We present a new limit theorem for random means: if the sample size is not deterministic but has a negative binomial or geometric distribution, the limit distribution of the normalised random mean is a t-distribution with degrees of freedom depending on the shape parameter of the negative binomial distribution. Thus the limit distribution exhibits exhibits heavy tails, whereas limit laws for random sums do not achieve this unless the summands have innite variance. The limit law may help explain several empirical regularities. We consider two such examples: rst, a simple model is used to explain why city size growth rates are approximately t-distributed. Second, a random averaging argument can account for the heavy tails of high-frequency returns. Our empirical investigations demonstrate that these predictions are borne out by the data.convergence, t-distribution, limit theorem

Christian Schluter

Mark Trede

English

Research Papers in Economics

 
Weak convergence to the t-distribution 
Christian Schluter
† and Mark Trede
‡ 
21/2011 
† DEFI, Aix-Marseille Université, France and University of Southampton, UK 
‡ Department of Economics, University of Münster, Germany 
wissen•leben 
 WWU Münster Weak convergence to the t-distribution
Christian Schluter y
DEFI, Aix-Marseille Universit e and University of Southampton
Mark Tredez
Westf alische Wilhelms-Universit at M unster
October 7, 2011
Abstract
We present a new limit theorem for random means: if the sample size is not deter-
ministic but has a negative binomial or geometric distribution, the limit distribution of
the normalised random mean is a t-distribution with degrees of freedom depending on
the shape parameter of the negative binomial distribution. Thus the limit distribution
exhibits exhibits heavy tails, whereas limit laws for random sums do not achieve this
unless the summands have innite variance.
The limit law may help explain several empirical regularities. We consider two
such examples: rst, a simple model is used to explain why city size growth rates are
approximately t-distributed. Second, a random averaging argument can account for the
heavy tails of high-frequency returns. Our empirical investigations demonstrate that
these predictions are borne out by the data.
We would like to thank Krzysztof Podg orski and Gerold Alsmeyer for fruitful discussions.
yEmail: C.Schluter@soton.ac.uk. DEFI, Aix-Marseille Universit e: 14 Avenue Jules Ferry, 13621 Aix-
en-Provence, France. University of Southampton: Southampton, SO17 1BJ, UK.
zCorresponding author. Email: mark.trede@uni-muenster.de. Institute for Econometrics, Cen-
ter for Quantitative Economics (CQE) and Center for Nonlinear Science (CeNoS), Westf alische Wilhelms-
Universit at M unster, Am Stadtgraben 9, 48143 M unster, Germany, Tel. +49 251 8325006, Fax. +49 251
8322012.
11 Introduction
Limit theorems are of great interest since the limit distribution does not depend on the
original sampling distribution. The most famous limit law is, of course, the central limit
theorem. As early as 1809 Laplace has shown that the standardised sum of independent ran-
dom variables converges in distribution to the Gaussian distribution (Stigler, 1986, p. 136),
and since then the theorem has been considerably generalised (see e.g. Davidson (1994)).
While the sample size in these two limit theorems is non-random, sample sizes in many
situations of interest are random. This has led to the discovery of limit laws for random
sums. A leading case is geometric sums where the sample size is geometrically distributed,
and independent of the summands. In particular, if the random summands are iid, non-
negative and with nite mean, then the (non-randomly) normalised sum converges weakly
to an exponential variate (R enyi (1957)). If the summands are symmetric random variables
with nte variance, the limit distribution is the Laplace distribution (Kotz, Kozubowski
and Podg orski (2001)). With no assumptions on the distribution of the summands, the
limit is a geometric stable law (e.g. Gnedenko and Korolev (1996)), and in the case of the
non-random sum a stable law (e.g. Gnedenko and Kolmogorov (1968)).
This paper considers limit laws for random means. We show that the asymptotic distri-
bution of the random mean does not belong to the same distributional family as the random
sum. In particular, if sample sizes are drawn from a negative binomial or a geometrical dis-
tribution, the random mean converges weakly to a t-distribution as the expected sample
size goes to innity. The number of degrees of freedom depends on the shape parameter of
the negative binomial distribution, and thus equals two for the geometric distribution.
This result is of signicant interest since many distributions of empirical interest exhibit
heavy tails, i.e. tails that decay slowly like power functions, which lead to higher moments
failing to exist. The t-distribution is, of course, heavy-tailed. Hence we have a limit law
that exhibits heavy tails but which does not require the summands to have a heavy-tailed
distribution. By contrast, existing limit laws for random sums do not exhibit heavy tails
unless the summands themselves have innite variance (Gnedenko and Korolev (1996)).
Thus, when applied to specic contexts, the new limit law might suggest the mechanism
which generates the heavy-tailed limit without presupposing it for the summands. We
illustrate this in two distinct applications, one taken from urban economics, the other from
nance. The rst application considers the distribution of the growth rates of cities. Each
city consists of economic sectors which grow randomly. Our limit law implies that growth
rates of cities should be approximately distributed as a t2 variate. We verify this prediction
using data for city sizes in Germany. The second example considers high frequency stock
returns and focusses on the number of transactions of a stock. The limit law implies that
the one-period stock return is approximately a t2 variate. This prediction is veried using
high frequency data from the German DAX stock index.
The paper is organised as follows. In section 2 we state the limit theorem and its proof.
Section 3 presents two applications. Section 4 concludes.
2 Limit distributions of random means
Let X1;X2;::: be a sequence of i.i.d. random variables with E(Xk) = 0 and nite variance
V ar(Xk) = 2 < 1 for k 2 N. To keep the notation simple, we assume without loss of
generality that 2 = 1. Let  have a negative binomial distribution, independent of the Xi,
2with parameters r > 0 and 0 < p < 1. The probability function is
P ( = k) =

k + r   1
k

pr (1   p)
k (1)
for k = 0;1;2;:::. This is one of the two leading cases for count models, it accommodates
the overdispersion typically observed in count data (which the Poisson model cannot), and
the special case of the geometric distribution (r = 1) is the leading case in the literature on
random sums. The expectation of  is E () = r(1   p)=p, so, for r given, E() ! 1 as
p ! 0. The random mean of a random number  of draws is1
 Xp =
1

 X
k=1
Xk: (2)
The subindex p indicates the parameter p of the negative binomial distribution. Then, by
standard arguments, p
   Xp ! U  N (0;1): (3)
However, if the normalising factor is changed from the random variable
p
 to the deter-
ministic constant
p
E () =
p
r(1   p)=p, the limiting distribution ceases to be Gaussian.
In particular, the normalised mean converges in distribution to a t-distribution with 2r
degrees of freedom:
Theorem 1 Under the assumptions given above, as p ! 0,
r
r
p
  Xp ! T  t2r: (4)
Before we prove the theorem, we show the following:
Lemma 2 As p ! 0, the random variates p and
p
  Xp are asymptotically independent.
Proof of Lemma 2. Consider the joint survival function
P
 
p > t;
p
  Xp > x

= P

 >
t
p
;
p
  Xp > x

=
X
n>t=p
P
 
 = n;
p
n  Xp > x

:
Since  and  Xp are independent, the joint probability can be factored as
X
n>t=p
P
 
 = n;
p
n  Xp > x

=
X
n>t=p
P ( = n)P
 p
n  Xp > x

:
Since n ! 1 as p ! 0, the central limit theorem applies to
p
n  Xp and
X
n>t=p
P ( = n)P
 p
n  Xp  x

!
X
n>t=p
P ( = n)  (1   (x))
= (1   (x))
X
n>t=p
P ( = n)
= (1   (x))  P (p > t):
1In case of  = 0, the mean is dened as 0. Alternatively, one may use a dierent parametrisation of the
negative binomial distribution with support fr;r + 1;r + 2;:::g and expectation r=p. The following results
do not depend on the parametrisation.
3Hence, the joint probability can be factorized asymptotically.
Proof of Theorem 1. Rewrite
r
r
p
  Xp =
r
2r
2p

p
  Xp:
According to (3)
p
  Xp is asymptotically N(0;1). In addition, for p ! 0, the random vari-
able 2p converges in distribution to V where V  2
2r. According to lemma 2,
p
2r=(2p)
and
p
  Xp are asymptotically independent. Hence, as p ! 0
r
r
p
  Xp !
r
2r
V
U
where U  N(0;1) and V  2
2r are independent. Since
q
2r
V U  t2r we conclude that the
normalised mean converges in distribution to a t-distribution with 2r degrees of freedom.
Corollary 3 Let  be geometrically distributed with parameter p. The normalised mean
converges to a t-distribution with 2 degrees of freedom,
r
1
p
  Xp ! T  t2:
The corollary follows immediately from the observation that the geometric distribution
equals the negative binomial distribution with parameter r = 1.
We end this Section with a few remarks. First, some of the assumptions have only been
made to clarify the exposition, and can be relaxed easily. If E(X1) =  and V ar(X1) = 2
then
p

   Xp   

= ! U  N(0;1) and
r
r
p

 Xp   

! T  t2r: (5)
Second, it is not necessary to assume that X1;X2;::: are i.i.d. Any version of the central
limit theorem with non-identical and/or dependent observations can be applied as long as
the sample size is independent of the X1;X2;:::. Third, the distributional assumptions
concerning the random sample size can also be relaxed. As long as 2p converges weakly to
a 2-distribution as p ! 0, the results still hold. In particular,  could be a shifted negative
binomial or geometric distribution. Fourth, the theorem can also be transferred to sample
statistics other than the mean. If a sample statistic can be estimated by a
p
n-consistent
estimator (e.g. quantiles), the theorem continues to hold.
3 Applications
We present two distinct applications, one taken from urban economics, the other from
nance, in order to demonstrate how the weak convergence theorem can be applied to
explain striking empirical regularities. In particular, both example show how heavy-tailed
distributions can emerge, and we demonstrate that the theoretical prediction is born out
by the data.
43.1 The growth rates of cities
The size distribution of cities is often found to be of the Pareto type, an empirical regularity
known as Zipf's law (see e.g. Gabaix (1999) or C ordoba (2008)). Accordingly, assume that
the size Xi of city i, measured as the number of inhabitants, follows a Pareto distribution
with scale parameter x0 > 0 and shape parameter  > 0,
P (Xi > x) =

x
0x  for x > x0
0 for x  x0
(6)
Next, assume that there are Si economic sectors in city i and that the number of sectors Si
depends on the city size in the following way
Si = C + lnXi: (7)
A simple model consistent with (7) is Christaller's central place theory Christaller (1966),
and empirical evidence in support of (7) is reported in (Mori et al., 2008, Figure 6). Equa-
tions (6) and (7) imply that Si has a shifted exponential distribution with cdf
P (Si  s) =

1   B exp
 
 
s

for s > C + lnx0
0 else
where B = x
0eC=.
The discrete counterpart of the (shifted) exponential distribution is a (shifted) geometri-
cal distribution. Hence, the limit theorem can be used to explain the unconditional growth
rate distribution of city sizes. In particular, if sector j grows at the random rate rj with
E(rj) =  and V ar(rj) = 2 < 1, then the average growth rate Ri of city i is the random
mean
Ri =
1
Si
Si X
j=1
rj:
According to Theorem 1,
 
p2 1=2 (Ri   )  t2; (8)
i.e. the city size growth rate distribution should follow approximately the t2-distribution.
The exact Pareto assumption of equation (6) has been made mainly for expositional
simplicity. This could be generalised to a domain of attraction assumption, so that (6)
holds for suciently large x, and x0 can be replace by any slowly varying function. Thus
we can accommodate weaker versions of Zipf's law (see also Schluter and Trede (2011) for
further discussions).
3.1.1 Empirical investigation
We consider the distribution of city growth rates using data for German cities (or, more
precisely, \Gemeinde"). The data are provided by the German Federal Statistical Oce,
and very accurate given the legal obligation of citizens to register with their respective
town halls. For a detailed description of the data, see Schluter and Trede (2011). In this
illustration, we select two representative years, 1995 and 1996, in which the number of cities
is n = 14551. Our earlier work conrms that the upper right tail of the city size distribution
is indeed heavy and thus of the Pareto type. The estimates of shape parameter  (or, more
precisely, of the extremal index) range between 1.24 and 1.31 depending on the estimation
method, and did not change much over the period under investigation, 1995-2006.
5Turning from the size data to the annual growth rate of cities, the mean growth rate is
1.02%. The proportion of observations with small absolute values is large, the 0.05-quantile
is  3:33% and the 0.95-quantile is 6:4%. However, the range is large with a smallest growth
rate of  57% and a largest one of +100%: Figure 1 shows the histogram of the growth rates.
−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15
0
5
1
0
1
5
2
0
2
5
City size growth rates 1995−1996
growth rate
Figure 1: Histogram of the city size growth rates and a tted t-distribution with two degrees
of freedom.
If the preceding hypotheses are met, growth rate are distributed as a t2 variate by (8).
We depict the density in Figure 1, having estimated the location parameter  and scale
parameter  in equation (8) by maximum likelihood, the estimates being ^  = 0:00521
(standard error se = 0:000175) and ^  = 0:01626 (se = 0:000190). It is evident that the
limit law ts the actual data closely.
If the degrees of freedom parameter is estimated along the location and scale parameters,
the maximum likelihood estimated degrees of freedom is b df = 2:005 with a standard error
of 0.043. Hence, the hypothesis that the number of degrees of freedom equals 2 cannot be
rejected at usual signicance levels.
We conclude by noting one implication of these results for estimation and inference.
Averaging a random number of sectoral growth rates with nite variances results, in the
limit, in the overall growth rate distribution having innite variance. This implies that
statistical procedures relying on nite second moments are invalid.
3.2 High-frequency stock returns
Stock returns distributions are well-known to exhibit heavy tails (see the discussion in e.g.
Schluter and Trede (2008)). We proceed to propose a transaction-based model which, in
conjunction with our limit theorem, yields a heavy-tailed returns distribution.
Let Kt;0 denote a stock price at the begin to period t = 1;2;:::. Suppose that there is a
random number t of market transactions during period t. For simplicity the transactions
are equally spaced over period t, and the random length of the sub-periods is t = 1=t.
Let each transaction cause a \periodised" return rt;i where rt;i is i.i.d. with E(rt;i) = 0 and
V ar(rt;i) = 2 < 1, i = 1;:::;t. The stock price after the jth transaction in period t is
Kt;j = Kt;0  exp
Pj
i=1 rt;it

, hence
Kt+1;0 = Kt;0  exp
 
t X
i=1
rt;it
!
:
6Note that the number of transactions t has an impact on the return distribution of each
single transaction. The more transactions occur, the lower their individual variances. This
may be thought of as an liquidity eect, as more transactions tend to make the price
movements smoother. Given the number of transactions, the variance over one entire period
is always 2.
The unconditional distribution of the one-period returns
Rt+1 = ln(Kt+1;0=Kt;0)
=
1
t
t X
i=1
rt;i
is a random mean of a random number of returns. If the number of transactions t has a
negative binomial distribution with shape parameter r (or geometric distribution, r = 1),
then  
p2 1=2 (Ri   )  t2r; (9)
as p ! 0. The limit distribution is thus heavy-tailed even though V ar(rt;i) = 2 < 1;
however, the distribution has a nite variance if the degrees of freedom exceed 2.
3.2.1 Empirical investigation
We consider high frequency returns of the German stock index DAX. The data are provided
by the EUREX database, the sample includes 2,612,189 observations (at the 1 second sam-
pling frequency) from 2nd January until 28th April 2006, observed for 90 trading days over
18 weeks. The daily trading phase starts at 9 am and ends at 5.45 pm. In order to prevent
noise due to the market's micro structure we compute 5 min returns. Overnight returns are
discarded, as are 84 zero returns. The remaining number of observations is 8,541. Figure 2
depicts the histogram of the 5 min returns.
−0.004 −0.002 0.000 0.002 0.004
0
2
0
0
4
0
0
6
0
0
8
0
0
High−frequency returns
return
Figure 2: Histogram of the 5 minute returns of the DAX index and density of a tted
t-distribution
The gure also depicts the tted density of a scaled t2r-distribution, based on equation
(9), with location and scale parameters ( and ) as well as the degrees of freedom (r) all
estimated by maximum likelihood. The estimates are ^  = 1:4146  10 5 (with standard
error se = 0:8498  10 5), ^  = 3:9716  10 4 (se = 0:1658  10 4), and ^ r = 2:4090
7(se = 0:0584). The limit law ts the actual data very closely. We conclude by observing that
the estimated degrees of freedom are signicantly larger than 2, indicating that although
the return distribution is heavy-tailed its variance is nite.
4 Conclusion
If the sample size is not deterministic but has a negative binomial or geometric distribution,
the limit distribution of the normalised random mean is a t-distribution with a number of
degrees of freedom depending on the shape parameter of the negative binomial distribution.
Hence the limit distributions of the random mean and the random sum belong to two
dierent families.
The limit distribution exhibits exhibits heavy tails without presupposing it for the sum-
mands, and may thus help explain several empirical regularities. We have considered two
such examples: rst, a simple model is used to explain why city size growth rates are ap-
proximately t-distributed. Second, a random averaging argument can account for the heavy
tails of high-frequency returns.
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Weak convergence to the t-distribution

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