$cov(X,Y)=E[XY]-E[X]\cdot E[Y]\mbox{ \ and \
}cor(X,Y)=\frac{cov(X,Y)}{sd(X)\cdot sd(Y)},$

where $sd(Z)=\sqrt{E[Z^{2}]-(E[Z])^{2}}$ denotes the standard deviation of
$Z$.

If $A,B\subset\mathbb{R}$ are Borel sets with $\mathbb{P}_{(X,Y)}(A\times
B)=\mathbb{P}[X\in A,Y\in B]>0$ then one can consider the corresponding
conditional covariance and the conditional correlation, i.e. for $U=\\{X\in
A,Y\in B\\}$ one may calculate those coefficients with respect to the
conditional probability $\mathbb{P}[\cdot|U]$. The conditional covariance
$cov_{(A,B)}(X,Y)$ is thus given by

$cov_{(A,B)}(X,Y)=E[XY|X\in A,Y\in B]-E[X|X\in A,Y\in B]\cdot E[Y|X\in A,Y\in
B],$

assuming that the above expectations exists, and the conditional correlation
$cor_{(A,B)}(X,Y)$ satisfies

$cor_{(A,B)}(X,Y)=\frac{cov_{(A,B)}(X,Y)}{sd_{(A,B)}(X)\cdot sd_{(A,B)}(Y)},$

where $sd_{(A,B)}(\cdot)$ is the standard deviation with respect to the
conditional probability $\mathbb{P}[\cdot|U]=\mathbb{P}[\cdot|X\in A,Y\in B]$.
Those conditional statistics are easy to estimate in practice: given the
sample of pairs of observations $(x_{i},y_{i})_{i=1}^{n}$, one just calculates
the standard covariance and correlation coefficients on the subsample of
$(x_{i},y_{i})_{i=1}^{n}$ determined by

$\\{i\in\\{1,\dots,n\\}\colon(x_{i},y_{i})\in A\times B\\}.$

The mathematical justification of this procedure follows directly from the
properties of the conditional probability: if the random vectors
$Z_{i}=(X_{i},Y_{i})$, $i=1,2,\dots$, are i.i.d. and $\mathbb{P}[Z_{1}\in
A\times B]>0$ then for

$N_{1}:=\min\\{n\colon Z_{n}\in A\times B\\}\mbox{ and
}N_{k+1}:=\min\\{n>N_{k}\colon Z_{n}\in A\times B\\}$

the random variables $Z_{N_{1}},Z_{N_{2}},\dots$ are independent and have
probability distribution equal to the conditional distribution
$\mathbb{P}_{Z_{1}}[\cdot|Z_{1}\in A\times B]$, i.e.

$\mathbb{P}[Z_{N_{k}}\in D]=\mathbb{P}[Z_{1}\in D|Z_{1}\in A\times B],\
D\in\mathcal{B}(\mathbb{R}^{2}),k\in\mathbb{N}.$

If the $X$ and $Y$ are not square - integrable (in practice, for instance, the
data may exhibit heavy tails), the tests based on the classical covariance
fail to work properly. At the same time, the conditional covariance is well
defined for any bounded rectangle $A\times B$ and one may consider the
conditional covariance instead of the classical one, see Jaworski et al.
(2024) for other applications.

If $X$ and $Y$ are independent in the sense $\mathbb{P}_{(X,Y)}(D_{1}\times
D_{2})=\mathbb{P}_{X}(D_{1})\cdot\mathbb{P}_{Y}(D_{2})$,
$D_{1},D_{2}\in\mathcal{B}(\mathbb{R})$, then $cov(X,Y)=0$ and
$cov_{(A,B)}(X,Y)=0$ for any admissible test set $A\times B$. The theorem
below states that this implication may be reversed.

######  Theorem 1.

Assume that $\mathcal{K}=\\{[a,b]\colon a<b\\}$. For any random variables $X$
and $Y$, the following conditions are equivalent:

  1. 1)

$X$ and $Y$ are independent,

  2. 2)

for any $A,B\in\mathcal{K}$ with $\mathbb{P}_{(X,Y)(A\times B)}>0$,

$cov_{(A,B)}(X,Y)=0.$

Under additional assumption: $E|X|<\infty$ and $E|Y|<\infty$, we may replace
the bounded intervals with the family $\mathcal{K}=\\{[t,\infty)\colon\
t\in\mathbb{R}\\}$ ( or with $\\{(-\infty,t]\colon\ t\in\mathbb{R}\\}$).

In conclusion, the square-integrable random variables $X$ and $Y$ are
independent iff they are uncorrelated conditioning on the tails $\\{X\geq
t\\}$ and $\\{Y\geq s\\}$, i.e. for any $t,s$ with $\mathbb{P}[X\geq t,Y\geq
s]>0$ the following function

$H(t,s)=cor_{([t,\infty),[s,\infty))}(X,Y)$ (1)

is constant and equal to zero.

Below, just for illustration, we consider the textbook example of $(X,Y)$ with
dependent but uncorrelated normal-distributed marginals, Broffitt (1986).

######  Example 1.

Let $X$ and $Z$ be independent and such that $\mathbb{P}_{X}=N(0,1)$ and
$\mathbb{P}[Z=1]=\mathbb{P}[Z=-1]=\frac{1}{2}$. Define $Y=Z\cdot X$. It is
easy to see that $\mathbb{P}_{Y}=N(0,1)$, $X$ and $Y$ are not independent and
at the same time they are uncorrelated:$E[YX]=E[ZX^{2}]=E[Z]\cdot
E[X^{2}]=0=E[Y]\cdot E[X].$ We will take a look on the behaviour of the
function

$h(t)=H[t,t]=cor_{((-\infty,t],(-\infty,t])}(X,Y).$

This function may be calculated analytically but instead we draw the sample
$(x_{i},y_{i})_{i=1}^{n}$ of size $n=10^{3}$ and calculate the conditional
correlations numerically, see Figure 1 below. The conditional correlation
coefficient increases until it reaches value 1 at $t=0$ as the condition
$\\{X\geq 0,Y\geq 0\\}$ implies full linear dependence $X=Y$.

Figure 1: estimated conditional correlation of $X,Y$ given $\\{X\geq t,Y\geq
t\\}$ , the sample size $n=10^{3}$

The proof of Theorem 1 may be found in the next section. It is based on the
properties of the Radon-Nikodym derivative and the conditional moments from
Navarro et al. (1998). The same technique will allow us to prove Theorem 2
which characterizes the mutual independence of $X_{1},\dots,X_{n}$ by the
conditional correlation matrix $\Sigma_{U}\in M(n,n)$ given by

$\Sigma_{U}[i,j]=cor_{U}(X_{i},X_{j}),\ i,j=1,\dots,n,$

where $U=\\{X_{1}\in A_{1},\dots,X_{n}\in A_{n}\\}$ satisfies
$\mathbb{P}[U]>0$ and $A_{i}\in\mathcal{K}$, $i=1,\dots,n$, and the
correlation $cor_{U}(X_{i},X_{j})$ is calculated with respect to the
conditional probability $\mathbb{P}[\cdot|U]$. If for some
$j\in\\{1,\dots,n\\}$ and $U=\\{X_{1}\in A_{1},\dots,X_{n}\in A_{n}\\}$ with
$\mathbb{P}[U]>0$ we have $sd_{U}(X_{j})=0$ then we shortly define

$cor_{U}(X_{i},X_{j}):=1_{\\{j\\}}(i)=\delta_{i,j}\ (\mbox{ Kronecker delta})$

which avoids unnecessary techniqualities.

######  Theorem 2.

Let $X_{1},\dots,X_{n}$ be random variables and $\mathcal{K}=\\{[a,b]\colon
a<b\\}$. The following conditions are equivalent:

  1. 1.

random variables $X_{1},\dots,X_{n}$ are mutually independent,

  2. 2.

for any $U=\\{X_{1}\in A_{1},\dots,X_{n}\in A_{n}\\}$ with
$A_{i}\in\mathcal{K}$ and $\mathbb{P}[U]>0$ the conditional correlation matrix
equals to the identity matrix

$\Sigma_{U}=I_{n},$

(in case $sd_{U}(X_{j})=0$ let $cor_{U}(X_{i},X_{j}):=\delta_{i,j}$).

If the second moments are finite $E|X_{i}|^{2}<\infty$, $i=1,\dots,n$, the
bounded intervals may be replaced with $\mathcal{K}=\\{[t,\infty)\colon
t\in\mathbb{R}\\}$ (or with $\mathcal{K}=\\{(-\infty,t]\colon
t\in\mathbb{R}\\}$).

We end this section with the following example.

######  Example 2.

Consider the following linear model

$Y=\alpha_{1}\cdot X_{1}+\dots+\alpha_{n}X_{n}+\varepsilon,$

where $(\alpha_{1},\dots,\alpha_{n})\in\mathbb{R}^{n}$ is a vector and
$(X_{1},\dots,X_{n})\in\mathbb{R}^{n}$ is a random vector. By Theorem 1, the
$\varepsilon$ is independent of the predictor $\hat{Y}:=\alpha_{1}\cdot
X_{1}+\dots+\alpha_{n}X_{n}$ iff for any admissible $t,s$ we have

$cor_{[t,\infty,),[s,\infty)}(\varepsilon,\hat{Y})=0.$

By Theorem 2, the variables $X_{1},\dots,X_{n}$ are mutually independent iff
the conditional correlation matrix equals to the identity matrix:
$\Sigma_{U}=I_{n}$, given any admissible condition $U=\\{X_{1}\geq
t_{1},\dots,X_{n}\geq t_{n}\\}$

##  3 Conditional uncorrelation equals independence

Before we start proving theorems 1 and 2 we will need to introduce two lemmas
and one observation. Let us start with the following lemma which follows
directly from the results of Navarro et al. (1998).

######  Lemma 3.

  1. 1.

For any random variable $X$, the function

$(a,b)\to E[X|X\in[a,b]]$ (2)

uniquely determines the probability distribution $\mathbb{P}_{X}$ (the domain
of the function is the set of pairs $a<b$ with $\mathbb{P}_{X}([a,b])>0$).

  2. 2.

For any random variable $X$ with $E|X|<\infty$, the function

$t\to E[X|X\geq t]$ (3)

uniquely determines the probability distribution $\mathbb{P}_{X}$ (the domain
of the function is the set of $t\in\mathbb{R}$ with $\mathbb{P}[X\geq t]>0$.)

Let $\mathcal{M}^{1}(\mathbb{R})$ denote the set of Borel probability measures
on $\mathbb{R}$. In the whole paper we assume that

$\mathcal{K}=\\{[a,b]\colon a<b\\}\mbox{ or }\mathcal{K}=\\{[t,\infty)\colon
t\in\mathbb{R}\\}$

although other types of intervals could be considered as well. For any
$P\in\mathcal{M}^{1}(\mathbb{R})$ let

$\mathcal{K}^{+}_{P}=\\{A\in\mathcal{K}\colon P(A)>0\\}.$

Paper Navarro et al. (1998) shows how the function

$\mathcal{K}^{+}_{P}\ni A\to\frac{\int_{A}xdP(x)}{P(A)}\in\mathbb{R}$

(uniquely) determines the distribution function $F_{P}(t)=P((-\infty,t])$ (in
case $\mathcal{K}=\\{[t,\infty)\colon t\in\mathbb{R}\\}$ it is assumed that
$P$ has a first moment: $\int\limits_{A}|x|dP(x)<\infty$ ). Observation 4 is a
consequence of Lemma 3.

######  Observation 4.

If $P_{1}$ and $P_{2}$ are two probability measures (with finite first moments
in case $\mathcal{K}=\\{[t,\infty)\colon t\in\mathbb{R}\\}$) and
$\mathcal{K}^{+}_{P_{2}}\subset\mathcal{K}^{+}_{P_{1}}$, then conditioning

$\frac{\int_{A}xdP_{2}(x)}{P_{2}(A)}=\frac{\int_{A}xdP_{1}(x)}{P_{1}(A)},\ \ \
A\in\mathcal{K}^{+}_{P_{2}}$ (4)

forces $\mathcal{K}^{+}_{P_{1}}=\mathcal{K}^{+}_{P_{2}}$ and, by Lemma 3,
$P_{2}(A)=P_{1}(A)$ for any $A\in\mathcal{B}(\mathbb{R})$.

###### Proof.

Let us assume that $X_{1}$ and $X_{2}$ are random variables with
$\mathbb{P}_{X_{i}}=P_{i}$, $i=1,2$, so we may use the convenient notation
$E[X_{i}|X_{i}\in A]=\frac{\int_{A}xP_{i}(dx)}{P_{i}(A)}$, $i=1,2$.

1. Consider the case $\mathcal{K}=\\{[a,b]\colon a<b\\}$ which does not assume the existence of the first moments. Assume for a contradiction that for some $c<d$ we have

$\mathbb{P}_{X_{1}}([c,d])>0\mbox{ and }\mathbb{P}_{X_{2}}([c,d])=0.$

We have $\mathbb{P}_{X_{2}}(\mathbb{R}\setminus[c,d])=1$ and without loss of
generality we assume that $\mathbb{P}_{X_{2}}((d,+\infty))>0$. Let

$x:=\inf\\{y>d\colon\mathbb{P}_{X_{2}}([c,y])>0\\}.$ (5)

For any $h>0$ we have $\mathbb{P}[X_{2}\in[c,x+h]]>0$ and, by (4),

$E[X_{1}|X_{1}\in[c,x+h]\ ]=E[X_{2}|X_{2}\in[c,x+h]\ ].$

From the assumptions $\mathbb{P}[X_{2}\in[c,x)]=0$, and thus
$\lim\limits_{h\to 0^{+}}E[X_{2}|X_{2}\in[c,x+h]\ ]=x$. Hence

$E[X_{1}|X_{1}\in[c,x]\ ]=\lim\limits_{h\to 0^{+}}E[X_{1}|X_{1}\in[c,x+h]\
]=\lim\limits_{h\to 0^{+}}E[X_{2}|X_{2}\in[c,x+h]\ ]=x.$

As $E[X_{1}|X_{1}\in[c,x]\ ]=x$, we have $\mathbb{P}[X_{1}\in[c,x)\ ]=0$. This
forces $x=d$ and

$\mathbb{P}[X_{1}\in[c,d]]=\mathbb{P}[X_{1}=d]>0.$

It remains to show that $\mathbb{P}_{X_{1}}(\\{d\\})>0$ contradicts
$\mathbb{P}_{X_{2}}(\\{d\\})\leq\mathbb{P}_{X_{2}}([c,d])=0$. By definition of
$d=x$ given by (5), and by $\mathbb{P}_{X_{2}}(\\{d\\})=0$, we have
$\mathbb{P}_{X_{2}}([d+h,d+1])>0$ for any small $h>0$. We thus may write
(recall that
$\mathcal{K}^{+}_{\mathbb{P}_{X_{2}}}\subset\mathcal{K}^{+}_{\mathbb{P}_{X_{1}}}$)
:

$E[X_{i}|X_{i}\in(d,d+1]\ ]=\lim\limits_{h\to 0^{+}}E[X_{i}|X_{i}\in[d+h,d+1]\
],\ i=1,2.$

Hence, by (4),

$E[X_{1}|X_{1}\in(d,d+1]\ ]=E[X_{2}|X_{2}\in(d,d+1]\ ].$ (6)

Let

$M:=E[X_{1}|X_{1}\in[d,d+1]\ ]=E[X_{2}|X_{2}\in[d,d+1]\ ].$

We have, by $\mathbb{P}[X_{2}=d]=0$, by $\eqref{exd}$ and next by
$\mathbb{P}[X_{1}=d]>0$,

$M=E[X_{2}|X_{2}\in[d,d+1]\ ]=E[X_{2}|X_{2}\in(d,d+1]\
]=E[X_{1}|X_{1}\in(d,d+1]\ ]>E[X_{1}|X_{1}\in[d,d+1]\ ]=M,$

a contradiction.
2. The proof of the case $\mathcal{K}=\\{[t,\infty)\colon t\in\mathbb{R}\\}$ (with $E[|X_{i}|]<\infty$, $i=1,2,$) is not very different than the above proof: first we show that for

$x_{i}:=\inf\\{y\colon\mathbb{P}[X_{i}\in[y,+\infty)\ ]>0\\},i=1,2,$

the conditioning (4) (and the equality $E[X_{1}]=E[X_{2}]$) force
$x_{1}=x_{2}$, and next it remains to show that the situation
$\mathbb{P}[X_{2}=x_{2}]=0$ and $\mathbb{P}[X_{1}=x_{1}]>0$ would lead to a
contradiction.

∎

Now we will introduce some notation and definitions from measure theory, see
Dudley (2018) or Rudin (1986) for more. Let $M^{1}(\Omega,\Sigma)$ denote the
set of probability measures on the measurable space $(\Omega,\Sigma)$ and let
$\mathcal{M}^{1}(\mathbb{R})$ denote the set of Borel probability measures on
$\mathbb{R}$. We will say that $Q$ is a nonnegative measure on
$(\Omega,\Sigma)$ if $Q$ is of the form $Q=c\cdot P$, where $c\geq 0$ and
$P\in M^{1}(\Omega,\Sigma)$. We will say that $Q$ is a signed measure if it is
of the form $Q=c_{1}\cdot P_{1}-c_{2}\cdot P_{2}$, where $c_{1},c_{2}\geq 0$
and $P_{1},P_{2}\in M^{1}(\Omega,\Sigma)$. We will say that a signed measure
$Q$ is absolutely continuous with respect to $P\in M^{1}(\Omega,\Sigma)$,
denoted by $Q<P$, if for any $A\in\Sigma$ with $P(A)=0$ we have $Q(A)=0$. For
example, if $X$ is a random variable with
$E[X]=\int\limits_{\Omega}|X|d\mathbb{P}<\infty$ then $Q$ given by

$Q(A)=E[1_{A}(X)X]=\int_{A}xd\mathbb{P}_{X}(x),\ A\in\mathcal{B}(\mathbb{R}),$

is a signed measure on the measurable space
$(\mathbb{R},\mathcal{B}(\mathbb{R}))$ with $Q<\mathbb{P}_{X}$. A Borel
measurable function $h\colon\Omega\to\mathbb{R}$ is Radon-Nikodym derivative
of a signed measure $Q$ with respect to a nonnegative measure $P$, denoted by
$\frac{dQ}{dP}(x)=h(x)$, if we have

$Q(A)=\int\limits_{A}h(x)P(dx),\ A\in\Sigma.$

By Radon-Nikodym theorem, for any signed measure $Q$ and a nonnegative measure
$P$, the Radon-Nikodym derivative $\frac{dQ}{dP}$ exists if and only if $Q<P$.
A useful property is that $Q_{2}<Q_{1}$ and $Q_{1}<P$, and
$\frac{dQ_{2}}{dP}(x)=h_{2}(x)$, $\frac{dQ_{1}}{dP}(x)=h_{1}(x)$, then

$\frac{dQ_{2}}{dQ_{1}}(x)=\frac{h_{2}(x)}{h_{1}(x)}\cdot 1_{\\{z\colon
h_{1}(z)>0\\}}(x).$ (7)

Now we are ready to prove the following lemma. As in the previous section, all
the random variables are defined on the probability space
$(\Omega,\Sigma,\mathbb{P})$.

######  Lemma 5.

Let $X$ and $Z$ be random variables with $Z\geq 0$, $E[Z]<\infty$ and
$E|XZ|<\infty$. Define the nonnegative measure $Q_{1}$ and the signed measure
$Q_{2}$, both on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, by:

$Q_{1}(A)=E[1_{A}(X)Z],\ A\in\mathcal{B}(\mathbb{R}),\mbox{ and
}Q_{2}(A)=E[1_{A}(X)XZ],\ A\in\mathcal{B}(\mathbb{R}).$

We have

$\frac{dQ_{2}}{dQ_{1}}(x)=x.$

###### Proof.

On the measurable space $(\Omega,\Sigma_{X})$, where
$\Sigma_{X}=\\{X^{-1}(C)\colon C\in\mathcal{B}(\mathbb{R})\\}$, we define two
measures $\hat{Q}_{1}$ and $\hat{Q}_{2}$ by

$\hat{Q_{1}}(X^{-1}(C))=Q_{1}(C)\mbox{ and }\hat{Q_{2}}(X^{-1}(C))=Q_{2}(C),\
C\in\mathcal{B}(\mathbb{R}).$

By the definitions of $\hat{Q_{1}}$ and $\hat{Q_{2}}$, $Q_{1}$ and $Q_{2}$,
and next by the definition of the conditional expectation
$E[Z|X]=E[Z|\Sigma_{X}],$

$\hat{Q_{1}}(X^{-1}(C))=Q_{1}(C)=\int\limits_{X^{-1}(C)}Zd\mathbb{P}=\int\limits_{X^{-1}(C)}E[Z|\Sigma_{X}]d\mathbb{P}$

and

$\hat{Q_{2}}(X^{-1}(C))=Q_{2}(C)=\int\limits_{X^{-1}(C)}XZd\mathbb{P}=\int\limits_{X^{-1}(C)}XE[Z|\Sigma_{X}]d\mathbb{P}$

which implies

$\frac{d\hat{Q_{1}}}{d\mathbb{P}}=E[Z|\Sigma_{X}]\mbox{ and
}\frac{d\hat{Q_{2}}}{d\mathbb{P}}=XE[Z|\Sigma_{X}].$ (8)

Above, from the formal perspective, $\mathbb{P}$ is treated as a probability
measure on the space $(\Omega,\Sigma_{X})$ which is smaller then
$(\Omega,\Sigma)$. From (8), by (7), we have:

$\frac{d\hat{Q_{2}}}{d\hat{Q_{1}}}=X.$

Hence, by the definition of the Radon-Nikodym derivative and by standard
change of variables,

$Q_{2}(C)=\hat{Q_{2}}(X^{-1}(C))=\int_{X^{-1}(C)}Xd\hat{Q_{1}}=\int_{\Omega}1_{C}(X)Xd\hat{Q_{1}}=$
$=\int_{\mathbb{R}}1_{C}(x)xdQ_{1}(x)=\int\limits_{C}xQ_{1}(dx)$

which implies $\frac{dQ_{2}}{dQ_{1}}(x)=x$. ∎

Now we are ready to prove the main results. We start with Theorem 1: _Assume
that $\mathcal{K}\subset\mathcal{B}(\mathbb{R})$ contains the family
$\\{[a,b]\colon a<b\\}$. For any random variables $X$ and $Y$, the following
conditions are equivalent:_

  1. 1)

$X$_and_ $Y$ are independent,

  2. 2)

_for any_ $A,B\in\mathcal{K}$ with $\mathbb{P}_{(X,Y)(A\times B)}>0$,

$E[XY|X\in A,Y\in B]=E[X|X\in A,Y\in B]\cdot E[Y|X\in A,Y\in B].$

_Under additional assumption: $E|X|<\infty$ and $E|Y|<\infty$, we may replace
the bounded intervals with the family $\\{[t,\infty)\colon\ t\in\mathbb{R}\\}$
( or with $\\{(-\infty,t]\colon\ t\in\mathbb{R}\\}$)._

###### Proof.

We will prove the theorem under the assumption $\mathcal{K}=\\{[a,b]\\}_{a<b}$
or $\mathcal{K}=\\{[t,\infty)\\}_{t\in\mathbb{R}}$.
1. We will start with the direct implication $"1)\Rightarrow 2)"$ . The independence:

$\mathbb{P}[X\in A,Y\in B]=\mathbb{P}[X\in A]\cdot\mathbb{P}[Y\in B],\
A,B\in\mathcal{B}(\mathbb{R}),$

implies

$E[X1_{A}(X)\cdot Y1_{B}(Y)]=E[X\cdot 1_{A}(X)]\cdot E[Y\cdot 1_{B}(Y)],\
A,B\in\mathcal{K},$

and, by the definition of the conditional expectation, for any
$A,B\in\mathcal{K}$ with $\mathbb{P}_{(X,Y)}(A,B)>0$,

$E[XY|X\in A,Y\in B]=\frac{E[XY1_{(A\times B)}(X,Y)]}{\mathbb{P}[X\in A,Y\in
B]}=$ $=\frac{E[X1_{A}(X)]\cdot E[Y1_{B}(Y)]}{\mathbb{P}[X\in A,Y\in B]}\cdot
1\cdot 1=\frac{E[X1_{A}(X)]\cdot E[Y1_{B}(Y)]}{\mathbb{P}[X\in A,Y\in
B]}\cdot\frac{E[1_{B}(Y)]}{\mathbb{P}[Y\in
B]}\cdot\frac{E[1_{A}(X)]}{\mathbb{P}[X\in A]}=$
$=\frac{E[X1_{A}(X)1_{B}(Y)]\cdot E[Y1_{A}(X)1_{B}(Y)]}{\mathbb{P}[X\in A,Y\in
B]\cdot\mathbb{P}[X\in A,Y\in B]}=E[X|X\in A,Y\in B]\cdot E[Y|X\in A,Y\in B].$

Now we will prove the second implication $"2)\Rightarrow 1)"$.

2. Assume that $E|X|<\infty$ and $E|Y|<\infty$.
2a. We will now consider the case $\mathbb{P}[Y>0]=1$. Fix $B\in\mathcal{K}$
with $\mathbb{P}[Y\in B]>0$ which implies that $E[Y1_{B}(Y)]>0$. More, for any
$A\in\mathcal{K}$ we have

$E[1_{A}(X)1_{B}(Y)]>0\Leftrightarrow E[Y1_{A}(X)1_{B}(Y)]>0.$

From condition 2) of the theorem, for any $A\in\mathcal{K}$ with
$\mathbb{P}_{(X,Y)}(A\times B)=E[1_{A}(X)1_{B}(Y)]>0$ we have

$\frac{E[XY1_{A}(X)1_{B}(Y)]}{\mathbb{P}_{(X,Y)}(A\times
B)}=\frac{E[X1_{A}(X)1_{B}(Y)]}{\mathbb{P}_{(X,Y)}(A\times
B)}\cdot\frac{E[Y1_{A}(X)1_{B}(Y)]}{\mathbb{P}_{(X,Y)}(A\times B)}$

and thus

$\frac{E[XY1_{A}(X)1_{B}(Y)]}{E[Y1_{A}(X)1_{B}(Y)]}=\frac{E[X1_{A}(X)1_{B}(Y)]}{E[1_{A}(X)1_{B}(Y)]}.$
(9)

Note that

$\hat{P_{1}}(A)=E[1_{A}(X)1_{B}(Y)]\mbox{ and
}\hat{Q_{1}}(A)=E[Y1_{A}(X)1_{B}(Y)],\ A\in\mathcal{B}(\mathbb{R}),$

are nonnegative Borel measures with the same class of null sets
$\hat{P_{1}}(A)=0\Leftrightarrow\hat{Q_{1}}(A)=0$. We now normalize those
measures to obtain the corresponding probability measures:

$Q_{1}(A):=\frac{1}{E[Y1_{B}(Y)]}\cdot E[Y1_{A}(X)1_{B}(Y)],\
A\in\mathcal{B}(\mathbb{R})$

and

$P_{1}(A):=\\\ P[X\in A|Y\in B]=\frac{E[1_{A}(X)1_{B}(Y)]}{E[1_{B}(Y)]},\
A\in\mathcal{B}(\mathbb{R}).$

We want to reformulate equation (9) in a way which will allow us to use Lemma
5. In this purpose define the following signed measures

$Q_{2}(A):=\frac{1}{E[Y1_{B}(Y)]}\cdot{E[XY1_{A}(X)1_{B}(Y)]},\ \
A\in\mathcal{B}(\mathbb{R}),$ $P_{2}(A):=\frac{1}{E[1_{B}(Y)]}\cdot
E[X1_{A}(X)1_{B}(Y)],\ \ A\in\mathcal{B}(\mathbb{R}),$

so, for any $A\in\mathcal{K}$ with $\mathbb{P}[X\in A,Y\in B]>0$, equation (9)
takes the form

$\frac{Q_{2}(A)}{Q_{1}(A)}=\frac{P_{2}(A)}{P_{1}(A)}.$ (10)

Now, we apply Lemma 5 to the pair $Q_{2}$ and $Q_{1}$ (with $Z:=Y1_{B}(Y)$)
which provides the Radon Nikodym-derivative $\frac{dQ_{2}}{dQ_{1}}(x)=x$. In
case $\mathcal{K}=\\{[a,b]\colon a<b\\}$ Equation $\eqref{RN}$ forces
$\frac{dP_{2}}{dP_{1}}(x)=\frac{dQ_{2}}{dQ_{1}}(x)$ (see the differentiation
of measures from Mattila (1995)) but in case $\mathcal{K}=\\{[t,\infty)\colon
t\in\mathbb{R}\\}$ we need to use Lemma 5 again: the pair $P_{1}$ and $P_{2}$,
by Lemma 5 with $Z:=1_{B}(Y)$, satisfies $\frac{dP_{2}}{dP_{1}}(x)=x$. Hence,
the equation (10) takes the form

$\frac{\int\limits_{A}xQ_{1}(dx)}{Q_{1}(A)}=\frac{\int\limits_{A}xP_{1}(dx)}{P_{1}(A)},\
\mbox{ for any }A\in\mathcal{K}\mbox{ with }P_{1}(A)>0.$

By Observation 4 (see also Navarro et al. (1998) for the details on how to
reconstruct the probability distribution from the conditional first moments ),
we have $Q_{1}(A)=P_{1}(A)$, $A\in\mathcal{B}(\mathbb{R})$. As
$B\in\mathcal{K}$ with $\mathbb{P}_{Y}(B)>0$ was fixed arbitrarily, by
definitions of $Q_{1}$ and $P_{1}$, we have obtained that for all
$B\in\mathcal{K}$ and $A\in\mathcal{B}(\mathbb{R})$ with
$\mathbb{P}_{(X,Y)}(A\times B)>0$, we have

$\frac{E[Y1_{A}(X)1_{B}(Y)]}{E[1_{A}(X)1_{B}(Y)]}=\frac{E[Y1_{B}(Y)]}{E[1_{B}(Y)]}.$

Thus, for $A\in\mathcal{B}(\mathbb{R}),B\in\mathcal{K}$ with $\mathbb{P}[X\in
A,Y\in B]>0$ we have ended up with the following

$E[Y|X\in A,\ Y\in B]=E[Y|Y\in B].$ (11)

As the conditioning $E[Y|X\in A,Y\in B]$ does not depend on $X$, it is easy to
believe that $X$ and $Y$ are independent. As Lemma 5 and Observation 4 and are
still warm, we will use them again: Equation (11) may be rewritten as:

$\frac{\frac{1}{\mathbb{P}[X\in A]}\cdot E[Y1_{A}(X)1_{B}(Y)]}{\mathbb{P}[Y\in
B|X\in A]}=\frac{E[Y1_{B}(Y)]}{\mathbb{P}[Y\in B]}\mbox{ if }\
\mathbb{P}_{(X,Y)}(A\times B)>0,$

which, for any $A\in\mathcal{B}(\mathbb{R})$ with $\mathbb{P}_{X}(A)>0$, by
Lemma 5, takes the form

$\frac{\int\limits_{B}y\mathbb{P}_{Y}[dy|X\in A]}{\mathbb{P}[Y\in B|X\in
A]}=\frac{\int\limits_{B}y\mathbb{P}_{Y}(dy)}{\mathbb{P}_{Y}[B]}\mbox{ for any
}B\in\mathcal{K}\mbox{ with }\mathbb{P}[Y\in B|X\in A]>0.$

By Observation 4, we have $\mathbb{P}_{Y}(\ \cdot\ |X\in
A)=\mathbb{P}_{Y}(\cdot)$, where $A\in\mathcal{B}(\mathbb{R})$ is arbitrary
with $\mathbb{P}[X\in A]>0$. This implies that $X$ and $Y$ are independent
(note that if $\mathbb{P}[X\in A]=0$ then the event $\\{X\in A\\}$ is
independent of $\\{Y\in B\\}$ for any $B\in\mathcal{B}(\mathbb{R})\ )$.

2b. Now consider the case $Y>-c$, where $c>0$. By the assumption, $X$ and $Y$
are conditionally uncorrelated in sense $cov_{(A,B)}(X,Y)=0$ for any
admissible $A$,$B\in\mathbb{K}$. We will show that $X$ and $Y+c$ are
conditionally uncorrelated too. Note that for any $A,B\in\mathcal{K}$ we have
$\mathbb{P}_{(X,Y+c)}(A\times
B)>0\Leftrightarrow\mathbb{P}_{(X,Y)}(A\times(B-c))>0$, where
$B-c=\\{b-c\colon b\in B\\}$. Furthermore, the covariance is translation-
invariant and the class $\mathcal{K}$ is translation invariant in sense
$B\in\mathcal{K}\Rightarrow B-c\in\mathcal{K}.$ Thus, for any
$A,B\in\mathcal{K}$ with $\mathbb{P}_{(X,Y+c)}(A\times B)>0$, we have:

$cov_{(A,B)}(X,Y+c)=cov(X,Y+c|X\in A,Y+c\in B)=$ $=cov(X,Y|X\in A,Y+c\in
B)=cov(X,Y|X\in A,Y\in B-c)=0.$

Now, as we see that $Y+c$ and $X$ are conditionally uncorrelated, by point 2a.
$X$ and $Y+c$ are independent. Hence, $X$ and $Y$ are independent.

2c. Now we do not assume that $Y$ is bounded from below. The random variables
$X$ and $Y$ are independent if and only if for any $c\in\mathbb{R}$ with
$\mathbb{P}[Y>c]>0$ the $X$ and $Y$ are independent with respect to the
conditional probability $\mathbb{P}_{c}=\mathbb{P}[\cdot|Y>c]$, i.e.

$\mathbb{P}[X\in A,Y\in B|Y>c]=\mathbb{P}[X\in A|Y>c]\cdot\mathbb{P}[Y\in
B|Y>c],\ A,B\in\mathcal{B}(\mathbb{R}).$

For any $c$ small enough let $cov_{\mathbb{P}_{c}}$ denote the covariance
calculated on the probability space $(\Omega,\Sigma,\mathbb{P}_{c})$. Note
that for any admissible $A,B\in\mathcal{K}$ we have
$B\cap[c,\infty)\in\mathcal{K}$ and

$cov_{\mathbb{P}_{c}}(X,Y|X\in A,Y\in B)=cov(X,Y|X\in A,Y\in
B\cap[c,\infty))=0.$

By 2b., the $X$ and $Y$ are independent with respect to the conditional
probability measure $\mathbb{P}_{c}=\mathbb{P}[\cdot|Y\geq c]$. As $c$ may be
arbitrarily small, $X$ and $Y$ are independent.

3. It remains to consider the case $\mathcal{K}=\\{[a,b]\colon a<b\\}$ without assuming the existence of the first moments of $X$ and $Y$. To prove that $X$ and $Y$ are independent it is sufficient to prove that they are independent with respect to the conditional probability

$\mathbb{P}_{c}=\mathbb{P}[\ \cdot\ |(X,Y)\in[-c,c]^{2}]$

for any $c>0$ large enough to have $\mathbb{P}[(X,Y)\in[-c,c]^{2}]>0$. On the
probability space $(\Omega,\Sigma,\mathbb{P}_{c})$ the $X$ and $Y$ are
integrable and conditionally uncorrelated by the assumptions (note that if
$A\in\mathcal{K}$ and $A\cap[-c,c]\neq\varnothing$ then
$A\cap[-c,c]\in\mathcal{K}$), which by point 2. implies that $X$ and $Y$ are
$\mathbb{P}_{c}$ \- independent for any large $c$. ∎

In the above proof, points 2c. and 3. have been done separately because the
reasoning from point 3. does not apply to the class
$\mathcal{K}=\\{[t,\infty)\colon t\in\mathbb{R}\\}$. The second theorem may be
proved by mathematical induction with use of the same techniques.
Theorem 2:_Assume that $X_{1},\dots,X_{n}$ are random variables and
$\mathcal{K}=\\{[a,b]\colon a<b\\}$. For any $U=\\{X_{1}\in
A_{1},\dots,X_{n}\in A_{n}\\}$ with $\mathbb{P}[U]>0$ and
$A_{i}\in\mathcal{K}$, $\Sigma_{U}$ denotes the conditional corelation matrix,
i.e. $\Sigma_{U}[i,j]=cor_{U}(X_{i},X_{j})$ (if $sd_{U}(X_{i})=0$ let
$cor_{U}(X_{i},X_{j}):=\delta_{ij}$ ) . The following conditions are
equivalent:_

  1. 1.

_random variables_ $X_{1},\dots,X_{n}$ are mutually independent,

  2. 2.

_the conditional correlation matrix equals to the identity matrix, i.e._

$\Sigma_{U}=I_{n},$

_for any_ $U=\\{X_{1}\in A_{1},\dots,X_{n}\in A_{n}\\}$ with
$\mathbb{P}[U]>0$.

_The bounded intervals may be replaced with $\mathcal{K}=\\{[t,\infty)\colon
t\in\mathbb{R}\\}$ under assumption that $X_{i}$ have finite second moments_.

###### Proof.

The implication $"1.\Rightarrow 2."$ is straightforward and we will focus on
$"2.\Rightarrow 1."$ The proof is based on the induction. The case $n=2$ is a
straightforward conclusion from Theorem 1. Assume that Theorem 2 holds true
for some natural $n\geq 2$ and assume that we are given the random vector
$(X_{1},\dots,X_{n+1})$ which satisfies condition 2., i.e.
$\Sigma_{U}=I_{n+1}$ for any admissible $U$. We may apply Theorem $2$ to any
$n$-dimensional vector of the form $(X_{i})_{i\neq
j}=(X_{1},\dots,X_{j-1},X_{j+1},\dots,X_{n})$ which implies the mutual
independence of $X_{1},\dots,X_{j-1},X_{j+1},\dots,X_{n}$. Thus, to show the
induction step it is enough to show that some marginal variable $X_{j}$ is
independent of the $n$-dimensional vector $(X_{i})_{i\neq j}$. The general
induction step from $"n"$ to $"n+1"$ is the same as in the case from $n=2$ to
$n=3$ but the notation is more complex. We will thus focus on the implication
from $"n=2"$ to $"n=3"$. Assume that $(X,Y,Z)=(X_{1},X_{2},X_{3})$ We will
show that $Y$ is independent of the vector $(X,Z)$. The proof is similar the
reasoning from the proof of Theorem 1 and we will be a little more concise.
1. Assume that $X,Y,Z$ are square-integrable.
1a) Assume that $Y>0$. If $(A,B,C)\in\mathcal{K}^{3}$ satisfy
$\mathbb{P}[(X,Y,Z)\in A\times B\times C]>0$, then for $U:=\\{(X,Y,Z)\in
A\times B\times C\\}$, by the assumptions, we have

$E[X\cdot Y|U]=E[X|U]\cdot E[Y|U]$

which may be rewritten as

$\frac{E[XY1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[Y1_{A}(X)1_{B}(Y)1_{C}(Z)]}=\frac{E[X1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[1_{A}(X)1_{B}(Y)1_{C}(Z)]}.$
(12)

We fix $B,C\in\mathcal{K}$ with $\mathbb{P}[(Y,Z)\in B\times C]>0$ and we
normalize the nonnegative measures $A\to E[Y1_{A}(X)1_{B}(Y)1_{C}(Z)]$ and
$A\to E[1_{A}(X)1_{B}(Y)1_{C}(Z)]]$ to obtain the following two probability
Borel measures

$Q_{1}(A)=\frac{E[Y1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[Y1_{B}(Y)1_{C}(Z)]}\mbox{ and
}P_{1}(A)=\frac{E[1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[1_{B}(Y)1_{C}(Z)]}.$

Now we rewrite (12) with use of $P_{1}$ and $Q_{1}$ and next, as in the proof
of Theorem 1, we use the Lemmas 3 and 5 to obtain $P_{1}=Q_{1}$, i.e.

$\frac{E[Y1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[Y1_{B}(Y)1_{C}(Z)]}=\frac{E[1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[1_{B}(Y)1_{C}(Z)]}$
(13)

for any $A\in\mathcal{B}(\mathbb{R})$ and $B,C\in\mathcal{K}$ with
$\mathbb{P}_{(Y,Z)}(B\times C)>0$. If $\mathbb{P}_{(X,Y,Z)}[A\times B\times
C]>0$ then we rewrite (13) to obtain

$\frac{E[Y1_{A}(X)1_{B}(Y)1_{C}(Z)]}{E[1_{A}(X)1_{B}(Y)1_{C}(Z)]}=\frac{E[Y1_{B}(Y)1_{C}(Z)]]}{E[1_{B}(Y)1_{C}(Z)]}.$
(14)

Now fix $A\in\mathcal{B}(\mathbb{R})$ and $C\in\mathcal{K}$ with
$\mathbb{P}_{(X,Z)}(A\times C)>0$ and normalize the nonnegative measures $B\to
E[1_{A}(X)1_{B}(Y)1_{C}(Z)]$ and $B\to E[1_{B}(Y)1_{C}(Z)]$ from the
denominators of $\eqref{E3}$, and again use Lemmas 5 and 3 to obtain:

$\frac{E[1_{A}(X)1_{B}(Y)1_{C}(Z)]}{1_{A}(X)1_{C}(Z)}=\frac{E[1_{B}(Y)1_{C}(Z)]]}{E[1_{C}(Z)]}$

for any $A,B\in\mathcal{B}(\mathbb{R})$ and $C\in\mathcal{K}$ with
$\mathbb{P}_{(X,Z)}(A\times C)>0$. In other words, for such $A$,$C$ and any
$B\in\mathcal{B}(\mathbb{R})$ we have

$\mathbb{P}[Y\in B|X\in A,Z\in C]=\mathbb{P}[Y\in B|Z\in C].$ (15)

By the assumptions, $Y$ and $Z$ are conditionally uncorrelated and thus, as
Theorem 2 holds true for $n=2$, $Y$ and $Z$ are independent. To sum up: for
any $B\in\mathcal{B}(\mathbb{R})$ and any
$A\in\mathcal{B}(\mathbb{R}),C\in\mathcal{K}$ with $\mathbb{P}_{(X,Z)}[A\times
C]>0$, by (15) and by the independence of $Y$ and $Z$,

$\mathbb{P}[Y\in B|X\in A,Z\in C]=\mathbb{P}[Y\in B|Z\in C]=\mathbb{P}[Y\in
B]$

which implies that $Y$ is independent of the vector $(X,Z)$.This finishes the
proof (recall that the marginals $X$ and $Z$ are independent by Theorem 2
applied to the $2$-dimensional vector $(X,Z)$).
1b) If there is $c\in\mathbb{R}$ such that $\mathbb{P}[Y>c]=1$ then note that
the random variables $X,Y+c,Z$ satisfy the assumptions of Theorem 2 and thus
they are mutually independent by point 1a). This forces the independence of
$X,Y,Z$.
1c) In general case $Y\in\mathbb{R}$, for any small $c\in\mathbb{R}$ with
$\mathbb{P}[Y>c]>0$ we consider the probability space
$(\Omega,\Sigma,\mathbb{P}[\cdot|Y\geq c])$ on which, as we may note, the
random vector $(X,Y,Z)$ satisfy $\Sigma_{U}=I_{3}$ for any admissible $U$. By
1b), the marginals $X,Y,Z$ are mutually independent with respect to
$\mathbb{P}[\cdot|Y\geq c])$ for any $c$ small enough wich implies that they
are independent with respect to $\mathbb{P}$.

2. At the end note that in case $\mathcal{K}=\\{[a,b]\colon a<b\\}$ we do not need the assumption that $X,Y,Z$ have finite moments as the conditional standard deviations are finite. Also, note that $X,Y,Z$ are bounded and conditionally uncorrelated on the probability space $(\Omega,\Sigma,\mathbb{P}_{c})$, where $c>0$ is large enough to have the following conditional probability well defined

$\mathbb{P}_{c}=[\ \cdot\ |X,Y,Z\in[-c,c]].$

By point 1., $X,Y,Z$ are mutually independent with respect to $\mathbb{P}_{c}$
for any large $c$ which implies that $X,Y,Z$ are mutually independent with
respect to $\mathbb{P}$. ∎

Funding. This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors.
Data availability statement. No datasets were generated or analysed during the
current study.
Conflict of interest. The author has no conflicts of interest to declare that
are relevant to the content of this article

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# Reconstruction of $\Lambda$CDM Universe in $f(Q)$ Gravity

Gaurav N. Gadbail 0000-0003-0684-9702<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Sanjay Mandal 0000-0003-2570-2335
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

In this manuscript, we present a number of fascinating explicit
reconstructions for the $f(Q)$ gravity from the background of Friedmann-
Laîmatre-Robertson-Walker (FLRW) evolution history. We find the more general
functions of non-metricity scalar $Q$ that admit exact $\Lambda$CDM expansion
history. Adding extra degrees of freedom to the matter sector is the only
method to get the scale factor to behave in this manner for more generic
functions of $Q$. In addition, a cosmological reconstruction for modified
$f(Q)$ gravity is constructed in terms of e-folding. It is shown how any FLRW
cosmology can arise from a specific $f(Q)$ theory. We also reconstruct the
well-known cosmological evolution for the specific examples of $\Lambda$CDM
cosmology.

Keywords: $f(Q)$ gravity, $\Lambda$CDM cosmology, Equation of state, perfect
fluid, dust matter

##  I Introduction

For the last two decades, one prominent debatable problem has been the current
acceleration of the universe. The recent advances in the observational
cosmology confirm this phenomenon of the universe; for example the type Ia
supernovae [1, 2, 3], Cosmic Microwave Background Radiation [4] and large-
scale structure observations [5, 6, 7]. But, the real issue arises when it
comes to the theoretical development to address this process of the universe.
It is well-known that, after more than one hundred years, general relativity
(GR) is still the most successful fundamental gravitational theory to describe
the large-scale structure of the universe. When we apply it to cosmology, the
FLRW spacetime and the matter source give some exact solutions for the scale
factor $a(t)$, and it helps us to study the expansion of the Universe. If
someone wishes to recover FLRW spacetime, then the universe needs to go
through two types of accelerated expansion. One is the early time inflation,
which dynamics can be studied by introducing the scalar field in the Einstein-
Hilbert action. And the present accelerated expansion can be explained by
adding a cosmological constant in the Einstein field equation. This is the
very simple model (called $\Lambda$CDM model), which is able to explain the
late-time cosmic acceleration of the universe and fit all the available
observations discussed formerly. But unfortunately, this beautiful model is
affected by fine-tuning problems related to the vacuum energy scale [8, 9,
10]. Therefore, it is important to investigate alternative approaches or
generalizations of fundamental theories of gravity.

The simplest generalization of GR is the so-called $f(R)$ gravity, which
generalizes the Hilbert-Einstein action to a general function of the Ricci
scalar $R$ [11, 12]. This modified theory of gravity is well-known for its
successful representation of cosmic acceleration, and also it can reproduce
the entire cosmological history, as well as the behavior of the cosmological
constant ($\Lambda$) [13, 14, 15]. In order to recover the properties of
$\Lambda$CDM through modified theories of gravity, there are several
methodologies adopted in the literature. But, among them, the cosmological
reconstruction schemes have a special place. For instance, an interesting
scheme for the cosmological reconstruction of $f(R)$ gravity in terms of
e-folding was developed by Nojiri et al. [16]. For an accurate reconstruction
of $\Lambda$CDM evolution in $f(R)$ gravity, Dunsby et al. [17] determined
that the matter components must have an extra degree of freedom. Carloni et
al. [18] reconstructed $f(R)$ gravity by utilizing cosmic parameters instead
of any form of scale factors. Goheer et al. [19] demonstrated that exact
power-law solutions in $f(G)$ gravity exist exclusively for a specific class
of models because many popular $f(G)$ models ( all examples addressed in [20])
do not permit any exact power-law solutions. This result is also to that found
for $f(R)$ and $f(R,T)$ gravity models [21, 22]. In the context of $f(R,T)$
gravity, Jamil et al. [23] have reconstructed cosmological models and
demonstrated that the dust fluid reproduces the $\Lambda$CDM, de Sitter
Universe, Einstein static Universe, phantom and non-phantom era, and phantom
cosmology. In [24], have represented the cosmological reconstruction of
$f(R,T)$ models and their stability. An analysis of the stability of $f(R,G)$
models is presented in [25] for power law and $\Lambda$CDM cosmology. Other
than these curvature-based modified theories of gravity, there are two types
of modified theories of gravity by which one can explain the universe:
teleparallel gravity and symmetric teleparallel gravity theories.

In this work, we aim to study the more recent modifications based on a
Lagrangian density that is a generic function of the non-metricity scalar $Q$
that has been investigated [26]. At the background level, the modified theory
of $f(Q)$ gravity leads to fascinating cosmological phenomenology, one can see
[27, 28, 29, 30, 31, 32]. Several studies in the literature indicate that the
$f(Q)$ theory is one of the favorable alternative gravity formulations for
explaining cosmological observations [33, 34, 35]. A class of $f(Q)$ theories
with Q coupled non-minimally to the matter Lagrangian was developed by Harko
et al. [36]. They demonstrate through a cosmic application that it can serve
as an alternate approach for understanding dark energy (DE). Testing against a
variety of recent background data, including Type Ia Supernovae, Pantheon
data, Hubble data, etc., has been done to provide observational limits on the
background behavior of numerous $f(Q)$ models [37, 38]. Furthermore, $f(Q)$
gravity comfortably passes the constraints imposed by Big Bang Nucleosynthesis
(BBN) [39]. Using reconstruction methods, F. Esposito et al. [40] investigated
exact isotropic and anisotropic cosmological solutions.

Inspired by the fascinating characteristics of $f(Q)$ gravity, we aim to
develop a class of $f(Q)$ functions, which will be able to mimic the
properties of the $\Lambda$CDM model. For this purpose, we adopt a number of
explicit reconstructions, which lead to a number of interesting results
categorized in two schemes. In the first scheme, we obtain some real values
$f(Q)$ functions that are able to retrieve the $\Lambda$CDM expansion history
of the universe filled with various matter components, respectively. In the
second scheme, in terms of e-folding, a cosmological reconstruction for
modified $f(Q)$ gravity is constructed. It is shown how any FLRW cosmology can
arise from a specific $f(Q)$ theory. We further show that a theory can be
constructed that can mimic the $\Lambda$CDM expansion history, which can be
impossible to distinguish from GR at the fundamental level of the FLRW
universe.

This manuscript is arranged as follows. A general action and motion equation
for $f(Q)$ gravity is presented in section II. In section III, we reconstruct
the cosmological models for the various kind of fluid. In section IV, in terms
of e-folding, we reconstruct the cosmological models for modified $f(Q)$
gravity. Lastly, discussion and conclusions are presented in Section V.

##  II Basic Formalism of $f(Q)$ gravity

Consider the most general $f(Q)$ gravity action, which is given by [26]

$S=\int\left[\frac{1}{2}f(Q)+\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x,$ (1)

where $Q$ is the non-metricity scalar, $f$ is an arbitrary function of $Q$,
$\mathcal{L}_{m}$ is the matter Lagrangian density, and $g$ is a determinant
of the metric tensor $g_{\alpha\beta}$. Here, we assume $8\pi G=1$.
$Q_{\sigma\alpha\beta}=\nabla_{\sigma}\,g_{\alpha\beta}$ is the definition of
nonmetricity tensor in $f(Q)$ gravity and the corresponding traces are

$Q_{\sigma}=Q_{\sigma\,\,\,\,\alpha}^{\,\,\,\,\alpha}\,,\,\,\,\,\,\,\,\,\tilde{Q}_{\sigma}=Q^{\alpha}_{\,\,\,\,\sigma\alpha}\,.$
(2)

Moreover, the superpotential tensor $P_{\,\,\mu\nu}^{\lambda}$ is given by

$4P_{\,\,\alpha\beta}^{\sigma}=-Q^{\sigma}_{\,\,\,\,\alpha\beta}+2Q^{\,\,\,\,\,\,\sigma}_{(\alpha\,\,\,\,\beta)}-Q^{\sigma}g_{\alpha\beta}-\tilde{Q}^{\sigma}g_{\alpha\beta}-\delta^{\sigma}_{(\alpha}\,Q\,_{\beta)},$
(3)

acquiring the nonmetricity scalar as

$Q=-Q_{\sigma\alpha\beta}P^{\sigma\alpha\beta}.$ (4)

The energy-momentum tensor for matter read as

$T_{\alpha\beta}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g})\mathcal{L}_{m}}{\delta
g^{\alpha\beta}}.$ (5)

The gravitational field equation of $f(Q)$ gravity shown below is derived by
varying the action (1) over $g_{\mu\nu}$:

$\frac{2}{\sqrt{-g}}\nabla_{\sigma}\left(f_{Q}\sqrt{-g}\,P^{\sigma}_{\,\,\alpha\beta}\right)+\frac{1}{2}f\,g_{\alpha\beta}+\\\
f_{Q}\left(P_{\alpha\sigma\lambda}Q_{\beta}^{\,\,\,\sigma\lambda}-2Q_{\sigma\lambda\alpha}P^{\sigma\lambda}_{\,\,\,\,\,\,\beta}\right)=-T_{\alpha\beta},$
(6)

where $f_{Q}=\frac{df}{dQ}$. The equation (1) can also be varied with respect
to the connection as follows:

$\nabla_{\alpha}\nabla_{\beta}\left(f_{Q}\sqrt{-g}\,P_{\,\,\,\,\,\,\,\sigma}^{\alpha\beta}\right)=0.$
(7)

Throughout the study, we will assume a spatially flat FLRW Universe, whose
metric is given by

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}).$ (8)

Here, $a(t)$ is a scale factor and $t$ is the cosmic time. The non-metricity
scalar in FLRW metric is obtained as $Q=6H^{2}$, where $H=\frac{\dot{a}}{a}$
is the Hubble function and the over-head dot indicates the derivative with
respect to $t$. In this case, the energy-momentum tensor of a perfect fluid
$T_{\alpha\beta}=(p+\rho)u_{\alpha}u_{\beta}+pg_{\alpha\beta}$, where $\rho$
and $p$ are energy density and pressure, respectively.
For the metric (8), the corresponding Friedmann equations are [26]:

$3H^{2}=\frac{1}{2f_{Q}}\left(-\rho+\frac{1}{2}f\right),$ (9)
$\dot{H}+3H^{2}+\frac{\dot{f}_{Q}}{f_{Q}}H=\frac{1}{2f_{Q}}\left(p+\frac{1}{2}f\right).$
(10)

Now, one can investigate various cosmological applications using above
Friedmann equations in the background of $f(Q)$ gravity. Further, we would
like to mention here that the $f(Q)$ gravity satisfies the conservation
equation

$\dot{\rho}+3H(\rho+p)=0.$

##  III Reconstruction of a $f(Q)$ theory that admits an exact $\Lambda$CDM
model

In this section, we reconstruct the $f(Q)$ gravity model for various epochs
such that it can precisely mimic the $\Lambda$CDM model. We can develop a
real-valued function for the non-metricity scalar that can provide the
particular cosmological evolution of the $\Lambda$CDM model. The observational
cosmology suggests that the Hubble rate in terms of redshift, which is
described in the $\Lambda$CDM model, is given by

$H(z)=\sqrt{\frac{\rho_{0}}{3}(1+z)^{3}+\frac{\Lambda}{3}},$ (11)

where $\Lambda$ is a cosmological constant and $\rho_{0}\geq 0$ is the matter
density. In follows, we try to construct the $f(Q)$ gravity theories that
exactly mimic the $\Lambda$CDM expansion.
Using the scale factor $a$ and redshift $z$ relationship as $\frac{1}{a}=1+z$,
the above equation can be demonstrated as

$\frac{\dot{a}}{a}=\sqrt{\frac{\rho_{0}}{3a^{3}}+\frac{\Lambda}{3}}.$ (12)

From the above equation, we may get the derivative of the scale factor $a(t)$
with respect to time ($t$) as

$\dot{a}=\sqrt{\frac{\rho_{0}}{3a}+\frac{\Lambda}{3}a^{2}}.$ (13)

Now, we can rewrite the non-metricity scalar $Q$ in terms of the scale factor
by plugging Eq.(12),

$Q(a)=\frac{2(\rho_{0}+\Lambda\,a^{3})}{a^{3}}.$ (14)

With the help of above equation, we write the scale factor in terms of non-
metricity scalar as

$a(Q)=\left(\frac{2\,\rho_{0}}{(Q-2\,\Lambda)}\right)^{\frac{1}{3}}.$ (15)

We would like to note here that so far, we have not presumed a priori and a
specific theory of gravity. Expression for Hubble parameter is obtained by
observations, and the non-metricity scalar is completely geometrical for FLRW
spacetime, independent of gravitational theory. Further, the scale factor in
Eq. (15) has one real and two complex roots. The scale factor has to be real.
Therefore, with the real value of the scale factor, the non-metricity scaler
reaches the value $2\Lambda$ in an infinite time. From Eq.(15), we can
calculate the Hubble parameter in terms of non-metricity scalar as

$H(Q)=\frac{1}{a(Q)}\sqrt{\frac{\rho_{0}}{3a(Q)}+\frac{\Lambda}{3}\,a^{2}(Q)}.$
(16)

Now, in order to find a class of $f(Q)$ functions, which mimic the
$\Lambda$CDM expansion, , we plug all of the above quantities represented as
functions of the non-metricity scalar into the Friedmann equation Eq.(9),
yielding the first order inhomogeneous differential equation for the function
$f(Q)$ in $Q$-space,

$Q\,\frac{df(Q)}{dQ}+\rho(Q)-\frac{f(Q)}{2}=0.$ (17)

Utilizing the energy conservation equation, the inhomogeneous term $\rho(Q)$
may be derived in terms of nonmetricity $Q$. The inhomogeneous term disappears
in the vacuum case, yielding the homogeneous first order differential
equation, and its solution is $f(Q)=c\,\sqrt{Q}$, where $c$ is a integration
constant.
In further study, we only focus on the the inhomogeneous term $\rho(Q)$ to
construct more theories for the $\Lambda$CDM expansion. let us explicitly
reconstruct the theories that would lead to a particular matter field obeying
a $\Lambda$CDM expansion history.

###  III.1 Reconstruction for dust-like matter

First, we reconstruct the $f(Q)$ model, which may represent the $\Lambda$CDM
era while taking dust into account as a matter content. For the dust like
case, the value equation of state $w=0$. Imposing the dust like case in the
conservation equation, we obtained the energy density in terms of $a(t)$ as

$\rho(a)=\frac{\rho_{0}}{a^{3}}.$ (18)

With the help of Eq.(15), rewrite above equation as

$\rho(Q)=\frac{Q-2\Lambda}{2}.$ (19)

Substitute the above value in differential Eq.(17), we get the general
solution as

$f(Q)=-Q-2\,\Lambda+c_{1}\sqrt{Q},$ (20)

where $c_{1}$ is a integration constant.
This finding is intriguing because it demonstrates that the only real-valued
Lagrangian $f(Q)$ capable of accurately simulating the $\Lambda$CDM expansion
history for a universe composed primarily of dust-like stuff. Also if we put
$\Lambda=0$ in Eq. (20), the function for $f(Q)$ recover the dust-dominated
universe other than GR.

###  III.2 Reconstruction for perfect fluid with EoS $p=-\frac{1}{3}\rho$

In this scenario, we reconstruct the $f(Q)$ model in which the universe is
accelerating, and the EoS value $w=-\frac{1}{3}$ is physically interesting
because it sits near the limit of the set of matter fields that obey the
strong energy condition. Imposing the value of EoS $p=-\frac{1}{3}\rho$ in
conservation equation, we obtained the energy density in terms of $a(t)$ as

$\rho(a)=\frac{\rho_{0}}{a^{2}}.$ (21)

With the help of Eq.(15), rewrite above equation as

$\rho(Q)=\left(\frac{\rho_{0}^{2}(Q-2\Lambda)}{2}\right)^{\frac{2}{3}}.$ (22)

Substitute the above value in differential Eq.(17), we get the general
solution as

$f(Q)=c_{1}\sqrt{Q}+\mu_{1}\left(\frac{Q-2\Lambda}{2\Lambda-Q}\right)^{\frac{2}{3}}\times_{2}F_{1}\left(-\frac{2}{3},-\frac{1}{2},\frac{1}{2};\frac{Q}{2\Lambda}\right),$
(23)

where ${}_{2}F_{1}$ is a Hypergeometric function and
$\mu_{1}=2\left(\frac{\rho_{0}^{2}}{\Lambda}\right)^{\frac{2}{3}}$.

###  III.3 Reconstruction for Multifluids

Our present knowledge of the Universe is based on the interactions of multiple
”matter” components, which primarily interact via gravity and electromagnetic
radiation. The nature of the various components and possible interactions is
often based on the concept of coupled ideal fluids. to shed some light in this
type of scenario, we start from the same place as in the previous sections,
but with two fluid species instead of one. Let us now assume that the universe
contains both dust-like matter and a non-interacting stiff fluid, with current
densities of $\rho_{0}$ and $\rho_{s}$, respectively. In this case, total
matter density is calculated using the conservation equation,

$\rho(a)=\frac{\rho_{0}}{a^{3}}+\frac{\rho_{s}}{a^{6}},$ (24)

With the help of Eq.(15), rewrite above equation as

$\rho(Q)=\frac{Q-2\Lambda}{2}+\frac{\rho_{s}}{\rho_{0}^{2}}\frac{(Q-2\Lambda)^{2}}{4}.$
(25)

Substitute the above value in differential Eq.(17), we get the general
solution as

$f(Q)=-Q-2\Lambda+\mu_{1}Q-\mu_{2}Q^{2}+\mu_{3}+c_{1}\sqrt{Q},$ (26)

where $\mu_{1}=\frac{2\rho_{s}\Lambda}{\rho_{0}^{2}}$,
$\mu_{2}=\frac{\rho_{s}}{6\rho_{0}^{2}}$,
$\mu_{3}=\frac{2\Lambda^{2}\rho_{s}}{\rho_{0}^{2}}$.
As a result, we conclude that the above-described theory of gravity would
perfectly reproduce the history of the $\Lambda$CDM expansion if the universe
is composed of minimally connected noninteracting massless scalar fields with
dust like stuff.

###  III.4 Reconstruction for nonisentropic perfect fluids

The equation of state is used to characterise nonisentropic perfect fluids as

$p=h(\rho,a).$ (27)

We may obtain the appropriate differential equation for $\rho$ by using the
energy conservation relation,

$\frac{d\rho}{da}=-\frac{3}{a}(\rho+h(\rho,a)).$ (28)

For much simplicity of above equation, we assume that the separable function
of $h(\rho,a)$ as $h(\rho,a)=w(a)\rho$. In this scenario, we can simply
integrate Eq.(28) to obtain

$\rho(a)=Exp[-3\int\frac{1+w(a)}{a}\,da].$ (29)

Let us use an example where the time-dependent barotropic index is represented
by

$w(a)=\frac{2\alpha-\beta a^{3}}{\alpha+\beta a^{3}},$ (30)

where $\alpha$ and $\beta$ are constants. Using Eq.(30) in Eq.(29), we get

$\rho(a)=\frac{(\alpha+\beta a^{3})^{3}}{a^{9}}.$ (31)

With the help of Eq.(15), rewrite above equation as

$\rho(Q)=\frac{(2\rho_{0}\beta+\alpha Q-2\Lambda\alpha)^{3}}{8\rho_{0}^{3}}.$
(32)

Substitute the above value in differential Eq.(17), we get the general
solution as

$f(Q)=-\mu_{1}Q+\mu_{2}Q^{2}-\mu_{3}Q^{3}-\mu_{4}+c_{1}\sqrt{Q},$ (33)

where,
$\mu_{1}=\frac{3\alpha(\Lambda\alpha-\beta\rho_{0})^{2}}{\rho_{0}^{3}}$,
$\mu_{2}=\frac{\alpha^{2}(\Lambda\alpha-\beta\rho_{0})}{2\rho_{0}^{3}}$,
$\mu_{3}=\frac{\alpha^{3}}{20\rho_{0}^{3}}$ and
$\mu_{4}=\frac{2(\Lambda\alpha-\beta\rho_{0})^{3}}{\rho_{0}^{3}}$.
Let us take a look at another type of nonisentropic perfect fluid whose
equation of state is

$p=w\rho+h(a).$ (34)

Integrating energy conservation Eq.(28), we obtained

$\rho(a)=\left(-\int 3a^{(2+3w)}h(a)\,da+c\right)a^{-3(1+w)}.$ (35)

Let us use the particular cases of $h(a)=a^{-12}$ and $w=0$. This suggests
that the universe’s matter field is made up of dust and a time-dependent
cosmological component that diverges at the Big Bang singularity and quickly
decays to zero at subsequent epochs.

After solving Eq.(35) we obtained matter density as

$\rho(a)=\frac{\rho_{0}}{a^{3}}+\frac{\rho_{1}}{a^{12}}.$ (36)

With the help of Eq.(15), rewrite above equation as

$\rho(Q)=\frac{Q-2\Lambda}{2}+\frac{\rho_{1}}{\rho_{0}^{4}}\frac{(Q-2\Lambda)^{4}}{16}.$
(37)

Substitute the above value in differential Eq.(17), we get the general
solution as

$f(Q)=-Q+\mu_{1}Q-\mu_{2}Q^{2}+\mu_{3}Q^{3}-\mu_{4}Q^{4}+\mu_{5}+c_{1}\sqrt{Q},$
(38)

where $\mu_{1}=\frac{4\Lambda^{3}\rho_{1}}{\rho_{0}^{4}}$,
$\mu_{2}=\frac{\Lambda^{2}\rho_{1}}{\rho_{0}^{4}}$,
$\mu_{3}=\frac{\Lambda\rho_{1}}{5\rho_{0}^{4}}$,
$\mu_{4}=\frac{\rho_{1}}{56\rho_{0}^{4}}$ and
$\mu_{5}=-2\Lambda+\frac{2\Lambda^{4}\rho_{1}}{\rho_{0}^{4}}$.

##  IV Cosmological reconstruction of $f(Q)$ gravity

The first Friedmann equation corresponding to the FLRW equation can be rewrite
as,

$0=\frac{f(Q)}{2}-6H^{2}f^{\prime}(Q)-\rho,$ (39)

where $Q=6H^{2}$ and $f_{Q}=f^{\prime}$. The above Friedmann equations are
written as functions of the number of e-foldings instead of the time $t$,
$N=log\frac{a}{a_{0}}$. The variable $N$ is related with the redshift $z$ by
$e^{-N}=\frac{a_{0}}{a}=(1+z)$. By adding the fluid densities with a constant
EoS parameter $w_{i}$, we may get the matter energy density $\rho$ as

$\rho=\sum_{i}\rho_{i0}\,a^{-3(1+w_{i})}=\sum_{i}\rho_{i0}\,a_{0}^{-3(1+w_{i})}e^{-3(1+w_{i})N}.$
(40)

Let us write the Hubble parameter in terms of $N$ via the function $g(N)$ as

$H=g(N)=g(-ln(1+z)).$ (41)

Then non-metricity scalar written as $Q=6H^{2}=6g(N)^{2}$, where $N=N(Q)$. By
using Eq.(40) and Eq.(41), Eq.(39) can be written as,

$0=\frac{f(Q)}{2}-6G(N)\frac{df(Q)}{dQ}-\sum_{i}\rho_{i0}\,a_{0}^{-3(1+w_{i})}e^{-3(1+w_{i})N}.$
(42)

Here we denote $G(N)=g(N)^{2}=H^{2}$. As an example, consider the CDM model.
As will be demonstrated below, for the gravity theory represented by action
(1), such evolution may be reconstructed without the need of a cosmological
constant factor, and the shift from a decelerated to an accelerated period is
achieved. The FLRW equation for $\Lambda$CDM cosmology in Einstein gravity is
given by

$\frac{3}{\kappa^{2}}H^{2}=\frac{3}{\kappa^{2}}H_{0}^{2}+\frac{\rho_{0}}{a^{3}}=\frac{3}{\kappa^{2}}H_{0}^{2}+\rho_{0}\,a_{0}^{-3}e^{-3N},$
(43)

$H_{0}$ and $\rho_{0}$ are constants in this case. The cosmological constant
is represented by the first component in the RHS, while cold dark matter is
represented by the second term (CDM). In the present universe, the (effective)
cosmological constant $\Lambda$ is given by $\Lambda=12H_{0}^{2}$. Then comes
the

$G(N)=H_{0}^{2}+\frac{\kappa^{2}}{3}\rho_{0}\,a_{0}^{-3}e^{-3N},$ (44)

and $Q=6G(N)=6H_{0}^{2}+2\kappa^{2}\rho_{0}\,a_{0}^{-3}e^{-3N}$, which can be
solved for $N$ as follows:

$N=-\frac{1}{3}\,ln\left(\frac{Q-6H_{0}^{2}}{2\kappa^{2}\rho_{0}\,a_{0}^{-3}}\right).$
(45)

With the help of above equation, Eq.(42) can be taken as

$0=\frac{f(Q)}{2}-Q\frac{df(Q)}{dQ}-K(Q-6H_{0}^{2})^{(1+w)}.$ (46)

where $K=\rho_{0}\left(\frac{1}{2\kappa^{2}\rho_{0}}\right)^{(1+w)}$. Solution
of above differential equation is

$f(Q)=c_{1}\sqrt{Q}-2K(6H_{0}^{2})^{(1+w)}\left(\frac{Q-6H_{0}^{2}}{6H_{0}^{2}-Q}\right)^{w}\\\
\times_{2}F_{1}\left(\frac{-1}{2},-1-w,\frac{1}{2};\frac{Q}{6H_{0}^{2}}\right),$
(47)

where ${}_{2}F_{1}$ is a Hypergeometric function.
As a result, we showed that modified $f(Q)$ gravity can characterise the
$\Lambda$CDM era without introducing the effective cosmological constant.

As another example, in Einstein gravity, we reconstruct $f(Q)$ gravity by
replicating the system with non-phantom matter and phantom matter, whose FLRW
equation is provided by

$\frac{3}{\kappa^{2}}H^{2}=\rho_{p}\,a^{-c}+\rho_{q}\,a^{c},$ (48)

where $\rho_{p}$, $\rho_{q}$ and $c$ are positive constants. We can verify
that the first component in the R.H.S. in this solution relates to a non-
phantom fluid with an equation of state (EoS) $w=-1+\frac{c}{3}>-1$, whereas
the second term has an equation of state (EoS) $w=-1-\frac{c}{3}<-1$, which
relates to a phantom fluid. Then since $G(N)=g(N)^{2}=H^{2}$, we find

$G(N)=G_{p}\,e^{-cN}+G_{q}\,e^{cN}.$ (49)

where $G_{p}$ and $G_{q}$ are constants. Then since $Q=6G(N)$,

$e^{cN}=\frac{Q\pm\sqrt{Q^{2}-144G_{p}G_{q}}}{12G_{q}}.$ (50)

Then above equation can be solved for $N$ as follows:

$N=\frac{1}{c}ln\left(\frac{Q\pm\sqrt{Q^{2}-144G_{p}G_{q}}}{12G_{q}}\right).$
(51)

We consider $c=4$ case. In the case, the non-phantom matter corresponding to
the first term in the RHS of Eq.(48) could be radiation with $w=\frac{1}{3}$.
Then Eq.(42) in this case is given by

$0=\frac{f(Q)}{2}-Q\frac{df(Q)}{dQ}-\frac{\rho_{r0}}{a_{0}^{4}}\left(\frac{Q\pm\sqrt{Q^{2}-144G_{p}G_{q}}}{12G_{q}}\right).$
(52)

Solution of above differential equation is

$f(Q)=-\mu_{1}Q+c_{1}\sqrt{Q}-\mu_{2}\sqrt{\frac{144G_{p}G_{q}-Q^{2}}{Q^{2}-144G_{p}G_{q}}}\\\
\times_{2}F_{1}\left(-\frac{1}{2},-\frac{1}{4},\frac{3}{4};\frac{Q^{2}}{144G_{p}G_{q}}\right).$
(53)

Where ${}_{2}F_{1}$ is a hypergeometric function and
$\mu_{1}=\frac{\rho_{r0}}{6a_{0}^{4}G_{q}}$,
$\mu_{2}=\frac{2\rho_{r0}}{a_{0}^{4}}\sqrt{\frac{G_{p}}{G_{q}}}$ are
constants.

##  V Discussion

Several modified theories of gravity have been proposed to fill the hole in
the GR. However, the $\Lambda$CDM model is very successful in describing the
accelerated expansion; but it has a few shortcomings. Therefore, in this
manuscript, we have looked at the modified theories of gravity that can mimic
the exact $\Lambda$CDM expansion history of the universe. We have adopted
several cosmological reconstruction techniques for $f(Q)$ gravity to carry out
this task. In our first approach, we have presumed different fluid components
and obtained a class of $f(Q)$ theories. We have also reconstructed the
$\Lambda$CDM universe in terms of e-folding without using an auxiliary scalar
in intermediate calculations. The great advantage of working on e-folding is
that one can reconstruct any type of requested FLRW cosmology, such as the
oscillating universe, transition from deceleration to phantom phase without
future singularity, the subsequent transition from deceleration to phantom
super-acceleration leading to Big Rip singularity, and $\Lambda$CDM epoch.

In our study, we have obtained a class of $f(Q)$ theories that exactly mimic
$\Lambda$CDM expansion history, even if it is impossible to distinguish from
the GR using measurements of the background cosmological parameters. Then, it
is an exciting problem to see how the perturbations studies (such as growth
factor, structure formations, or cosmological gravitational waves from GR) in
these $f(Q)$ theories can be verified experimentally [27, 41, 42]. In fact,
the $f(Q)$ theories presented in this manuscript can be tested through the
solar system test, which rules out the Lagrangians to have a modified theory
that works for both local and cosmological scales. Furthermore, we have
restricted the reconstruction scheme to the flat FLRW cases, whereas in non-
flat FLRW cases, one additional term for curvature will appear in the Hubble
parameter, non-metricity scalar $Q$, and the Friedmann equations. As a result,
a highly non-linear differential equation will arise for $f(Q)$, which is an
open problem for readers. Hence, developing these types of modified theories
adds a strong agreement in favor of inflation, dark matter, and dark energy in
the context of a unified gravitational alternative theory.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

GNG acknowledges University Grants Commission (UGC), New Delhi, India for
awarding Junior Research Fellowship (UGC-Ref. No.: 201610122060). S.M.
acknowledges Department of Science & Technology (DST), Govt. of India, New
Delhi, for awarding INSPIRE Fellowship (File No. DST/INSPIRE
Fellowship/2018/IF180676). We are very much grateful to the honorable referee
and to the editor for the illuminating suggestions that have significantly
improved our work in terms of research quality, and presentation.

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# Delay Optimization of Conventional Non-Coherent Differential CPM Detection

Anouar Jerbi, Karine Amis, Frédéric Guilloud, and Tarik Benaddi
This work has been financially supported by Thales Alenia Space and Brest
Métropole.
Anouar Jerbi is with the Mathematical and Electrical Engineering (MEE)
Department, IMT Atlantique, Lab-STICC, UMR CNRS 6285, 29238 Brest, France. He
is also with TéSA, 31000 Toulouse, France. (e-mail: anouar.jerbi@imt-
atlantique.fr).
Karine Amis and Frédéric Guilloud are with the Mathematical and Electrical
Engineering (MEE) Department, IMT Atlantique, Lab-STICC, UMR CNRS 6285, 29238
Brest, France (e-mail: firstname.lastname@imt-atlantique.fr). Tarik Benaddi is
with Thales Alenia Space, 31000 Toulouse, France (e-mail :
<EMAIL_ADDRESS>
###### Abstract

The conventional non-coherent differential detection of continuous phase
modulation (CPM) is quite robust to channel impairments such as phase and
Doppler shifts. Its implementation is on top of that simple. It consists in
multiplying the received baseband signal by its conjugate and delayed version
of one symbol period. However it suffers from a signal-to-noise ratio gap
compared to the optimum coherent detection. In this paper, we improve the
error rate performance of the conventional differential detection by using a
delay higher than one symbol period. We derive the trellis description as well
as the branch and cumulative metric that take into account a delay of $K$
symbol periods. We then derive an optimization criterion of $K$ based on the
minimum Euclidean distance between two differential signals. We finally
determine optimized delays for some popular CPM formats whose values are
confirmed by error rate simulations.

###### Index Terms:

CPM, differential detection, Doppler shift, phase shift

##  I Introduction

Continuous Phase Modulations (CPM) are a class of non-linear constant-envelope
modulations with limited spectral occupancy. The constant envelope is
interesting when the channel includes a strong non-linearity like e.g. in
satellite communications. Moreover, non-coherent CPM detection enables to face
the possible phase distortion introduced by the channel, by doing without any
phase synchronisation. Combined with the energy efficiency of CPM, these
properties make this kind of waveform a good candidate for Internet of Things
(IoT) [1, 2]. Non-coherent CPM detectors can be grouped into two families
depending on the criterion they are based on. The first one is derived from
the generalized maximum-likelihood criterion [3, 4, 5] and only requires the
knowledge of the phase distribution. Algorithms proposed either in [6] or in
[7] with an uniformly-distributed phase assumption belong to it. The second
one preprocesses the received signal to neutralize the phase contribution
making possible the application of the maximum-likelihood criterion for
coherent detection on the resulting signal.

Numerous papers based on differentially-preprocessed signals can be found in
the state-of-the-art. The common feature is the use of the product of the
received baseband signal and a conjugate time-delayed version of it, yielding
a signal that we will refer to as differential signal in the remaining of the
paper. Different algorithms are proposed and apply either on time-discrete
differential signals (see e.g. [8]) or on time-continuous differential signals
(see e.g. [9, 10, 11, 12]). In [9] applied to tamed frequency modulation (TFM)
and in [12] extended to CPM signals, a detection metric is defined from a set
of multiple differential signal versions of the original one (differing from
the delay value). Simulations are used to compare different set definitions
(with a maximum delay equal to three symbol periods) for the TFM and the
Gaussian minimum shift keying (GMSK). However, the most used representative of
the second class is the conventional differential detection as defined in [10,
11] and applied with one symbol period as the delay value. The latter is on
top of that insensitive to Doppler shifts, which motivated us to focus on this
kind of receiver given our interest for SAT-IoT. Its main drawback is the
signal to noise ratio (SNR) gap as compared to the optimum coherent detection.
In this paper, for the purpose of reducing the SNR gap, we propose to modify
the differential detection as defined in [10]. Our contributions are
threefold, (i) the extension of the usual differential detection algorithm to
consider a delay higher than one symbol period (including the definition of
the phase trellis and of the branch and cumulative metric), (ii) the
proposition of an optimization criterion of the delay value based on the
minimum Euclidean distance between two differential signals and, (iii) the
optimized delay values for different CPM formats (modulation index, phase
shaping pulse length, frequency pulse).

The remainder of this article is organized as follows: in Section II, the
system model is presented and the notations are introduced. In Section III,
the differential detection using a delay of $K$ symbol periods is exposed,
followed by the optimization criterion of $K$ in Section IV. The simulations
and the resulting tables for different CPM formats are presented in Section V.
A conclusion is drawn in Section VI which ends the paper.

##  II System Model and Notations

We consider a sequence of $N$ independent and identically distributed (i.i.d.)
information symbols $\mathbf{a}=\\{a_{i}\\}_{0\leq i\leq N-1}$ to be
transmitted. Given $M$ an even positive integer, $a_{i}$ takes on values in
the $M$-ary alphabet $\mathcal{M}=\\{\pm 1,\pm 3,..\pm(M-1)\\}$ with equal
probabilities. The complex envelope of the CPM-modulated signal is given by:

$s(t,\mathbf{a})=\sqrt{\frac{E_{s}}{T_{s}}}\mathrm{e}^{j\theta(t,\mathbf{a})},$
(1)

where $E_{s}$ is the average symbol energy, $T_{s}$ is the symbol period and
$\theta(t,\mathbf{a})$ is the signal phase which depends on the information
symbols. It is defined by:

$\theta(t,\mathbf{a})=2\pi h\displaystyle\sum_{i=0}^{N-1}a_{i}q(t-iT_{s}),$
(2)

where $h$ is the modulation index and $q(t)$ is the phase shaping pulse whose
expression is $q(t)=\int_{-\infty}^{t}g(u)du$ with $g(t)$ the frequency
waveform. In practice, $g(t)$ has a finite duration $LT_{s}$ and it must
satisfy the following conditions:

$\begin{cases}g(t)=g(LT_{s}-t),&0\leq t<LT_{s}\\\
\int_{-\infty}^{t}g(\tau)\mathrm{d}\tau=q(LT_{s})=\frac{1}{2},&\forall t\geq
LT_{s}\end{cases}$ (3)

We assume that the modulated signal is transmitted over a Gaussian channel.
The equivalent baseband received signal, denoted by $r(t)$, is given by:

$r(t)=s(t,\mathbf{a})e^{j\psi}+n(t),$ (4)

where $\psi$ is an arbitrary phase introduced by the channel supposed to be
uniformly distributed in $[0,2\pi)$ and $n(t)$ is the realization of a zero-
mean wide-sense stationary complex circularly symmetric Gaussian noise,
independent of the signal, and with double-sided PSD $\frac{N_{0}}{2}$.

##  III Differential detection of CPM

###  III-A $K$-delay based differential receiver

At the receiver side, a differential signal denoted by $R_{K}(t)$ is generated
using the received signal $r(t)$ and its $K-$symbol periods delayed version
$r(t-KT_{s})$ and is expressed as the sum of two signals:

$\displaystyle
R_{K}(t)=\frac{1}{2}r(t)r^{*}(t-KT_{s})=S_{K}(t,\mathbf{a})+N_{K}(t),$ (5)

where the first term does not include any noise part:

$\displaystyle
S_{K}(t,\mathbf{a})\frac{1}{2}s(t,\mathbf{a})s^{*}(t-KT_{s},\mathbf{a})=\frac{E_{s}}{2T_{s}}\mathrm{e}^{j\Theta_{K}(t,\mathbf{a})}$
(6)

with
$\Theta_{K}(t,\mathbf{a})=\theta(t,\mathbf{a})-\theta(t-KT_{s},\mathbf{a})$.

The second term which includes noise-dependent component $N_{K}(t)$ is
decomposed as $N_{K}(t)=U_{K}(t)+W_{K}(t)$ with

$\displaystyle U_{K}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(s(t,\mathbf{a})n^{*}(t-KT_{s})+n(t)s^{*}(t-KT_{s},\mathbf{a})\right),$
$\displaystyle W_{K}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(n(t)n^{*}(t-KT_{s})\right).$ (7)

The noise component autocorrelation can be written as:

$\displaystyle E[N_{K}(t)N^{*}_{K}(t-\tau)]$ $\displaystyle=$
$\displaystyle\frac{1}{4}(N_{0}^{2}+2A^{2}N_{0})\delta(\tau)$ (8)

with $A^{2}=|s(t,\mathbf{a})|=\sqrt{\frac{E_{s}}{T_{s}}}$ and $\delta(t)$ the
delta function. The noise process $N_{K}(t)$ is wide-sense stationary with
zero mean and constant PSD equal to $\frac{1}{4}(N_{0}^{2}+2A^{2}N_{0})$. From
now on, it will be assumed to follow a Gaussian distribution as in [12].

###  III-B Phase trellis description

Let $t~{}=~{}\tau~{}+~{}nT_{s}$, with $0\leq\tau<T_{s}$. Taking into account
the properties of the frequency pulse given in (3), the phase introduced in
(6) can be decomposed as the sum of a time-independent term and a time-
dependent term:

$\displaystyle\Theta_{K}(\tau+nT_{s},\mathbf{a})=\phi_{n}+2\pi
ha_{n}q(\tau)+\varphi_{n}(\tau),$ (9)

with $\phi_{n}=\pi h\displaystyle\sum_{i=0}^{K-1}a_{n-L-i}$ and

$\varphi_{n}(\tau)~{}=~{}2\pi
h\left(\displaystyle\sum_{i=1}^{L-1}(a_{n-i}-a_{n-K-i})q(\tau+iT_{s})-a_{n-K}q(\tau)\right).$
(10)

$\varphi_{n}(\tau)$ represents the time-dependent contribution which
corresponds to the last $L$ memory symbols of both the signal and its delayed
version. The term $\phi_{n}$ represents the time-independent part.
$\varphi_{n}(\tau)$ and $\phi_{n}$ are completely determined by the set of
symbols $(a_{n-i})_{1\leq i\leq L+K-1}$. We can thus define the state
$\Sigma_{n}=[a_{n-L-K+1},...,a_{n-1}]$ of the CPM encoder. Note that there are
$M^{K+L-1}$ different possible states. The symbol modulation corresponds to a
path representing $\Theta_{K}$ in the trellis made of states $\Sigma_{n}$.

###  III-C Maximum likelihood (ML)-based detection

The ML criterion is applied to detect the information symbols from $R_{K}(t)$.
Given the constant amplitude property of CPM, it consists in maximizing the
correlation between $R_{K}(t)$ and all possible realizations of $S_{K}(t)$.
The inner product between $R_{K}(t)$ and $S_{K}(t)$, denoted by
$\Gamma_{N}\tilde{\mathbf{a}})$, is defined as

$\Gamma_{N}(\tilde{\mathbf{a}})=\int_{-\infty}^{NT_{s}}R_{K}(t)S_{K}(t,\tilde{\mathbf{a}})dt,$
(11)

which can be recursively computed:

$\Gamma_{n}(\tilde{\mathbf{a}})=\Gamma_{n-1}(\tilde{\mathbf{a}})+\Lambda_{n}(\tilde{\mathbf{a}})$
(12)

with

$\Lambda_{n}(\tilde{\mathbf{a}})=\int_{nT_{s}}^{(n+1)T_{s}}R_{K}(t)S_{K}(t,\tilde{\mathbf{a}})dt.$
(13)

The Viterbi algorithm is applied on the trellis. At the $n-$th section, it
computes for each state the maximum cumulative metric (12) among all the paths
arriving at this state.

##  IV Proposed delay optimization based on the minimum Euclidean distance
criterion

In this Section, we aim at tuning $K$ to improve the detection error
probability. Let us consider the following error event when $s(t,\mathbf{a})$
is transmitted and $s(t,\tilde{\mathbf{a}})$ is detected and
$\mathbf{a}\neq\tilde{\mathbf{a}}$. Given the ML-based detection criterion and
the independence between $R_{K}$ and $N_{K}$, it means that:

$\int_{0}^{NT_{s}}|R_{K}(t)-S_{K}(t,\tilde{\mathbf{a}})|^{2}dt\leq\int_{0}^{NT_{s}}|R_{K}(t)-S_{K}(t,\mathbf{a})|^{2}dt$
(14)

which can be reformulated as:

$Z_{K}\geq\frac{1}{2}\Delta^{2}_{K}(\mathbf{a},\tilde{\mathbf{a}}),$ (15)

where
$Z_{K}=\int_{0}^{NT_{s}}\mathrm{Re}\left[\left(S_{K}(t,\mathbf{a})-S_{K}(t,\tilde{\mathbf{a}})\right)N^{*}_{K}(t)\right]dt$.
$\Delta_{K}(\mathbf{a},\tilde{\mathbf{a}})=\sqrt{\int_{0}^{NT_{s}}\left|S_{K}(t,\mathbf{a})-S_{K}(t,\tilde{\mathbf{a}})\right|^{2}dt}$
is the Euclidean distance between the two differential signals
$S_{K}(t,\mathbf{a})$ and $S_{K}(t,\tilde{\mathbf{a}})$ corresponding to the
symbol sequences $\mathbf{a}$ and $\tilde{\mathbf{a}}$. $Z_{K}$ has zero mean.
Assuming that $Z_{K}$ is Gaussian, the probability of an error event is given
by

$P_{e}(\mathbf{a};\tilde{\mathbf{a}})=Q\left(\sqrt{\frac{4\varepsilon_{b}}{N_{0}^{2}+2A^{2}N_{0}}d^{2}_{K}(\mathbf{a},\tilde{\mathbf{a}})}\right)$
(16)

where $Q$ is the $Q-$function and where
$d_{K}(\mathbf{a},\tilde{\mathbf{a}})=\frac{\Delta_{K}(\mathbf{a},\tilde{\mathbf{a}})}{\sqrt{2\varepsilon_{b}}}$
is the normalized Euclidean distance, $\varepsilon_{b}$ denoting the average
energy per information bit in the differential symbol sequence. Proceeding as
in [13] (Chapter 2, Paragraph 2.1.2), a union bound on the probability of
error is obtained at reasonably high SNR. The error probability is thus
approximated by

$P_{e}\propto
Q\left(\sqrt{\frac{4\varepsilon_{b}}{N_{0}^{2}+2A^{2}N_{0}}d_{\mathrm{min}}^{2}(K)}\right)$
(17)

where $d_{\mathrm{min}}(K)$ denotes the minimum Euclidean distance between two
differential signals:

$d_{\mathrm{min}}^{2}(K)=\min_{\begin{subarray}{c}\mathbf{a},\mathbf{\tilde{a}}\\\
a_{0}\neq\tilde{a_{0}}\end{subarray}}\left(d^{2}_{K}(\mathbf{a},\tilde{\mathbf{a}})\right)$
(18)

By applying the same reasoning as in [13], we obtain another formulation of
the Euclidean distance:

$d^{2}_{K}(\mathbf{a},\tilde{\mathbf{a}})=\frac{\log_{2}(M)}{T_{s}}\int_{0}^{NT_{s}}[1-\cos\left(\Theta_{K}(t,\mathbf{e})\right)]dt$
(19)

where $\mathbf{e}=\mathbf{a}-\tilde{\mathbf{a}}$ is the so-called difference
symbol sequence.

Finding the minimum Euclidean distance is done by searching over all possible
pairs of sequences $\mathbf{a}$ and $\tilde{\mathbf{a}}$. In practice, these
pairs are those whose respective paths on a phase tree diverge at time 0 and
merge again as soon as possible. Proceeding as in [13], the phase difference
tree is a good method to determine the difference symbol sequences to be
considered and the corresponding pairs of symbol sequences.

For each value of the delay, a corresponding value of the minimum Euclidean
distance $d_{\mathrm{min}}$ is obtained. Since we are looking for minimizing
the error probability, the best choice of the delay is the value that yields
the highest $d_{\mathrm{min}}$.

##  V Numerical results

In this Section, we study different CPM formats. In all cases, we consider
short frames communications (symbol sequence length $N=120$) and a symbol
duration $T_{s}=10^{-4}$ s.

###  V-A Influence of $K$ on the detection performance

We first illustrate the influence of $K$ on the detection performance. We
consider a CPM format with rectangular frequency pulse, $L=3$ and $h=0.75$.
The delay $K$ takes on values in {1,2,3,4}. The Bit Error Rate (BER) is
plotted as a function of $E_{b}/N_{0}$ in Fig. 1. We observe that $K=3$ is the
delay that yields the best BER. A gain of 3 dB is obtained compared to the
receiver with $K=1$ and almost 1 dB compared to the receiver with $K=2$ while
the receiver with $K=4$ exhibits a slight degradation of performance.

Figure 1: BER of differential detection for the CPM scheme 3REC with
modulation modulation index $h=0.75$ for different values of delay $K$

###  V-B Optimization of $K$ from the minimum Euclidean distance criterion

The optimization over $K$ of the Euclidean distance in (19) is run by Monte-
Carlo simulations by considering several possible pairs of sequences yielding
different possible realizations of $\mathbf{e}$ for several CPM families and
several parameters. The optimized value of $K$ is provided in Tables I,II,III
for Raised Cosine, rectangular and Gaussian frequency shaping pulses
respectively, for several modulation indices and several pulse lengths. The
coherence of these optimized delays has also been checked by BER simulations,
but the curves are omitted in this article due to space limitations. Note that
when several values of $K$ provide the best error rate, then the displayed
value is simply the lowest one to reduce the complexity of the decoder.

TABLE I: Optimized values of $K$ for RC CPM  Freq. pulse | Modulation index
---|---
length $L$ | $h=1/3$ | $h=1/2$ | $h=3/4$
$1$ | $K=2$ | $K=2$ | $K=3$
$3$ | $K=3$ | $K=3$ | $K=3$
$5$ | $K=4$ | $K=4$ | $K=4$
TABLE II: Optimized values of $K$ for REC CPM  Freq. pulse | Modulation index
---|---
length $L$ | $h=1/3$ | $h=1/2$ | $h=3/4$
$1$ | $K=2$ | $K=2$ | $K=4$
$3$ | $K=4$ | $K=4$ | $K=3$
$5$ | $K=5$ | $K=5$ | $K=5$
TABLE III: Optimized values of $K$ for GFSK $(BT=0.3)$ Freq. pulse | Modulation index
---|---
length $L$ | $h=1/3$ | $h=1/2$ | $h=3/4$
$3$ | $K=3$ | $K=3$ | $K=4$
$5$ | $K=3$ | $K=3$ | $K=4$

###  V-C Comparison with some state-of-the-art receivers

In Fig. 2, we show a comparison between the optimized differential receiver,
the $K=1$ differential receiver, and also the coherent receiver. This
comparison is performed for 2 different CPM families: GMSK with BT=0.3, and
5RC with $h=0.5$. For GMSK, there is almost $4$dB between the coherent BER and
the $K=1$ differential detection. Using the optimized $K=3$ reduces this gap
by almost $2$dB. For the 5RC CPM, using the optimized $K=4$ delay reduces the
gap to coherent BER from around $6$dB down to $2$dB. Note that the curves for
the coherent and the optimized differential receivers are quasi-parallel which
means that the diversity gain is almost the same and the difference between
the two is mainly in the noise variance which is higher for the differential
receiver.

Figure 2: BER comparison between coherent and differential detection for two
CPM schemes: GMSK with $BT=0.3$, and 5RC with $h=0.5$

###  V-D Comparison in presence of Doppler shift

The differential detector is especially interesting in applications where the
Doppler shift affects the communication. In Fig. 3, a performance comparison
between the differential detector and the coherent one in terms of BER is
illustrated for the rectangular pulse with $h=0.5$ and $L=5$ in presence of a
small Doppler shift. We see a huge degradation of performances for the
coherent detector whereas the differential detector is not affected, whatever
the Doppler shift value.

Figure 3: BER comparison between coherent and differential detection for 5REC
with $h=0.5$ in the presence of a Doppler shift

##  VI Conclusion

In this article, the increase of the delay used in the conventional non-
coherent differential detection of CPM is shown to have an impact on the error
rate. We therefore propose an optimization criterion of this delay based on
the minimum Euclidean distance between two differential signals. This
optimized delay ranges from 2 to 5 symbol periods depending on the considered
CPM format. Simulations confirm the choice of the optimized delay value which
offers a gain from 2 to 4 dB on the error rate performance compared to a
single symbol duration delay. CPM with optimized differential detection may be
an alternative candidate waveform in the perspective of limited-power
Satellite IoT where Doppler shift is an issue.

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