What Is the Difference between Gamma and Gaussian Distributions? ()
1. Introduction
1.1. Problem
We first introduce some notations. Denote Gamma distribution function as
(1)
for 
and
, where 
is the Gamma function, i.e.,
Assume 
for
. The density of chisquare distributed random variable 
with 
degrees of freedom is
It is well-known that the random variable 
can be interpreted by 
with 
independent and identically distributed (i.i.d.) random variables 
where 
denotes the standard Gaussian distribution. The mean and variance of 
is respectively
Then, by simple change of variable we find
(2)
On the other side, by the Berry-Esseen inequality to
, it is easy to find a bound
such that
(3)
where 
is the standard Gaussian distribution function, i.e.,
(4)
Then, by Equations (2) and (3) it follows
(5)
which describes the distance between Gamma and Gaussian distributions. The purpose of this paper is to derive asymptotic sharper bound 
in Equation (5), which much improves the constant 
by directly using Berry-Esseen inequality. The main framework of analysis is based on Gil-Pelaez formula (essentially equivalent to Levy inversion formula), which represents distribution function of a random variable by its characteristic function.
The main result of this paper is as following.
Theorem 1.1 A relation of the Gamma distribution (1) and Gaussian distribution (4) is given by
(6)
where
with 
and 
for any
.
Clearly, 
as
. Thus, the asymptotical bound is
as
. To check the tightness of the limit value of
, we plot in Figure 1 the multiplication
for
, where the straight line is the limit value
. From this experiment it seems that 
is the best constant. The tendency of the theoretical formula 
is plotted for 
in Figure 2, which also shows the tendency to the limit value
. The slow trend is due to that some upper bounds formulated over interval 
have been weakly estimated, e.g., the third and fourth terms of
.
1.2. Comparison to the Bound Derived by Berry-Esseen Inequality
Let 
be a sequence of independent identically distributed random variables with EX1 = 0 
and finite third absolute moment
. Denote
By classic Berry-Esseen inequality, there exists a finite positive number 
such that
(7)
The best upper bound 
is found in [1] in 2009. The bound is improved in [2] at some angle in a slight different form as
(8)
with
The inequality (8) will be sharper than Equation (7) for
.
Now let us derive the constant 
in (5) by applying Berry-Esseen inequality to
. It is difficult to calculate the exact value of third absolute moment of the random variable
. Thus, it is approximated as
by using Matlab to integrate over interval 
divided equivalently 100,000 subinterval for its half value.
By Equation (7) with 
we have
and by Equation (8) we have
Hence, the best constant 
in Equation (5) by applying Berry-Esseen inequality is
. Obviously, the limit bound
found in this paper for chi-square distribution is much better.
The technical reason is that the Berry-Esseen inequality deals with general i.i.d. random sequences without exact information of the distribution.
2. Proof of Main Result
Before to prove the main result, we first list a few lemmas and introduce some facts of characteristic function theory.
2.1. Some Lemmas
Lemma 2.1 For a complex number 
satisfying
, 
Proof First show that
By Taylor’s expansion and noting
, we have
Together with
the assertion follows.
Lemma 2.2 For a real number 
satisfying
,
where 
is the imaginary unit and
Clearly,
Proof. By Taylor expansion for complex function, for 
we have
where 
is shown above. By further noting the two alternating real series above, it follows the upper bound.
We cite below a well-known inequality [3] as a lemma.
Lemma 2.3 The tail probability of the standard normal distribution satisfies
for
.
2.2. Characteristic Function
Let us recall, see e.g., [4], the definition and some basic facts of characteristic function (CF), which provides another way to describe the distribution function of a random variable. The characteristic function of a random variable 
is defined by
where 
is the imaginary unit, and 
is the argument of the function. Clearly, the CF for random variable 
with real numbers 
and 
is
Another basic quality is
for 
with 
and 
independent to each other.
It is well-known that the CF of standard Gaussian 
is
(9)
and the CF of chi-square distributed variable 
is
Thus, the CF for 
is
(10)
The CF is actually an inverse Fourier transformation of density function. Therefore, distribution function can be expressed by CF directly, e.g., Levy inversion formula. We use another slightly simpler formula. For a univariate random variable
, if 
is a continuity point of its distribution
, then
(11)
which is called Gil-Pelaez formula, see, e.g., page 168 of [4].
2.3. Proof of Main Result
We are now in a position to prove the main result.
Proof of Theorem 1.1 First analyze CF of 
given by Equation (10). Denote
. For
, i.e., 
, by Lemma 2.2,
(12)
where
Clearly,
To make sure 
for some
denote
. Then, it is easy to see that
(13)
for
. Hence, by Equations (12) and (13) and Lemma 2.1,
(14)
for
.
Now let us consider the difference between 
and
, i.e., the CF (9) of Gaussian distribution, over the interval
. By Equation (14)
Note that
it follows
(15)
Similarly,
(16)
Below let us analyze the residual integrals over the interval
. By Lemma 2.3,
(17)
Similarly,
(18)
It is somewhat difficult to analyze the residual integral over 
for
. We divide it into two subintervals as following:
where
.
Observe that 
decreases on interval 
and 
for
, we have
where
The fact 
is used in above formula. Thus,
(19)
For the other interval
, we proceed as
(20)
By Equations (19) and (20)
(21)
Similarly,
(22)
By Equation (15), Equation (17), Equation (21) and Equation (16), Equation (18), Equation (22)
where
In view of Formula (11) , the formula to be proved follows directly.