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Teacher’s Corner
Understanding Convergence Concepts: A VisualMinded and
Graphical SimulationBased Approach
Pierre L
AFAYE DE
M
ICHEAUX
and Benoit L
IQUET
This article describes the difﬁcult concepts of convergence in
probability, convergence almost surely, convergence in law, and
convergence in
r
th mean using a visualminded and graphical
simulationbased approach. For this purpose, each probability
of events is approximated by a frequency. An
R
package that
reproduces all of the experiments cited in this article is available
in CRAN. See the online Supplement for details.
KEY WORDS: Convergence almost surely; Convergence in
law; Convergence in probability; Convergence in
r
th mean; Dy
namic graphics; Monte Carlo simulation; R language; Visual
ization.
1. INTRODUCTION
Most departments of statistics teach at least one course on the
difﬁcult concepts of convergence in probability (
P
), almost sure
convergence (
a.s.
), convergence in law (
L
), and convergence in
r
th mean (
r
) at the graduate level (see Sethuraman
1995
). In
deed, as pointed out by Bryce et al. (
2001
), “statistical theory is
an important part of the curriculum, and is particularly impor
tant for students headed for graduate school.” Such knowledge
is prescribed by learned statistics societies (e.g., the Accredi
tation of Statisticians by the Statistical Society of Canada and
Curriculum Guidelines for Undergraduate Programs in Statisti
cal Science by the American Statistical Association). The main
textbooks (e.g., Chung
1974
; Billingsley
1986
; Ferguson
1996
;
Lehmann
2001
; Serﬂing
2002
) devote about 15 pages to deﬁn
ing these convergence concepts and their interrelations. Very
often, these concepts are provided as deﬁnitions, and students
are exposed only to some basic properties and to the universal
implications displayed in Figure
1
.
The aim of this article is to clarify these convergence con
cepts for master’s students in mathematics and statistics, and
Pierre Lafaye de Micheaux is Associate Professor, Départment Mathéma
tiques et de Statistique, Universitè de Montréal, Canada (Email:
[email protected]
dms.umontreal.ca
). Benoit Liquet is Associate Professor, INSERM U897,
ISPED, University of Bordeaux 2, Bordeaux, France (Email:
[email protected]
isped.ubordeaux2.fr
). The authors thank R. Drouilhet, M. Lejeune, and the ref
erees, the editor, and the associate editor for their many helpful comments.
Figure 1.
Universally valid implications of the four classical modes
of convergence. (See Ferguson
1996
for proofs.)
also to provide software useful for learning these concepts.
Each convergence notion provides an essential foundation for
further work. For example, convergence in law is used to
obtain asymptotic conﬁdence intervals and hypothesis tests
using the central limit theorem; convergence in probability
is used to obtain the limiting distribution of the
Z
test re
placing an unknown variance with its estimate (through Slut
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 Fall '09
 Zivot
 Probability theory, Convergence, FN, Convergence of random variables, The American Statistician

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