Sergio Da Silva

English

Research Papers in Economics

Autocorrelation and the sum of stochastic
variables
Annibal Figueiredo a Iram Gleria b Raul Matsushita c
Sergio da Silva d;1
aDepartment of Physics, University of Brasilia, Brazil
bDepartment of Physics, Federal University of Alagoas, Brazil
cDepartment of Statistics, University of Brasilia, Brazil
dDepartment of Economics, Federal University of Rio Grande do Sul, Brazil and
National Council for Scientic and Technological Development, Brazil
Abstract
We examine the role of nonlinear autocorrelations in the convergence to the Gaus-
sian equilibrium and put forward an attempt to generalize the central limit theorem.
Our results are illustrated with data coming from the British pound-US dollar rate.
Key words: Central Limit Theorem; L evy distributions; Econophysics.
05.40.+j;02.50.-r
1 Introduction
We have previously suggested that scaling power laws associated with trun-
cated L evy ights (TLFs) [1] can be explained on the basis of particular fea-
tures of autocorrelation in data [2]. Another interesting property of the TLF
is a sluggish convergence to a Gaussian, observed in real data. Here we discuss
the role of nonlinear autocorrelations in the convergence of the process and
employ data from the British pound-US dollar rate to illustrate our point.
The paper is organized as follows. Section 2 presents our novel results. Section
3 exemplies them, and Section 4 concludes.
1 Corresponding author, e-mail:sergiodasilva@angelre.com
Preprint submitted to Elsevier Science 2 April 20042 Autocorrelation and the sum of stochastic variables
Consider the sum of random variables xi, i.e.:
Sn 
n X
i=1
xi
with zero mean for xi. L evy [3] shows that, for reduced variables  x = x=
p
2
(where 2 is the variance of x), the characteristic function (CF)  (z) of a
process with nite second moments can be written as  (z) = e z2(1+w(z))=2,
w(0) = 0. For the CF of xi we can thus write
 i(z) = e
 
i2
2 z2(1+wi(
1=2
i2 z))
For Sn, the CF is
	n(z) = e
 
n2
2 z2(1+
n(
1=2
n2 z))
where n2 is the variance of Sn. The statistical moments of order p of xi
and Sn are ip =< x
p
i > and np =< Sp
n > respectively. Also consider that
np =
P
i ip.
For independent variables it holds true that 	n(z) =  1(z) n(z) . But
this does not hold for autocorrelated processes. Here we focus on a class of
autocorrelated processes for which the CF of the sum variable Sn is such that
	n(z) = Cn(z) 1(z) n(z) (1)
where Cn(z) = 1 for an independent process.
Expanding the CF of xi in series obtains
 i(z) = 1 +
{2
2!
i2z
2 +
{3
3!
i3z
3 +  (2)
We assume that
Cn(z) = e
  z2
2 ( 2Cn2+Wn(z)) = 1 + Cn2z
2 + Cn3z
3 +  (3)
We can do the same for the CF of Sn. Performing the expansion of 	n(z)
gives:
	n(z) = 1 +
{2
2!
n2z
2 +
{3
3!
n3z
3 +  (4)
2Plugging (2), (3) and (4) in (1), and comparing equal order terms we obtain:
Cn2 =  
1
2
(n2   n2);
Cn3 =  
{
2
(n3   n3);
Cn4 =
1
4!
(n4   n4)  
1
2!2!
(n2(n2   n2) + n) (5)
where n =
Pn 1
i=1
Pn
j=i+1 i2j2. If we write Wn(z) = {Wn1z + Wn2z2 + O(z3),
from (3) and (5) one obtains
Wn1 =
1
3
(n3   n3);Wn2 =
1
4
(n2   n2)
2   2Cn4 (6)
Plugging (5) and (6) back in the CF of Sn yields
	n(z) = e
  1
2z2

n2+
Pn
i=1 i2wi(
1=2
i2 z)+Wn(z)

(7)
After writing the CF of the reduced variable as
 	n(z) = e
  1
2z2

1+

(1)
n (z)+

(2)
n (z)

and reminding that
 	n(z) = 	n(
z

1=2
n2
)
one has


(1)
n (z) =
1
n2
n X
i=1
i2wi

(
i2
n2
)
1=2z



(2)
n (z) =
1
n2
Wn
 
z

1=2
n2
!
(8)
Function 
(1)
n matches the one for uncorrelated series, i:e: as n ! 1 it ap-
proaches w(0) = 0 according to the central limit theorem (CLT). Term 
(2)
n (z)
is related to the autocorrelations. It gives precisely the CF of the sum vari-
able, which in turn can be used to obtain the probability distribution function
(PDF) as n ! 1. This generalizes the CLT for autocorrelated processes as
in (1).
3Now we relate 
(2)
n (z) to nonlinear correlation terms, which can be captured
by
< p1p2 pk >n=
n X
i1ik
(< x
p1
i1 x
pk
ik >   < x
p1
i1 >  < x
pk
ik >) (9)
where p1p2 pk are positive integers, and i1 6= i2 6=  6= ik. After writing

(2)
n = 

(2)
n1z{ + 

(2)
n2z2 it can be shown that


(2)
n1 =
1
3
n3   n3

3=2
n2
=
1
3
< 111 >n +3 < 12 >n

3=2
n2
(10)


(2)
n2 =
1
4
(1  
2
n2
2
n2
)  
1
12
n4   n4   6n
2
n2
= (11)
1
4
(1  
2
n2
2
n2
)  
1
12
< 1111 >n +6 < 112 >n +4 < 13 >n +3 < 22 >n
2
n2
where 

(2)
n1 and 

(2)
n2 are functions of third- and fourth-order autocorrelations
respectively.
Due to the presence of nonlinear correlations, 

(2)
n1 and/or 

(2)
n2 may remain
bounded above zero as n becomes larger. From these results it turns out that
the limit distribution may not be a Gaussian. Furthermore, a suitable measure
of the distance of a PDF to the Gaussian can be dened as follows. Given an
arbitrary process with nite variance and moments p =< xp >, let the CF
of the reduced variable be written as
  (z) = e
 z2(1+W(z))=2
For a given , the distance between a given distribution f and the Gaussian
can then be estimated as
D(f;Gauss) =
 Z
 
q
Wr(z)2 + W{(z)2dz (12)
Expression W(z) = Wr(z) + {W{(z) can be expanded in series to give [2]
Wr(z) =  
z2
12
K + O(z
4);
W{(z) =
z
3
Sk + O(z
3)
4where
K 
4
2
2   3
is the kurtosis and
Sk 
3
2
3=2
is the skewness. Thus the leading terms in 
n are the kurtosis and skewness
of the sum variable Sn. After remembering that such quantities are zero for a
Gaussian, our results mean that the distance to the Gaussian is given by how
far K and Sk are from zero, which sounds quite reasonable.
It has to be said that the data analysis is carried out by assuming a sta-
tionary process. This is particularly needed for the second moment to exist.
Stationarity of the process is also implicitly presumed when the correlation
structure between x(t) and x(t+k) is assumed to be the same, regardless of t
and depending only on the lag k. And here it would really make sense average
these values. Thus our methodology considers in particular stationary pro-
cesses with long memory correlation structure. We leave for future research
the issue of what is going on with conditionally heteroscedastic series that
show volatility clusters.
3 Applications
Now we illustrate our approach with data coming from the daily variations
of the British pound-US dollar rate. The data set contains 8033 data points,
covering the time period from 1 April 1971 to 1 September 2003. We take
returns Z rather than raw data as our stochastic variable, i:e:
Zt(t) = Y (t + t)   Y (t)
where Y (t) is a rate at day t. Note that Zt(t)  Sn and t  n.
Fig. 1 shows the curve of (12) for  = 1. It can be seen that the function is
somewhat constrained to some real value which prevents termalization (w(0) =
0) to take place.
Fig. 2(a) presents the kurtosis and Fig. 2(b) the skewness. These are the
leading terms in the expansion of w(z). The curve of an IID process is shown
for comparison. The skewness is clearly constrained to some real value; this
in turn limits 

(2)
n1 and then 
(2)
n = 

(2)
n1z{+

(2)
n2z2. From (8) we conclude that
the system cannot converge to the Gaussian in the time window from 1 until
500 trading days.
Figs. 3(a) and 3(b) present 

(2)
n1 and 

(2)
n2. From (8) and (11) it can be seen
that when 
(2)
n !  6= 0 the limit distribution is not Gaussian. From Figs.
53 one cannot say for sure that this is the case of our example, because we
have stopped at n = 500. However, the fact that 
(2)
n is always dierent from
zero in that time window do provide an explanation for the slow convergence
in terms of nonlinear autocorrelations and the behavior of the kurtosis and
skewness.
4 Conclusions
This paper examines the role of statistical autocorrelations in the conver-
gence to the Gaussian equilibrium by focusing on the characteristic function;
incidentally we generalize the central limit theorem. We explain the slow con-
vergence in terms of both nonlinear autocorrelations and the behavior of the
kurtosis and skewness. Our results are endorsed by exchange rate data from
the pound-dollar daily returns.
5 Acknowledgements
We thank the valuable comments from an anonymous referee. Iram Gleria
acknowledges support from FAPEAL.
6Fig. 1. (a) Distance to Gaussian distribution versus n (in trading days)
Fig. 2. (a) Behavior of the kurtosis for daily returns and an IID process. (b) Behavior
of the skewness for daily returns and an IID process
Fig. 3. (a) Norm of 

(2)
n1 versus n. (b) Norm of 

(2)
n2 versus n
References
[1] R. Mantegna, H. E. Stanley, An Introduction to Econophysics, Correlations and
Complexity in Finance, (Cambridge University Press, Cambridge, 2000).
[2] A. Figueiredo, I. Gleria, R. Matsushita, S. Da Silva, Autocorrelation as a source
of truncated L evy ights in foreign exchange rates,Physica A 323 (2003) 601-
625.
[3] D. Dugu e, Oeuvres de Paul L evy, Vol. III: El ements Al eatoires, (Gauthiers-
Villars, Paris, 1976).
7

An Introduction to Econophysics, Correlations and Complexity in Finance,

Autocorrelation as a source of truncated L evy ights in foreign exchange rates,Physica A 323

Autocorrelation and the Sum of Stochastic Variables

http://128.118.178.162/eps/fin/papers/0405/0405020.pdf

Autocorrelation and the Sum of Stochastic Variables

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