The 2009 Framework defines the joint Dutch national learning outcomes to be attained by medical students after completing their three-year master programme in medicine. The Framework thus helps to guarantee to society at large and to patients in particular that medical graduates who are starting out as practitioners have attained a certain professional level. This level is the aggregate of the physicians’ target profile, physicians’ (sub-) competencies to be achieved and the list of issues relating to illness and health. In addition, the Framework also defines the profile of the bachelor and the learning outcomes of the bachelor programme in medicine

Laan, Roland F. J. M.

Leunissen, Ron R. M.

van Herwaarden, C. L. A.

English

PubMed

R.F.J.M. Laan

R.R.M. Leunissen

C.L.A. van Herwaarden

Springer - Publisher Connector

10
Tijdschrift voor Medisch Onderwijs, maart 2010 | Vol. 29, nr. 1, p. 10-15
Undergraduate Medical Education
Introduction
In February 2007, the Board and the As-
sembly of Deans of Medical Schools of the
Dutch Federation of University Medical
Centres decided to institute a Project
Group to revise the 2001 Medical Educa-
tion Blueprint. The main reasons for un-
dertaking this project were developments
in medical scientific disciplines, such as
biotechnology and genetics, and develop-
ments in medical education, such as the
introduction of the bachelor-master struc-
ture and the ongoing modernization of
postgraduate speciality training pro-
grammes. In revising the 2001 Blueprint
the project group was specifically asked to
establish the attainment levels of both the
bachelor and the master programme in
medicine, to consider whether the Can-
MEDS1 model or a similar competency
profile could also be serviceable in revis-
ing the new Framework for medical edu-
cation and to reconsider the level of detail
of the revised 2001 Blueprint, aiming to
find formulations that are realistic and
testable and that facilitate external ac-
countability for programme content. (It
was decided to change the terminology
from ‘Blueprint’ to ‘Framework’. The term
‘Framework’ was thought to reflect bettter
the nature of the document).
On August 12, 2009, the 2009 Frame-
work for Undergraduate Medical Educa-
tion in the Netherlands was presented to
the Dutch minister for Health. A transla-
tion in English is available from the web-
site of the Dutch Federation of University
Medical Centres (http://www.nfu.nl/file
admin/documents/Raamplan2009engelstal
ige_versie.pdf).
The 2009 framework for undergraduate medical 
education in the Netherlands
R.F.J.M. Laan, R.R.M. Leunissen, C.L.A. van Herwaarden (on behalf of the Project Group)*
Summary
The 2009 Framework defines the joint Dutch national learning outcomes to be attained
by medical students after completing their three-year master programme in medicine.
The Framework thus helps to guarantee to society at large and to patients in particular
that medical graduates who are starting out as practitioners have attained a certain pro-
fessional level. This level is the aggregate of the physicians’ target profile, physicians’
(sub-) competencies to be achieved and the list of issues relating to illness and health. In
addition, the Framework also defines the profile of the bachelor and the learning out-
comes of the bachelor programme in medicine. (Laan RFJM, Leunissen RRM, Herwaar-
den CLA van. The 2009 framework for undergraduate medical education in the Nether-
lands. Netherlands Journal of Medical Education 2010;29(1):10-15)
* The project group consisted of professor C.L.A. van Herwaarden (chairperson), professor R.F.J.M. Laan (project
coordinator), drs. R.R.M. Leunissen (secretary), dr. W.M.C. Mulder, dr. H. de Vries, professor F.G.M. Kroese, pro-
fessor J.H. Bolk, professor B.G.M. van Engelen, professor A.C. Nieuwenhuijzen, professor T.J.M. Helmerhorst,
professor E.A.M Sanders and professor N.S. Klazinga. Advisory members were dr. L. Wigersma (Royal Dutch Med-
ical Association), dr. M.A.G. van den Berg (Clients Council of University Medical Centres), drs. L. Schöffer (Na-
tional Medical Students Platform), dr. R.A.F. de Lind van Wijngaarden (National Interns Platform) and D.I.M.
Hoefnagel LL.M (Ministry of Health, Welfare and Sports).
11
The 2009 framework for undergraduate medical education | R.F.J.M. Laan et al. 
Undergraduate Medical Education
The 2009 Framework consists of nine
chapters:
1. Introduction
2. Explanation
3. Developments in Dutch and European
Legislation
4. Stages, Cycles and Levels in Medical
Education in the Netherlands
5. Master’s Degree Programme in Medi-
cine: Target Profile
6. Master’s Degree Programme in Medi-
cine: Competencies
7. Master’s Degree Programme in Medi-
cine: Issues Relating to Illness and
Health
8. Bachelor’s Degree Programme in Med-
icine: Profile and Learning Outcomes
9. Basic Sciences in the Medical Curricu-
lum
In an appendix the final document also
includes a skills list.
The structure of medical education
in the Netherlands
In 2002, the bachelor-master structure
was introduced in the Dutch system for
higher education, establishing bachelor’s
degree and master’s degree programmes
as independent study programmes. This
change followed in the wake of the sign-
ing of the Bologna Declaration in 1999.
This Declaration, signed by 29 European
countries, aims to promote greater overall
convergence in higher education in Eu-
rope, based on a two-cycle degree struc-
ture. In the transitional period, two-cycle
and one-cycle programmes have been
available simultaneously.
Permission to use the protected title of
physician and qualification to perform
certain actions reserved for physicians are
linked to registration in the Individual
Health Care Professions Act register. Reg-
istration is only open to those who have
completed the degree programme in med-
icine and therefore to those who, follow-
ing the introduction of the bachelor-mas-
ter structure, have obtained the 
Master of Science (MSc) degree on com-
pletion of the master’s degree programme
in medicine.
Important choices in the 2009
framework
The Project Group decided to define stu-
dents’ learning outcomes upon comple-
tion of the master’s degree programme in
terms of competencies. A competency is
defined as the ability to adequately per-
form a professional activity in a specific,
authentic context, using an integrated
body of knowledge, understanding, skills
and professional behaviour. After careful
consideration, the CanMEDS model was
chosen, because, firstly, this model in-
volves an excellent and useful division
into physicians’ roles and competencies in
a variety of professional situations, and
secondly, this model is also used in the
framework of the modernization of post-
graduate specialty training. Together, the
seven roles or competency domains in the
CanMEDS model constitute the target
profile in chapter 5 of the 2009 Frame -
work (Table 1). In chapter 6, the profile is
elaborated in terms of key competencies
and their constituent sub-competencies.
Table 1. Physician’ roles included in the Profile.
1. Medical Expert
2. Communicator
3. Collaborator
4. Manager
5. Health Advocate
6. Scholar
7. Professional
As the Project Group felt that bachelor
students in medicine do not yet operate in
12 Undergraduate Medical Education
an authentic professional setting, the
bachelor’s degree programme is described
in terms of knowledge, skills, and behav-
iour and not competencies. Knowledge
and understanding relate to the basic sci-
ences, the foundation of medicine in the
natural sciences, and aspects of behav-
ioural and social sciences.
The 2001 Blueprint included a list of
clinical conditions which comprised
about 330 clinical conditions in all. The
Project Group wanted to trim down this
number to a more manageable set. Hav-
ing considered the issue at length, the
Project Group concluded that the phrase
‘issues relating to illness and health’ was
more appropriate to the wide-ranging
field in which today’s physicians are oper-
ating, as a fair share of physicians’ work is
not directly patient or disease related. The
list of issues described in chapter 7 has
been subdivided into several categories.
The first and most voluminous one com-
prises the complaints that induce patients
to consult their physician. The next two,
smaller, categories deal with findings
upon physical and additional examina-
tions. The final category comprises a list
of so-called healthcare issues.
The introduction of a list of issues
meant that the list of clinical conditions
that was included in the 2001 Blueprint
was not to return in the 2009 Framework:
on the one hand, the contemplation of
each issue would involve a series of clini-
cal presentations anyway, and, on the
other hand, a list of clinical conditions
would require ceaseless maintenance in
the wake of ongoing developments in
medical disciplines.
In the 1994 Blueprint and the 2001 
Blue print, the domain of knowledge only
came up to a fairly limited extent. The
Project Group felt that the aspect of re-
quired knowledge had been somewhat
eclipsed by lists of problems, clinical con-
ditions and skills. Especially the basic sci-
ences had remained underexposed, de-
spite their importance. Right from the
start, the Project Group wished to give the
basic sciences a more prominent place in
medical education. The model of compe-
tencies selected by the Project Group also
carried the risk of outlining required
knowledge only implicitly. Therefore, in
the 2009 Framework, a separate chapter
is devoted to basic sciences in the medical
curriculum dealing with both the founda-
tion of medicine in the natural sciences
and aspects of the behavioural and social
sciences.
The Appendices to the 2001 Blueprint
included a skills list. Spot-checking at the
eight Dutch medical faculties showed that
all medical faculties still used this list, 
sometimes with minor adjustments. After
discussion in the Project Group, the skills
list from the previous Blueprint, with
minor adjustments, was added to the 
Appendices to the 2009 Framework.
Stages, Cycles and Levels in Medical
Education in the Netherlands
The Project Group was asked to pay spe-
cial attention to finding formulations that
are realistic and testable and that allow
external accountability for medical edu-
cation programme content. This
prompted the Project Group to clearly de-
fine the competencies in terms of levels
that are to be attained.
Medical education in the Netherlands
involves a training sequence with two
stages that medical students pass through
in succession. The first stage of this train-
ing sequence comprises two cycles: a
three-year bachelor’s degree programme
in medicine, followed by a three-year
master’s degree programme in medicine.
In the 
second stage of the training sequence, 
following upon completion of the bache-
The 2009 framework for undergraduate medical education | R.F.J.M. Laan et al. 
13
The 2009 framework for undergraduate medical education | R.F.J.M. Laan et al. 
Undergraduate Medical Education
lor and master programmes, medical
graduates undertake postgraduate spe-
cialist training to become medical special-
ists in primary or secondary healthcare.
Having completed that stage, medical
practitioners then engage in lifelong
learning activities both in practice and in
formal continuing education courses.
The levels linked to the successive
stages can be called starter level, begin-
ning practitioner level and experienced
practitioner level.
After completion of their bachelor pro-
gramme in medicine, medical students
can be characterized as starters. These
students have mastered a basic body of
knowledge and understanding of scien-
tific disciplines that are relevant to their
subsequent professional practice. They
have also acquired a set of basic skills and
are able to show appropriate professional
behaviour in training-related situations.
They are able to apply knowledge, skills
and behaviour in dealing with issues that
involve a relatively low level of complex-
ity.
After completion of the master pro-
gramme in medicine, medical graduates
can be characterized as beginning practi-
tioners. Newly graduated physicians
demonstrate a basic competence in prac-
tice, possess an integrated body of knowl-
edge, skills and professional behaviour
and can handle issues involving higher
levels of complexity. Medical graduates
are capable of performing independent
consultations but only perform these
under supervision during postgraduate
medical training.
After a number of subsequent years of
further schooling, training and experience
in medical specialist training in the disci-
pline of their choice, physicians operate
independently without supervision. They
are then experienced practitioners or 
medical specialists. An element of inde-
pendent practice is participation in inter-
professional coaching activities and life-
long continuing-education courses.
In the 2009 Framework, the levels of
proficiency to be achieved in the bachelor
and master programmes in medicine are
defined in a five-level structure (Table 2).
In discussing the levels that must be at-
tained, the Project Group also had to con-
sider the level of difficulty of the issues that
are presented to students. This level of dif-
ficulty is chiefly determined by the pres-
ence of contextual factors. On the one
hand, these are medical factors, such as
typical and atypical presentations or co-
morbidity problems; on the other hand,
these are contextual factors from the psy-
chosocial model. The level of difficulty is
then codetermined by the availability of
standard solutions to a problem and the
applicability of protocols and guidelines.
The level of difficulty also lies in the degree
to which different physician roles must be 
integrated in dealing with the issue in
question. Graduates in medicine must be
able to handle issues involving varying −
including higher − levels of difficulty, with
the proviso that the degree of required 
supervision increases with the degree of
difficulty of the case presented. In this re-
spect, there is one sub-competency incor-
porated in the role of medical expert that is
of particular importance: students’ ability
to recognize and name the personal limits
to their knowledge and skills and to decide
in time if, and, if so, when a third party
needs to be called in, including their 
supervisor.
The 2009 Framework states that gradu-
ates in medicine must have mastered all
issues related to illness and health that
are included in the list at least at level II.
Some of the issues, depending on the
specifics of individual training pro-
grammes, will be included in additional
competency testing at higher levels. The
14 Undergraduate Medical Education
The 2009 framework for undergraduate medical education | R.F.J.M. Laan et al. 
Table 3. Examples of (sub)competencies and levels assigned per physician’s role.
Physician’s role Competency (Medical graduates are able to …) Level
Medical expert take a medical history V
draft a plan for diagnosis and / or therapy V
implement a diagnostic and / or therapeutic plan IV
Communicator guide the chronically or incurably ill and guide patients in palliative care III
Collaborator show an understanding of group processes and their impact on the 
healthcare process V
Manager apply a quality model in practice if this is relevant III
Health Advocate apply knowledge of epidemiology in practice at the individual patient level V
Scholar draft clinical protocols for the benefit of patient care III
Professional accept responsibility for their own professional performance V
terminate a doctor-patient relationship in a professional way IV
Table 2. Levels defined up to the beginning practitioner level.
Level I A Students have knowledge and understanding of those branches of science that are relevant
to medicine.
B Students demonstrate in standardized situations that they have skills that are relevant to
medicine.
C Students demonstrate that they possess the basic skills required for professional behaviour.
Level II Students use integrated knowledge, skills, and professional behaviour in an adequate* approach
to the issues of illness and health incorporated in this Framework. They demonstrate such com-
petence in context-rich training situations.
Level III Students adequately* perform the professional activities defined in the physicians’ competen-
cies in purpose-designed training situations and/or in simulated professional situations.
Level IV Students adequately* and independently perform the professional activities defined in the
physicians’ competencies in authentic professional situations, having received prior case-spe-
cific instruction and being intensively supervised by an experienced practitioner.
Level V Students adequately* and independently perform the professional activities defined in the
physicians’ competencies. An experienced practitioner is immediately available on stand-by and
always provides supervision after the event.
* The performance of professional activities is considered adequate if such performance is in line with the current state of science and the
prevailing standards and guidelines of the profession. Wherever the Framework mentions activities that, as a rule, span longer periods
of time, as in guiding the chronically ill, students demonstrate their competency by adequately performing partial activities.
15
The 2009 framework for undergraduate medical education | R.F.J.M. Laan et al. 
Undergraduate Medical Education
formulations of sub-competencies specify
the levels (III, IV, or V) that have to be at-
tained. To illustrate the assignment of lev-
els some examples are given in Table 3.
Conclusion
The 2009 Framework differs substantially
from the 2001 Blueprint. Major changes
are the definition of competencies, the in-
dication of the levels of competency to be
attained and the introduction of a list of
issues related to health and disease. The
knowledge domains of basic sciences rel-
evant to medicine are described in more
detail. The 2009 Framework also includes
the attainment levels of the bachelor pro-
gramme in medicine. The 2009 Frame-
work will be used as the basis for accred-
itation and for a revision of the legislation
that outlines the educational require-
ments of physicians.
References
1. Jason R. Frank, MD MA (Ed) FRCPC. The Can-
MEDS 2005 Physician Competency Framework:
Better standards. Better physicians. Better care.
The Royal Colege of Physicians and Surgeons of
Canada; 2005.
The authors:
R.F.J.M. Laan, MD PhD, is professor of rheumatology
and medical education, Radboud University Medical
Centre, Institute for Medical Education,Nijmegen, the
Netherlands.
R.R.M. Leunissen, MSc, is senior staff member, Rad-
boud University Medical Centre, Institute for Medical
Education, Nijmegen, the Netherlands.
C.L.A. van Herwaarden, MD PhD, is professor of pul-
monology and former dean of the Radboud University
Medical Centre, Nijmegen, the Netherlands.
Correspondence
Prof. dr. R.F.J.M. Laan, Radboud University Medical
Centre, P.O. Box 9101, 6500 HB Nijmegen, the Nether-
lands. E-mail: r.laan@reuma.umcn.nl 
No potential conflict of interest relevant to this article was reporte


The 2009 framework for undergraduate medical education in the Netherlands

Berbee, H.C.P.

Hollander, W.T.F. den

Leiden University Scholary Publications

Tail triviality for sums of stationary random variables

Berbee, Henri

Hollander, Frank

NARCIS 

Berbee, H.C.P. (Henri)

Hollander, W.T.F. (Frank) den

CWI's Institutional Repository

Centrum voor Wiskunde en lnformatica Centr8 for Mathematics and Computer Science H.C.P. Berbee, W.Th.F. den Hollander Tail triviality for sums of stationary random variables Department of Mathematical Statistics Report MS-R8810 June Bibl10theek Centrurn voor Wi!3t~unde en lnformetica Amst&rdl!iW The Centre for Mathematics and Computer Science is a research institute of the Stichting Mathematisch Centrum, which was founded on February 11, 1946, as a nonprofit institution aim-ing at the promotion of mathematics, computer science, and their applications. It is sponsored by the Dutch Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.0.). Copyright © Stichting Mathematisch Centrum, Amsterdam • Tail Triviality for Sums of Stationary Random Variables H.C.P. Berbee Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands W.Th.F. den Hollander Delft University of Technology Faculty of Mathematics and Informatics Julianalaan 132, 2628 BL Delft, The Netherlands We study tail o-fields and loss of memory associated with sums of stationary integer-valued random vari-ables. An application concerns convergence in distribution of interarrival times in 0-1 sequences. AMS 1980 Subject Classification: Primary 60F20, secondary 60G10, 28020, 28005, 60J15. Keywords and phrases: Tail o-field for sums, entropy, strongly aperiodic, random walk with stationary incre-ments, loss of memory, zero-two theorem, coupling, interarrival times. Report MS-R8810 Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands 1. Statement of results Let X 1 , X 2 , • •• be a strictly stationary sequence of integer-valued random variables and let S1 , S2 , • • • be the sums (n;:::: 1). If the increments are independent, then according to the Hewitt-Savage zero-one law (Breiman (1968)) the tail a-field n a(Sn : n;:::: N) N>O is trivial. Without the independence this need no longer be true. The main question that will be addressed in this paper is what can be said about the tail behavior of the sums in this more general setting. It is natural to extend (Xn)n~1 to a double-sided process X := (Xn)nEZ and to extend also the sums to a double-sided process (Sn) n E 'JI., by requiring that (1.1) The process X is defined on the probability space (0, J", P) with 0 = Z'Jl,, X is the identity on 0, J" is the product a-field generated by the discrete topology on 'JI.,, and P is a T-invariant probability measure with T the shift defined by (T X)n = Xn+ 1 (n E Z). We shall be interested in the following tail a-fields associated with (Sn): 9:,;v := n a(Sn - Sm : m :S -M, n;:::: N). M,N~O Note that 9;;-v = nN r-N 900' so 9;;-v is a T-invariant a-field included in the double tail a-field 900 . The following example shows that 9;,:-v and 900 may be different. EXAMPLE (1.2): Let (Yn) be i.i.d. with P(Yn = -1) = P(Yn = 1) = i and let Xn = Yn - Yn-1(n E 'JI.,). Then Sn - Sm = Xm+i + ··· + Xn = Yn - Ym and so 9;,:-v is trivial by the Kolmogorov zero-one law (Breiman (1968)). On the other hand, Sn= X 1 + ··· + Xn = Yn - Yo and hence limN-oo N- 1 L:=l Sn= -Y0 a.s. Thus Y0 is 900 -measurable, so 900 is not trivial. In fact, 900 = a(Y0 ) a.s. Let loo := n a(Xn : m::; -M, n > N) M,N~O be the double tail a-field associated with X. Since a(Sn - Sm : m :S -M,n;:::: N) = a(SN - S_u) V a(Xn: m :S -M,n > N) for all M, N;:::: 0, it is clear that 9;,:-v ~ 100 . In section 2 we prove: 2 THEOREM ( 1.3). Let (Xn) be stationary integer-valued and let (Sn) be given by (1.1). If X 0 has finite entropy, then G!:;" = 100 a.s. COROLLARY( 1.4). If X 0 has finite entropy, then G!::" is trivial iff 100 is trivial. As example (1.2) shows, to obtain triviality of the larger tail a-field Goo we need to assume more than triviality of 100 • Also, it will not be enough to impose stronger mixing conditions on X. In section 3 we prove the following zero-two theorem which will be seen to be the key to our study of G00 • THEOREM( 1.5). Let (Xn) be stationary, ergodic and real-valued and let (Sn) be given by (1.1). For every real h, either II p ( s N E · 11; ) - p ( s N + h E · 11; ) II= 2 for all N ~ 1 a.s. or else lim II P(SN E ·11;) - P(SN + h E ·11;) II= 0 a.s. N-oo Here 1; := a(Xn : n ~ {l, ... , N}) and 11 · 11 denotes total variation. For i.i.d. sequences the corresponding zero-two theorem was proved by Stam (1967) and by Ornstein (1969), while Berbee {1979) gave a proof for mixing sequences. Most zero-two theorems in the literature relate to Markov processes. Theorem(l.5) leads us in a natural way to associate a group H with (Sn) as follows: H := { h E R : there exists N ~ 1 such that with positive probability P(SN E ·11;) and P(SN + h E ·11;) have mass in common}. By theorem (1.5), the set H is a group. For (Xn) integer-valued, we say that (Sn) is strongly aperiodic if H = '/J,. Theorem (1.6) below gives sufficient conditions for triviality of 9.00 as well as of the right tail a-field Let G! := n u(Sn: n ~ N). N ?_O 1:, := n a(Xn : n > N) N ?_O and note that G! ::J l;I,. In section 4 we prove: THEOREM( 1.6). Let (Xn) be stationary integer-valued and let (Sn) be given by (1.1). Assume that (Sn) is strongly aperiodic. If l;I, is trivial then G! is trivial. If 100 is trivial then Goo is trivial. Note that in example (1.2) triviality of 100 holds because (Xn) is one-dependent but strong aperiodicity fails because SN = YN - Y0 while Y0 and YN are 1;-measurable, so H = {O}. ,,, 3 As an application of theorem (1.6) we consider a stationary ergodic 0-1 sequence ( Xn). Let T1 , T2 , • • • be the random positive times at which Xn assumes the value 1: T1 = inf{n 2: 1: Xn = 1}, (1.7) Tk = inf { n > Tk _ 1 : Xn = 1} ( k > 1). By stationarity and ergodicity, if P(X1 = 1) > 0 then Tk < oo a.s. for all k 2: 1. We call Tk+i - Tk the kth interarrival time. In section 5 we prove: COROLLARY( 1. 7). Let (Xn) be a stationary 0-1 sequence with P(X1 = 1) > 0 and let (Sn) be given by (1.1} and (Tk) by (1.7). If .9! is trivial, then for any integer t 2: 0 lim P(n+ 1 - Tk > t) = P(T2 - Ti > tlX1 = 1). k-+ 00 Corollary (1.7), together with theorem (1.6), extends a result by Janson (1984) form-dependent sequences. Janson (1984) contains an example of a I-dependent sequence with non-converging interarrival times. Corollary (1.4) has recently been applied by Den Hollander (1987) in a paper on mixing properties for random walk in random scenery. 2. Proof of theorem (1.3) We start by recalling a few properties of entropy (see e.g. Smorodinsky (1971) or Parry (1981)). Let Z be a discrete random variable on our probability space (0, J,P) taking values in a countable set. The entropy of Z is defined as H(Z) := - L P(Z = z) log P(Z = z) z (0 log 0 = 0). For a sub-a-field Jl C 1 the conditional entropy of Z given A is defined as H(ZjJl) := -EL P(Z = zjJl) log P(Z = zjA). z For two sub-a-fields A, B C J we denote by Jl V B the smallest a-field containing A and B. The following properties will be used below: (2.1) (2.2) (2.3) (2.4) (2.5) H(ZJA) :::; H(ZIB) if B c A, H(ZIA) :S H(Z), H(Z1, Z2 IA) = H(Z1 IA) + H(Z2 IA V a(Zi)) = H(Z2 IA) + H(Z1 IA V a(Z2)), H(Z1 + Z2) :S H(Zi) + H(Z2), H(Z) :Slog j{z: P(Z = z) > O}I. 4 PROOF OF THEOREM( 1.3): We introduce the notation Sm,n :=Sn - Sm = Xm+i + · · · + Xn (m,n E :Ii; m < n), 1N := u(Xn : n::; -N,n > N), 9N := u(Sm,n: m::; -N,n 2: N) = u(S-N,N) V 1N. Note that loo =nw?.olN andg;;" =nN"?.o9N· By the martingale convergence theorem {Smorodinsky (1971)) H(Xo I loo) = lim H(Xo llN ), N-+ oo H(Xol9!:") = lim H(XoJ9N ). N-+ oo Here the monotonicity of 1N and 9N is used together with (2.2) and the assumption H(Xo) < oo. As a first step towards proving g;,;:v = 100 a.s. we claim that (2.6) To get (2.6) we note that (2.7) H(Xm l9N)::; H(Xol9N+lml) for all m and N. This follows from stationarity, (2.1) and the inclusion T-m 9N+lml C gN. Now fix two positive integers K, L and consider the following quantity: H((xjK )1il:'.'::L-1,s-LK.LK 11LK ), i.e. the conditional entropy given 1LK of the sum S_ LK ,LK and of the Xn with lnl ::; (L - l)K but sampled over gaps of size K. The trick is to write this out in two different ways, using (2.3), I. H((XiK )lil:'.'::L-1 llLK) + H(S-LK,LK llLK V u((XiK )Iii:'.':: L-i)), II. H(S-LK ,LK llLK) + H((XiK )Iii:'.':: L-1 llLK V u(S_ LK ,LK )), to derive a lower bound for I and an upper bound for II, and thus to get an inequality between these two bounds which can then be exploited to prove (2.6). Indeed, if we iterate the first term of I using (2.3), then we get 2L -1 terms each of which is bounded below by H(Xo llK) because of stationarity and (2.1). Hence, by ignoring the second term of I we get I 2: (2L - l)H(Xo JlK ). In II, on the other hand, the first term is bounded above by H ( s _ L K ,L K ) because of (2.2), while the second term equals H((XiK )lil:'.'::L-il9LK) and is bounded above by Elil~L- iH(XiK 19LK) via (2.1) and (2.3). Together with (2.7) this gives II::; H(S-LK ,LK) + (2L - l)H(Xo 192LK ). Since 1=11 we thus have H(Xo llK) ::; (2L - It 1 H(S_LK ,LK) + H(Xo J92LK ). Next let L ~ oo while keeping K fixed, and use the following lemma which we shall prove below. ,, 5 LEMMA (2.8). If H(Xo) < oo, then H(S-N,N) = o(N). Finally we let K ~ oo and we end up with H(Xo l.100 ) :S H(X0 19~" ). But this implies (2.6) since the reverse inequality holds by (2.1) and .100 C 9~". Now, (2.6) is only the first step in proving 9~" = .100 a.s. The next step is to prove that for any positive integer M (2.9) This can be done in exactly the same way as above. Instead of sampling 2L - 1 single variables Xn over gaps of size K, we must now sample 2L - 1 blocks of variables of fixed length 2M + 1 separated by gaps of size K and then again take the limit L ~ oo followed by K ~ oo. The key inequality is the obvious modification of (2.7). These steps are left for the reader to verify. The assertion now follows from (2.9) and the following lemma which we shall prove below. LEMMA(2.10). Assume H(X0 ) < oo. Let A, B C .1 be two sub-a-fields such that Ac B. If then A = B a.s. This completes the proof of theorem (1.3). Ill PROOF OF LEMMA(2.8): We follow a standard truncation argument. Fix a positive integer M and define y = { 0 n X n if lXnl :'.SM if IXnl > M (n E 1l). Since Xn - Yn can at most take 2M + 1 distinct values, the last term is at most log(2N(2M + 1)) by (2.5). Letting N ~ oo while keeping M fixed, we get limsup (2Nt 1 H(S-N,N) :'.S H(Yo). N-+ oo But H(Yo) can be made arbitrarily small by letting M ~ oo, because H(X0 ) < oo. Ill! ,, 6 PROOF OF LEMMA (2.10): Let EE 8. Select Em E a((Xn)lnl::;m) such that (2.11) lim P(Em ilE) = 0. m-+ oo From (2.3}, for all m ~ 0 and by assumption this equals Because A C B there must be term by term equality, so H(lEm IA) = H(lEm IB). But (2.11) implies limm ..... 00 H(lEm IB) = H(lE IB) = 0, hence limm.-+oo H(lEm IA) = H(lE IA) = 0. This in turn implies P(EIA) = 0 or 1 a.s., and so E E A a.s. Thus B c A. Ill 3. Proof of theorem(l.5) We follow the proof of the contraction lemma in Berbee (1986), which goes back to an idea of Bradley (1983). Berbee (1979) contains a proof for mixing sequences using coupling techniques. The ergodic theoretic proof below is considerably shorter but also less transparant. Let us write P(X E ·IZ) for the conditional distribution of X given a(Z). The following facts will be used below: (3.1) E II P(X E ·IZ) - P(Y E ·IZ) II= II P((X, Z) E ·) - P((Y, Z) E ·) 11, (3.2) II P(f (X) E ·) - P(f (Y) E ·) II::; II P(X E ·) - P(Y E ·) 11, where X and Y are random variables taking values in the same measurable space, Z is any other random variable and f is any measurable function. PROOF OF THEOREM ( 1. 5): Let P1 and P2 be the two probability measures on 1 defined by where uh : n -> n is the map We compare the restrictions of P1 and P2 to the nested sequence of a-fields 9M.N := a(Sm,n: m::; -M,n ~ N), M,N ~ 0. 7 Since 9;.;v = nM ,N ~ 0 9M ,N, it follows via the martingale convergence theorem that (3.3) Because 9;_;v is T-invariant and P is T-invariant and ergodic on J", it is immediate that P 1 is T-invariant and ergodic on 9;.;v. The same is true, however, for P2 • This follows from the commutation rule UhT- l = r- l Uh Which holds on 9M ,N for any M, N ~ 1. Indeed, by this rule we have while ergodicity follows because if A is T-invariant so is Uh A by this rule. By an elementary application of the ergodic theorem we get from these ergodic properties that P1 and P2 on 9;_;v are either identical or mutually singular and thus the right hand side of (3.3) equals 0 or 2. The left hand side of (3.3) depends only on N + M because of stationarity. If we take M = 0, note that 9o.N = J";, V a(SN) and use (3.1), then (3.3) becomes equivalent to limN ..... 00 E Z N = 0 or 2 with Because SN+ 1 is (SN, XN +i)-measurable, we get from (3.1) and (3.2) that ZN is a reverse submartingale, i.e. ZN+1 ~ E(ZN 11";+ 1 ). Since 0 ~ ZN ~ 2, it follows that ZN converges a.s. with limit either 0 or 2. In the latter case we must have ZN = 2 a.s. for all N ~ l because EZN is decreasing in N. I 4. Proof of theorem (1.6) We need the following technical corollary of theorem (1.5). Once this has been proved theorem (1.6) will follow easily. COROLLARY( 4.1). Let (Xn) be stationary, ergodic and integer-valued and let (Sn) be given by (1.1). If (Sn) is strongly aperiodic and if U and V are integer-valued (Sn)-measurable random variables, then for every K ~ 0 lim II P((Sn + u,s_n + V)n~N E ·l(Sn)lnl::;K) N-+ oo - P((Sn,S-n)n~N E ·l(Sn)lnJ::;K) II= 0 a.s. PROOF OF COROLLARY ( 4. 1): Because U is integer-valued we can approximate U by random variables Um which are (Sn)lnl::;m-measurable such that limm-+oo P(Um i= U) = 0. Hence it is enough to prove the assertion for U measurable with respect to (Sn)JnJ::;m for every fixed m ~ 0, and the same for V. By theorem (1.5) and property (3.1) we have that for any integer h 8 By stationarity this is the same as lim llP((Sn+m - Sm+ h)n>N E ·l(Sn+m - Sm)n<o) N-+oo - --P((Sn+m - Sm)n?:N E ·J(Sn+m - Sm)n~o) II= 0 a.s. for every m. Now, (Sn+m - Sm)n~o and (Sn)n~m determine each other when m 2: 0 (because S0 = 0). Hence we may replace h by any (Sn)n~m- measurable random variable, and since moreover for all N 2: -m also (S_n + V)n?:N is (Sn)n~m-measurable we obtain )~~ IJP((Sn + U, S_n + V)n?:N E ·l(Sn)n~m) - P((Sn' s_n + V)n?:N E ·l(Sn )n~ m) II= 0 a.s. By (3.1) and (3.2) it follows that for all K::; m )~~ llP((Sn + U, S_n + V)n?:N E ·J(Sn)lnl~K) -P((Sn,S-n + V)n?:N E ·l(Sn)lnl~K) JI= 0 a.s. The same type of argument shows that V may be replaced by 0 and that the role of U and V may be interchanged. The assertion now follows by using the triangle inequality. I In the proof of theorem (1.6) below we use the notation {3(X, Y) to denote the dependence coefficient of random variables X and Y defined as {3(X, Y) := E II P(Y E ·IX) - P(Y E ·) II . We shall need the inequality (4.2) {3(X, Y) ::; 2E II P(Y E ·IX) - P(Z E ·) II which holds for any random variable Z taking values in the same measurable space as Y. This follows from the inequality E II P(Y E ·) - P(Z E ·) II::; E II P(Y E ·IX) - P(Z E ·) II which is easily deduced from (3.1) and (3.2). PROOF OF THEOREM (1.6): Let K ::; m. Apply corollary (4.1) with U = -Sm and V = -S-m, and combine this with E JIP((Sn - Sm,S-n - S_m)n?:N E ·l(Sn)lnl~K) - P((Sn - Sm' s_n - s_m )n?:N E ·) II ::=; {3((Xn)lnl~K' (Xn)lnl?:m) =: Cm,K' 9 where we use (3.1) and (3.2). This yields (4.3) N-+ oo - P((Sn' s_n)n?N E ·J(Sn)lnl:5K) lls Cm,K. Now use (4.2) and (4.3) to obtain (4.4) limsup,B((Sn)lnl:5K, (Sn)lnl?N) S 2c:m,K · N-+ oo But if 100 is trivial then limm.-+ 00 cm ,K = 0 for every fixed K (by a standard martingale argument) and so By the same token, this is equivalent to triviality of 900 • Thus the second assertion is proved. The proof of triviality of 9! from triviality of 1:, is an easy adaptation of the above proof. I 5. Proof of Corollary ( 1. 7) The proof of Corollary (1.7) uses coupling. Let (Z~)n?i and (z;)n?i be two arbitrary processes defined on probability spaces (1l1, 1i, pi) and (i"l2, 1 2 , P 2 ), re-spectively. A coupling is a construction of these processes on a common probability space (oi x 0 2 ,li x 1 2 ,Pi· 2 ) such that pi. 2 1.t• =pi and P 1 •2 1.t• = P 2 • Given two coupled processes taking values in the same measurable space (ni, 1i) = (02 , 1 2 ), we say that the coupling is successful if pi. 2 (Z~ = z; for all n sufficiently large) = 1. The key to corollary (1.7) is the following maximal coupling theorem of Goldstein (1979). MAXIMAL COUPLING THEOREM. Let (Z~)n?i and (z;)n?l be two arbitrary pro-cesses taking values in the same Borel space. There exists a successful coupling pi. 2 iff pi and P 2 agree on the tail a-field. The idea is to apply this theorem to two copies ( S~) and ( s;) of our sum process (Sn) obtained by conditioning on {Xi = O} and {Xi = 1}, respectively. That is pi ((S~) E ·) = P((Sn) E · JX1 = 0), P2 ((S~) E ·) = P((Sn) E ·JXi = 1). If 9! is trivial, then obviously and so by Goldstein's theorem there exists a successful coupling such that pi.2 (S~ = S~ for all n sufficiently large) = 1. 10 Now let (Tf h ~ 1 and (T,; h ~ 1 denote the corresponding sequences of random positive times at which X~ and X~ assume the value 1. Then we have for any t ~ O (5.1) where IP(Tk+l - n > t!X1 = o) - P(Tk+l -Tk > tlX1 = 1)1 = IP1 (Tf+ 1 -Tf > t) - P 2 (r:+ 1 - r: > t)I ::; p1.2 (Tf < Ti.2 or T: < T1,2) r1•2 = inf{m ~ 1: S! = s;, for all n ~ m} denotes the coupling time. Since P(limk-.oo Tk = oo) = 1, we have P 1 •2 (limk-.oo Tf = limk-. oo Tf = oo) = 1 and so (5.2) lim p1.2(Tf < T1,2 or T; < T1,2) = p1.2(r1,2 = oo) = 0. k-+ 00 Conditioning on {X1 =1} is natural because the process (Tk+ 1 -Tk)k~i is stationary given {X1 = 1}, i.e. (5.3) P(Tk+ 1 - Tk > tjX1 = 1) is independent of k. For this well-known fact and its implications we refer to Kakutani (1943) and Kac (1947). If we now combine (5.1-3), then the assertion follows. I ACKNOWLEDGMENT: The authors would like to acknowledge discussions with J. Aaronson and M. Smorodinsky (Tel Aviv) and B. Weiss (Jerusalem). The proof of theorem (1.3) as given in section 2 was suggested by B. Weiss and improves an original proof by the authors. 11 References Berbee, H.C.P. (1979). Random walks with stationary increments and renewal theory. Math. Centre Tract 112. Berbee, H.C.P. (1986). Periodicity and absolute regularity. Israel J. Math. 55, 289-304. Bradley, R.C. (1983). Absolute regularity and functions of Markov chains. Stoch. Process. Appl. 14, 67-77. Breiman, L. (1968) Probability. Addison-Wesley, Reading. Goldstein, S. (1979). Maximal coupling. Z. Wahrscheinlichkeitstheorie verw. Gebiete 46, 193-204. Hollander, W. Th.F. den ( 1988). Mixing properties for random walk in random scenery. Ann. Probab. Janson, S. (1984). Runs in m-dependent sequences. Ann.Probab. 12, 805-818. Kakutani, S. (1943). Induced measure preserving transformations. Proc. Japan Acad. Ser. A Math. Sci. 19, 635-641. Kac, M. (1947). On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53, 1002-1010. Ornstein, D.S. (1969). Random walks I. Trans. Amer. Math. Soc. 138, 1-43. Parry, W. (1981). Topics in Ergodic Theory. Cambridge Tracts in Mathematics 75. Cambridge University Press. Smorodinsky, M. (1971). Ergodic Theory, Entropy. Lecture Notes in Mathematics 214. Springer. Stam, A.J. (1967). On shifting iterated convolutions. Compositio Mathematica 17, 268-280, and 18, 201-228. Henry Berbee Center for Mathematics and Computer Science Kruislaan 413, 1098 SJ Amsterdam The Netherlands. Frank den Hollander Delft University of Technology Faculty of Mathematics and Informatics Julianalaan 132, 2628 BL Delft The Netherlands. 12 

We study tail $\sigma$-fields and loss of memory associated with sums of stationary integer-valued random variables. An application concerns convergence in distribution of interarrival times in zero-one sequences

Den Hollander, F.

Tail Triviality for Sums of Stationary Random Variables

The 2009 Framework defines the joint Dutch national learning outcomes to be attained by medical students after completing their three-year master programme in medicine. The Framework thus helps to guarantee to society at large and to patients in particular that medical graduates who are starting out as practitioners have attained a certain professional level. This level is the aggregate of the physicians' target profile, physicians' (sub-) competencies to be achieved and the list of issues relating to illness and health. In addition, the Framework also defines the profile of the bachelor and the learning outcomes of the bachelor programme in medicine

Laan, RF

Leunissen, RR

van Herwaarden, CL

German Medical Science

[english] The 2009 Framework defines the joint Dutch national learning outcomes to be attained by medical students after completing their three-year master programme in medicine. The Framework thus helps to guarantee to society at large and to patients in particular that medical graduates who are starting out as practitioners have attained a certain professional level. This level is the aggregate of the physicians’ target profile, physicians’ (sub-) competencies to be achieved and the list of issues relating to illness and health. In addition, the Framework also defines the profile of the bachelor and the learning outcomes of the bachelor programme in medicine

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The 2009 framework for undergraduate medical education in the Netherlands []

Better standards. Better physicians. Better care. Canada: The Royal College of Physicians and Surgeons of Canada;

URN: urn:nbn:de:0183-zma0006726 This article is freely available from http://www.egms.de/en/journals/zma/2010-27/zma000672.shtml Received: 2009-09-21 Revised: 2009-11-06 Accepted:

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The 2009 framework for undergraduate medical education in the Netherlands

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