Dependence in probability, analysis and number
theory: The mathematical work of Walter Philipp(1936 – 2006)
István Berkes and Herold Dehling

Introduction
On July 19, 2006, Professor Walter Philipp passed away during a hike in the Austrian Alps as a result of a sudden heart attack. By the time of his death, Walter
Philipp had been for almost 40 years on the faculty of the University of Illinois at
Urbana-Champaign, for the last couple of years as professor emeritus. He is survived by his wife Ariane and his four children, Petra, Robert, Anthony and André.
Walter Philipp is sorely missed by his family, but also by his many colleagues, coauthors and former students all over the world, to whom he was a loyal and caring
friend for a long time, in some cases for several decades.
Walter Philipp was born on December 14, 1936 in Vienna, Austria, where hegrew up and lived for most of the first 30 years of his life. He studied mathematics
and physics at the University of Vienna, where he obtained his Ph.D. in 1960 and hishabilitation in 1967, both in mathematics. From 1961 until 1967 he was scientificassistant at the University of Vienna. During this period, Walter Philipp spent two
years as a postdoc in the US, at the University of Montana in Missoula and at the
University of Illinois. In the fall of 1967 he joined the faculty of the University of
Illinois at Urbana-Champaign, where he would stay for the rest of his life. Initially,
Walter Philipp was on the faculty of the Mathematics Department, but in 1984 hejoined the newly created Department of Statistics at the University of Illinois. From1990 until 1995 he was chairman of this Department. While on sabbatical leavefrom the University of Illinois, Walter spent longer periods at the University of North
Carolina at Chapel Hill, at MIT, at Tufts University, at the University of Göttingen
and at Imperial College, London.
Walter Philipp received numerous recognitions for his work. Most outstandingof these was his election to membership in the Austrian Academy of Sciences.
As a student and postdoc at the University of Vienna, Walter Philipp worked
under the guidance of Professor Edmund Hlawka, founder of the famous postwar
1

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I. Berkes and H. Dehling

Austrian school of analysis and number theory. Other former students of Professor
Hlawka include Johann Cigler, Harald Niederreiter, Wolfgang M. Schmidt, Fritz
Schweiger and Robert Tichy. It was here that Walter Philipp got in touch with theclassical topics from analysis and number theory that would guide a large part ofhis research for the rest of his life. Uniform distribution, discrepancy of sequences,
number-theoretic transformations associated with various expansions of real numbers, additive number-theoretic functions, Diophantine approximation, lacunary series became recurrent themes in Walter Philipp’s subsequent work. He studied these
themes using techniques from probability theory, e.g. for mixing processes, martingales and empirical processes. He contributed greatly to the development of several
branches of probability theory and solved much investigated, difficult problems inanalysis and number theory with the help of the tools he developed.

Themes from analysis and number theory
A full understanding and appreciation of Walter Philipp’s research requires the background of some topics from analysis and number theory. In what follows we shallbriefly introduce the topics that were recurrent themes in Walter Philipp’s work.
Uniform distribution mod 1. A sequence (xn )n≥1 of real numbers is called uniformly distributed mod 1 if for all x ∈ [0, 1]
lim

1

N→∞ N

#{1 ≤ i ≤ N : {xi } ≤ x} = x,where {x} denotes the fractional part of x. More generally, a sequence (xn )n≥1 ofvectors in Rd is called uniformly distributed mod 1 if
lim

1

N→∞ N

#{1 ≤ i ≤ N : {xi } ∈ A} = λ(A)for all rectangles A ⊂ [0, 1]d , where λ denotes Lebesgue measure and for x ∈ Rd thesymbol {x} is interpreted coordinatewise. The famous Weyl criterion (1916) statesthat a sequence (xn )n≥1 in Rd is uniformly distributed mod 1 iff
1 N 2πihk,xn i
=0
∑e
N→∞ N
n=1
lim

for all k ∈ Zd \ {0}. An immediate consequence is the uniform distribution of thesequence (n α)n≥1 ⊂ R for irrational α, but with suitable methods this leads to theuniform distribution of many other sequences in one and higher dimensions, forexample, a large class of sequences of the type {nk α} for increasing sequences(nk )k≥1 of positive integers.

The mathematical work of Walter Philipp

3Discrepancies. Given a sequence (xn )n≥1 of real numbers uniformly distributed
mod 1, one can study the discrepancy
DN :=1#{1 ≤ i ≤ N : a ≤ {xi } < b} − (b − a) .
N0≤a<b<1
sup

A Glivenko–Cantelli type argument shows that DN → 0, and one may then ask forthe exact rate of convergence of DN to 0. In the case of a d-dimensional sequence(xn )n≥1 , one can define the discrepancy with respect to a class C of subsets of [0, 1]dby1
DN (C ) := sup #{1 ≤ i ≤ N : {xi } ∈ A} − λ(A) .
A∈C N
In this case, the additional issue of the choice of suitable class C arises. Note that
DN (C ) → 0 does not necessarily hold even if the sequence (xn )n≥1 is uniformly
distributed.
θ-adic expansion of real numbers. Let θ > 1, not necessarily an integer. Everyreal number ω ∈ [0, 1) can be written as an infinite series∞ω = ∑ θ−n xnn=1

where 0 ≤ xn < θ are integers. Clearly xn = [T n ω] where the transformation T :[0, 1] → [0, 1] is defined by T ω := {θ ω} and [x] denotes the integer part of x. Also,−k
T n ω = ∑∞k=1 θ xn+k . The basic asymptotic question here is the distribution of digits
in the expansion, for example, one can ask if the limits of relative frequencies
1 N∑ 1{xn =k}
N→∞ N
n=1

Fk := lim

exist. If θ is an integer, the transformation T is ergodic and has Lebesgue measure
as an invariant measure, thus the ergodic theorem implies that the limits Fk exist andare equal to θ−1 for k = 0, 1, . . . , θ − 1 and almost every real number ω. With theusual terminology, almost every real number is normal with respect to base θ. This
statement, proved first by Borel in 1909, is a typical result in the metric theory of
numbers, stating that a certain property holds for almost every real number, without
specifying the exceptional set. In fact, to determine whether a given number ω
is normal is a very hard problem and only√very few normal numbers are known
explicitly. We do not know, for example, if 2, e or π are normal in any base. Notethat the normality of ω in base a ∈ N is equivalent to the statement that the sequence(an ω)n≥1 is uniformly distributed mod 1.
If θ is not an integer, the transformation T is still ergodic, but Lebesgue measure
is not an invariant measure. It is known from work of Alfréd Rényi (1957) that there

4

I. Berkes and H. Dehling

exists a unique invariant measure µ which is equivalent to Lebesgue measure. In
this case, the sequence (θn ω)n≥1 is no longer uniformly distributed, but
lim

1

N→∞ N

#{1 ≤ n ≤ N : T n ω ≤ x} = µ([0, x]),for almost every ω.
Continued fraction expansion. Every real number ω ∈ (0, 1] can be expressed as
an infinite continued fractionω=1
x1 + x + 1 1
2= [x1 , x2 , . . .],x3 +...

where xi ∈ {1, 2, . . .}. Closely related is the transformation T : (0, 1] → (0, 1], defined byT ω := {1/ω}.

The n-th digit in the continued fraction expansion is given by xn = [T n ω]. As in thecase of the θ-adic expansion, T n ω can be written as a function of xn+1 , xn+2 , . . . byT n ω = [xn+1 , xn+2 , . . .].
T is an ergodic transformation with invariant measure given by the Gauss measureµ((a, b]) =

1
log 2

Z b

1
dx.
a 1+x

As a consequence, the asymptotic distribution of the sequence (T n ω)n≥1 is governedby the Gauss measure, i.e.
11#{1 ≤ n ≤ N : T n ω ≤ x} =
N→∞ N
log 2
lim

Z x

1
dt.
1
+
t
0

From here, one obtains that the integer k occurs in the continued fraction expank
k+1− log k+2
, a factsion of a random number ω with relative frequency log1 2 log k+1
already conjectured by Gauss.
Additive functions in number theory. A function f : N −→ R is called additiveiff (mn) = f (m) + f (n),whenever m and n are coprimes. A simple example of an additive function is ω(n),the number of prime divisors of n. Hardy and Ramanujan studied this function andshowed that ω(n) is of the order log log n. More precisely, if PN denotes the uniform
distribution on the first N integers, thenω(n)PN n ≤ N :− 1 ≥ ε −→ 0
log log N

The mathematical work of Walter Philipp

5

for any ε > 0. Turán observed that the Hardy–Ramanujan theorem is a simple consequence of an easily verifiable inequality for the second moment of ω(n) and Chebychev’s inequality and thus initiated the subject of probabilistic number theory. The
Hardy–Ramanujan theorem was later strengthened by Erdős and Kac, who proveda central limit theorem


Z x
2
ω(n) − loglog N
PN n ≤ N : √≤ x −→
e−y /2 dy.
log log N−∞
Diophantine approximation. Let f be a positive, continuous, nonincreasing function on R+ . By a classical result of Khinchin, for almost all real α the inequality|qα − p| ≤

f (q)
q

(1)

has infinitely many or only finitely many solutions in integers p, q according asf (k)∑ k diverges or converges. Probabilistically, this is the Borel–Cantelli lemma forcertain dependent events; the main difficulty is to deal with the dependence in thef (k)case ∑ k = ∞. Khinchin’s result has been generalized and improved upon in many
directions by Cassels, W. M. Schmidt, Erdős, LeVeque, Szüsz, Gallagher, Ennola,
Billingsley and many others. The simplest proof depends on a connection of the
problem with continued fraction theory. Call a fraction p/q a best approximation
to α if it minimizes |q′ α − p′ | over fractions p′ /q′ with denominator q′ not exceeding q. The successive best approximations to α are the convergents pn (α)/qn (α),
n = 1, 2, . . . of its continued fraction expansion and thus the value of |qα − p| for then-th in the series of best approximation is dn (α) = |qn (α)α − pn (α)|. Thus the studyof number of solutions of (1) is equivalent to the study of growth of the sequencedn (α). Khinchin proved that1
π2
log dn (α) → −nlog2
for almost every α.
Lacunary sequences. Probabilistic methods play an important role in harmonic
analysis and there is a profound connection between probability theory and trigonometric series. From a purely probabilistic point of view, the trigonometric system (cos 2πnx, sin 2πnx)n≥1 is a sequence of orthogonal (i.e. uncorrelated) randomvariables over [0, 1], which, however, are strongly dependent. For example, ther.v.’s sin 2πnx have the same distribution, but their partial sums ∑n≤N sin 2πnx remain bounded for any fixed x, a behavior very different from that of i.i.d. randomvariables. However, it has been known for a long time that for rapidly increasing(nk )k≥1 , the sequences (sin 2πnk x)k≥1 and (cos 2πnk x)k≥1 behave like sequences of
independent random variables. For example, Salem and Zygmund (1947) proved

6

I. Berkes and H. Dehlingthat if (nk )k≥1 satisfies the Hadamard gap condition nk+1 /nk ≥ q > 1 (k = 1, 2, . . .),then (sin 2πnk x)k≥1 obeys the central limit theorem, i.e.
lim λ{x ∈ (0, 1) : ∑ sin 2πnk x < t

N→∞

k≤N

Z t
p
2
N/2} = (2π)−1/2
e−u /2 du.

(2)

−∞

Erdős (1962) proved that the CLT√(2) remains valid if the Hadamard gap condition
is weakened to nk+1 /nk ≥ 1 + ck / k, ck → ∞ and this result is sharp. Similar results
hold if sin nk x is replaced by f (nk x), where f is a real measurable function satisfying
f (x + 1) = f (x),Z 1

f (x)dx = 0,

0

Z 1
0

f 2 (x)dx < ∞.

For example, if f ∈ Lip (α), α > 0 and nk+1 /nk → ∞, then the CLT and LIL holdfor f (nk x). (Takahashi (1961, 1963)). If we assume only nk+1 /nk ≥ q > 1, both
the CLT and LIL can fail, a fact discovered by Erdős and Fortet. Gaposhkin (1970)
showed that the validity of the CLT for f (nk x) is closely connected with the numberof solutions (k, l) of the Diophantine equation
ank + bnl = c,l ≤ k, l ≤ N.

Walter Philipp’s work
Given that Walter Philipp published close to 90 research papers, it is impossible tomention every single result he ever obtained. We will instead try to focus on themain lines of his research. Philipp’s earliest work, originating from his Ph.D. thesis,
concerns uniform distribution mod 1. Weyl (1916) had shown that the sequence(an ω)n≥1 is uniformly distributed modulo 1 for almost all ω ∈ [0, 1], if (an )n≥1 ⊂ R
is a sequence of positive numbers satisfying an+1 − an ≥ δ > 0 (n = 1, 2, . . .) forsome δ > 0. Walter Philipp studied this question for d-dimensional sequences,i.e. for the sequence (An ω)n≥1 where ω ∈ Rd and (An )n≥1 is a sequence of ddimensional matrices satisfying some growth condition. As a corollary, uniform
distribution of the sequence (An ω)n≥1 for almost all ω ∈ Rd can be obtained, provided the matrix A has all eigenvalues strictly larger than 1.
In 1967 Walter Philipp published the first in a series of papers on the asymptotic
behavior of weakly dependent stochastic processes. This topic would dominate his
research interests for the next 15 years and keep his close attention for the restof his life. In the second half of the 1960s and during all of the 1970s Walter
Philipp was internationally recognized as a leader in the development of new limit
theory for weakly dependent processes and their applications to problems in analysis
and number theory. In this period the focus of his research changed substantially:
instead of an analyst and number theorist using tools from probability theory hebecame a probabilist applying his results to problems in analysis and number theory.
In all of the topics from analysis and number theory mentioned above, there aresequences of random variables in the background, most of them defined on the probability space [0, 1], equipped with some measure equivalent to Lebesgue measure. In

The mathematical work of Walter Philipp

7Weyl’s almost sure equidistribution theory, we have the random variables ω 7→ an ω.
In each of the different expansions of real numbers ω ∈ (0, 1], the n-th digit maps
ω → xn = xn (ω) are random variables, and so are the n-th iterates ω 7→ T n ω. With
the notable exception of the digits in the expansion to an integer base a, none ofthese random variables are independent. But the dependence is weak, in some sense
yet to be defined. Walter Philipp soon realized that the theory of weakly dependent stochastic processes, then only recently created by publications of Rosenblatt(1956), Ibragimov (1962) and Billingsley (1968), provides the right framework forthe problems he wanted to attack.
There is no such thing as a universal definition of weak dependence that would
imply the validity of all limit theorems known for independent processes. There aremany notions, each of them allowing, under additional technical assumptions, theproof of some of the classical limit theorems of probability theory. The stronger
the notion, the more limit theorems can be established, but at the same time fewer
examples satisfy the conditions. The earliest and most classical notions of weak
dependence are α-mixing (also called strong mixing, but not to be confused withthe same notion in ergodic theory) and φ-mixing (also called uniform mixing). Let(Xn )n≥1 be a stochastic process, and define for integers k, l with k ≤ l the σ-fields
Fkl = σ(Xk , . . . , Xl ). We then define the mixing coefficientsα(k):= supφ(k):= supsup∞n≥1 A∈F1n , B∈Fn+ksup|P(A ∩ B) − P(A)P(B)|,∞ ,P(A)>0n≥1 A∈F1n ,B∈Fn+k|P(A ∩ B) − P(A)P(B)|.
P(A)

The process (Xn )n≥1 is called α-mixing if limn→∞ α(n) = 0 and φ-mixing if
limn→∞ φ(n) = 0. Rosenblatt (1956) and Ibragimov (1962) established central limit
theorems for α- and φ-mixing random variables, requiring a combination of moment conditions and conditions on the speed at which the mixing rates converge tozero. Later, several other related mixing concepts (ψ, ρ mixing, absolute regularity, etc.) were introduced and studied in detail. For stationary φ-mixing processes(Xn )n≥1 , Ibragimov conjectured that the central limit theorem holds if EX12 < ∞and Var (∑nk=1 Xk ) → ∞, but until today this conjecture has not been verified. Thisconjecture inspired some of Walter Philipp’s deepest results in the field of mixing
random variables: his 1986 joint paper with Dehling and Denker, giving a necessary and sufficient condition for the CLT for ρ-mixing sequences without any rate
or moment conditions and his 1998 joint paper with Berkes, giving a complete characterization of the law of the iterated logarithm and the domain of partial attractionof the Gaussian law for φ-mixing sequences, again without any moment or rate conditions.
In his first papers on limit theorems for weakly dependent processes, culminatinghis 1975 AMS memoir with William Stout, Philipp solved the central limit problem(characterizing the limit distributions of arrays with corresponding criteria for convergence to specific limits) in the case of bounded variances, proved Berry–Esseen

8

I. Berkes and H. Dehlingbounds for the speed of convergence in the CLT and obtained laws of the iterated
logarithm under various weak dependence conditions. In addition to the mixing
conditions already mentioned, he studied the ψ-mixing coefficients defined byψ(k) := supsup∞n≥1 A∈F1n ,B∈Fn+k|P(A ∩ B) − P(A)P(B)|,
P(A)P(B)and ψ-mixing processes. The notion of ψ-mixing is stronger than any of the othermixing conditions and ψ-mixing processes satisfy most of the classical i.i.d. limit
theorems. The digits of a random number ω ∈ (0, 1] in the continued fraction expansion form a ψ-mixing process. A second weak dependence condition investigatedby Walter Philipp is a correlation condition for mixed products, requiring that for
all integers 0 ≤ i1 < . . . < ir , 1 ≤ j ≤ r, pν ≥ 0



  p

p
j+1
E Xip1 1 · . . . · Xipr r − E Xip1 1 · . . . · Xi j j E Xi j+1· . . . · Xipr r≤ L( j)c(i j+1 − i j ) sup E|Xi |∑ pν .
1≤i≤ir

The standard method to prove limit theorems for mixing processes, employed in
the pioneering works of Rosenblatt (1956) and Ibragimov (1962), was the Bernsteinblocking technique, giving an approximation of the characteristic function of partial sums of mixing sequences by the characteristic function of sums of independent
random variables via suitable correlation inequalities. This method leads to sharp
results in the case of the CLT and LIL, but its applicability beyond them is rather
limited: for example, upper-lower class refinements of the LIL require delicate tailestimates for the considered r.v.’s which are beyond the scope of the method. In
his 1975 AMS Memoir with William Stout, Walter Philipp showed that sufficiently
separated block sums of weakly dependent sequences are, after suitable centering,close to a martingale difference sequence, and thus using Skorohod embedding andStrassen’s strong approximation technique, the partial sums of such sequences can
be closely approximated by a Wiener process. This observation not only opens theway to prove a vast class of refined asymptotic results for mixing sequences, but
the near martingale property can be easily verified for several other types of weak
dependent processes for which the previous theory does not work, or leads to great
difficulties: Markov processes, retarded mixing sequences, Gaussian processes, lacunary series, etc. Note that a different type of martingale approximation was usedearlier by Gordin (1969) in case of stationary sequences; the two methods complement each other and lead to sharp asymptotic results in many important situations.
In his 1979 joint paper with Berkes, Philipp made a further important step in the
study of weak dependent behavior, showing that block sums of weakly dependent
processes can be directly approximated by independent random variables, via theStrassen–Dudley existence theorem. This observation frees the investigations frommoment conditions and works not only for real valued random variables, but for
random variables taking values in abstract spaces. In the context of Banach spacevalued random variables, this method yields new results even for i.i.d. random variables, as a 1980 joint paper of Philipp and Kuelbs shows. This paper was the first

The mathematical work of Walter Philipp

9in a long series of papers of Walter Philipp dealing with limit theorems of independent and weakly dependent B-valued random variables and Hilbert space valued
martingales. The infinite dimensional setup also opens the way to study unform
Glivenko–Cantelli type results and uniform limit theorems for random variables indexed by sets, a popular and much studied topic in the 1970’s and 1980’s. Walter
Philipp’s contribution in this field is very substantial; see e.g. his profound joint
paper with Dudley (1983). In his last papers, Walter Philipp returned again to this
topic, showing that metric entropy can be used to provide deep information on pseudorandom behavior and in the theory of uniform distribution. This completes a long
circle in Philipp’s mathematical work and at the same time opens a new direction in
the study of weakly dependent behavior.
The importance of Walter Philipp’s contributions to the asymptotic theory forweakly dependent processes can only be appreciated in the light of the many applications to problems in analysis and number theory. Some early applications are
given in two papers entitled Some metrical theorems in number theory which areentirely devoted to such applications. In these papers Walter Philipp investigatedthe distribution of the sequence (T n ω)n≥1 for the maps T : (0, 1] → (0, 1] associated with the θ-adic expansion and the continued fraction expansion. If (In )n≥1 is asequence of intervals, In ⊂ (0, 1], one can study the quantity
N

A(N, ω) := ∑ 1{T n ω∈In } .
n=1

If µ denotes the invariant measure associated with T , then the expected value ofA(N, ω) becomesN

φ(N) = ∑ µ(In ).
n=1

If all the intervals are identical, i.e. In = I, we obtain from the ergodic theorem that
A(N, ω) = φ(N) + o(N) a.s. Walter Philipp sharpened this result considerably byshowing that for any ε > 0,
A(N, ω) = φ(N) + O(φ1/2 (N) log3/2+ε φ(N)),(3)

for almost all ω ∈ (0, 1]. While the accuracy of this approximation is limited by thesecond order method used in the proof, shortly thereafter Philipp went much further:he observed that the considered sequences (T n ω)n≥1 are φ-mixing with exponential
rate and thus using blocking techniques and applying some of his asymptotic theorems obtained earlier, one can prove a whole series of highly attracting limit theorems for the digits in various expansions and Diophantine approximation. These not
only improve several earlier results in the literature, but actually provide the precise
asymptotics in a number of important questions of metric number theory. Let us formulate a few such results here. Let x = [a1 (x), a2 (x), . . .] be the continued fraction
1
expansion of x ∈ (0, 1) and let ϕ(n) → ∞ be a sequence of integers with ∑ ϕ(n)
= ∞.

10

I. Berkes and H. Dehling

Denote by A(N, x) the number of integers n ≤ N with an (x) ≥ ϕ(n) and put

1
1
φ(N) =
log1+.
∑
log 2 n≤N
ϕ(n)
Then

)
Z z
A(N, x) − φ(N)
1
p
√
λ x:
<z →
exp(−t 2 /2)dt
2π −∞
φ(N)
(

and for almost all x|A(N, x) − φ(N)|= 1.
lim sup p
2φ(N) log log φ(N)N→∞

Also, letting LN (x) = max1≤k≤N an (x), Philipp proved

log log N
1
LN (x) =
N→∞
N
log 2

lim inf

for almost all x, verifying an old conjecture of Erdős. Further, let f be a continuous,
positive, nonincreasing function on R+ such that
f (k)
→∞
k≤n k

φ(n) = 2 ∑and let Nα, f (n) denote the number of solutions (p, q) of (1) in integers q ≤ n and p.
Then under mild additional regularity conditions on f , Walter Philipp proved()
Z z
Nα, f (n) − φ(n)1pλ α:<z → √exp(−t 2 /2)dt
2π −∞
φ(n)
and for almost all α|Nα, f (n) − φ(n)|= 1.
lim sup pn→∞2φ(n) log log φ(n)A further much studied connection between probability theory and number theory is the distribution of values of additive functions. Walter Philipp’s contribution
in his field can be found in his 1971 AMS Memoir written with the ambitious goal
to unify probabilistic number theory and to deduce at least the most typical results
as special cases of limit theorems for mixing random variables. Because of the very
different type of weak dependence conditions in number theory, there is little hopethat all applications of probability to number theory could be put in a general framework. But at least for applications in Diophantine approximation, continued fraction and related expansions, discrepancies and the distribution of additive functions,
Walter Philipp succeeded in this program to a remarkable degree. He also studied
weak convergence of additive function paths to Brownian motion, extending earlier
results of Billingsley.

The mathematical work of Walter Philipp

11

The basic motivation for the introduction of mixing conditions was to understandthe asymptotic properties of weakly dependent structures in stochastics and driven
by its intrinsic needs, the theory made a tremendous progress starting from the 60’s
and by now it is a closed, complete and beautiful theory, giving a nearly complete
answer for the basic asymptotic questions connected with mixing structures. For acomprehensive treatise of the theory see the recent monograph of R. Bradley (2007).
While questions on weak dependence kept Walter Philipp’s attention in his whole
career, this did not prevent him from making fundamental contributions in otherareas of probability theory, e.g. in the classical theory of independent random variables. In a short paper with M. Lacey in 1990 he proved that if (Xn )n≥1 is a sequenceof i.i.d. random variables with mean 0 and variance 1 then letting Sn = ∑nk=1 Xk we
have


Z x
2
1 N 1
Sk
1
√
√
I
lim
≤
x
=
e−t /2 dt∑k
N→∞ log N
−∞
2π
k
k=1
with probability 1 for all x ∈ R. This remarkable ‘pathwise’ form of the central
limit theorem was already stated (without proof and without specifying conditions)by Lévy in 1937 and was proved independently by Brosamler (1988), Fisher (1987)
and Schatte (1988) under the assumption of higher moments. Due to these papers,
almost sure central limit theory became extremely popular overnight and has not lostits attraction until today. The paper of Philipp and Lacey not only yields the final,
optimal form of this theorem, but the method they used became the basic method inthis field.
In a series of three papers, published in the mid 1980s jointly with Dehlingand Denker, Walter Philipp investigated the asymptotic behavior of degenerate Ustatistics. Given a symmetric function h : Rm → R and an i.i.d. process (Xn )n≥1 , them-variate U-statistic with kernel h is defined as
Un (h) =∑h(Xi1 , . . . , Xim ).

1≤i1 <...<im ≤n

The kernel is called degenerate if E(h(X, x2 , . . . , xm )) = 0 for almost all x2 , . . . , xm .
Dehling, Denker and Philipp proved a strong approximation of Un (h) by m-fold
Wiener–Itô integrals
In (h) =

Z

...

Z

h(x1 , . . . , xm ) dW (x1 ) . . . dW (xm ),where W (t) is a mean-zero Gaussian process. The results of this research enabled
Dehling in a subsequent paper to establish the functional law of the iterated logarithm for degenerate U-statistics and for multiple Wiener–Ito integrals. In thecourse of their work, Dehling, Denker and Philipp also investigated the empiricalU-process, defined by√ 1
n n  #{1 ≤ i1 < . . . < im ≤ n : h(Xi1 , . . . , Xim ) ≤ t}−P(h(X1 , . . . , Xm ) ≤ t) t∈R ,m

12

I. Berkes and H. Dehlingand established an almost sure invariance principle for this process.
The investigations on the asymptotic behavior of degenerate U-statistics leaddirectly to questions concerning the asymptotic behavior of certain Hilbert spacevalued martingales. For the U-statistic applications, a bounded law of the iterated
logarithm was sufficient. In later work, carried out jointly with Monrad, Walter
Philipp established a Skorohod embedding of Hilbert space valued martingales.
Another favorite topic of Walter Philipp’s research was lacunary series: he investigated such series already in his early papers in the 1960’s and in his very last papers in 2006 he returned once more to this topic. By Weyl’s (1916) theorem quoted
above, given any sequence (nk )k≥1 of positive numbers with nk+1 − nk ≥ δ > 0(k = 1, 2, . . .), the sequence {nk ω} is uniformly distributed mod 1 for almost all ω.
In contrast to the simplicity of this result, proving sharp bounds for the discrepancyof {nk ω} is very difficult, and the only precise results known before 1970 were theresults of Khinchin (1924) and Kesten’s (1964) for the case nk = k. Kesten’s result
states that the discrepancy DN (ω) of the sequence {kω} satisfies2
log N log log N
DN (ω) = 2
N→∞
N
π
lim

in probability.

In 1975 Walter Philipp proved that if (nk )k≥1 is a sequence of integers satisfying
the Hadamard gap condition nk+1 /nk ≥ q > 1 (k = 1, 2, . . .), then the discrepancy
DN (ω) of the sequence {nk ω} satisfies the law of the iterated logarithm, i.e.
s1N√ ≤ lim sup
DN (ω) ≤ C(4)log log N
4 2
N→∞for almost all ω, where C = C(q) = 166 + 664(q1/2 − 1)−1 . This remarkable result
verified a long standing conjecture of Erdős and Gál, and showed that, as far as its
discrepancies are concerned, {nk ω} behaves like a sequence of independent random variables. For the partial sums of sin nk x such phenomena have already been
observed by Salem and Zygmund in 1947, but the discrepancy situation is much
more delicate: as in a later paper Philipp (1994) showed, for a suitable sequence(nk )k≥1 the limsup in (4) is greater than C log log 1q with an absolute constant C andthus for q close to 1 the limsup can be as large as we wish. Very recently, Fukuyama(2008) succeeded in computing the limsup for the sequences nk = θk , θ > 1.
For sequences (nk )k≥1 growing slower than exponentially, the LIL (4) is generally false, and the behavior of the discrepancy of {nk ω} becomes very complicated. R. C. Baker (1981) proved that for any nk , NDN (ω) = O((log N)3/2+ε ) for
almost all ω and Philipp and Berkes (1994) showed that the constant 3/2 here cannot be replaced by any number less than 1/2. Despite these fairly precise results on
the extremal behavior of discrepancies, the exact order of magnitude of DN (ω) for“concrete” nk remains open. In one of his last papers, written jointly with Berkes
and Tichy, Walter Philipp made a substantial step in clearing up this phenomenon
as well: he showed that the asymptotic behavior of the discrepancy of {nk ω} is intimately connected with the number of solutions of Diophantine equations of theThe mathematical work of Walter Philipp

13

type a1 nk1 + · · · + a p nk p = b. The discovery of this remarkable arithmetical connection is again a characteristic achievement of Walter Philipp, linking probabilistic
phenomena with asymptotic results in analysis and number theory.
In conclusion we mention an interesting result of Philipp on the extremal behavior of exponential sums. From the Carleson convergence theorem for Fourier seriesin L2 it follows that if f is a nondecreasing positive function on R+ satisfying∞

1∑ k f (k)2 < ∞

(5)

k=1

then for any increasing sequence (nk )k≥1 of positive integers we haveN

∑ e2πink x = O(N 1/2 f (N))

a.e.

(6)

k=1

In particular, we have for any (nk )k≥1N

∑ e2πink x = O(N 1/2 (log N)1/2+ε)

a.e.

k=1

for any ε > 0. As early as 1930, Walfisz proved that for nk = k2 the left hand side of(6) exceeds N 1/2 (log N)1/4 for infinitely many N, but this does not reach Carleson’s
upper bound. Walter Philipp proved that if f is a nondecreasing positive function
on R+ satisfying mild regularity conditions such that the sum in (5) diverges, thenthe exponential sum in (6) exceeds N 1/2 f (N) a.e. for infinitely many N. This provides a complete solution of the problem of extremal speed of exponential sums
and provides yet another example for the power of weak dependence techniques inproblems of classical analysis.

Physics
In the last years of his life, Walter Philipp became interested in certain problems
concerning the foundation of physics. Nearly 100 years after the birth of quantum
mechanics, the problem of existence of “hidden parameters” in the theory (a question first investigated in depth by John von Neumann in 1932) is still not settled, due
to unsatisfactory probabilistic models traditionally used to disprove the existence ofsuch parameters. In a series of papers written jointly with Karl Hess, Walter Philipp
provided refreshing new ideas in this field, inevitably causing great controversy inphysics circles. It is a great loss to science that Walter Philipp’s death in 2006 put
an end to these investigations, leaving the solution of this important problem to the
future.

14

I. Berkes and H. Dehling

References[1] R. C. Baker, Metric number theory and the large sieve, J. London Math. Soc.
24 (1981), 34–40.
[2] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
[3] R. C. Bradley, Introduction to Strong Mixing Conditions, Volumes 1–3,
Kendrick Press, 2007.
[4] G. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561–574.
[5] P. Erdős, On trigonometric sums with gaps, Magyar Tud. Akad. Mat. Kut. Int.
Közl. 7 (1962), 37–42.
[6] A. Fisher, Convex-invariant means and a pathwise central limit theorem, Adv.
in Math. 63 (1987), 213–246.
[7] K. Fukuyama, The law of the iterated logarithm for discrepancies of {θn x},
Acta Math. Hungar. 118 (2008), 155–170.
[8] V. F. Gaposhkin, The central limit theorem for weakly dependent sequences,
Theory Probab. Appl. 15 (1970), 649–666.
[9] M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad.
Nauk SSSR 188 (1969), 739–741.
[10] I. A. Ibragimov, Some limit theorems for stationary processes, Theory Probab.
Appl. 7 (1962), 349–382.
[11] H. Kesten, The discrepancy of random sequences {kx}, Acta Arith. 10(1964/65), 183–213.
[12] A. Khinchin, Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125.
[13] P. Lévy, Theorie de L’Addition des Variables Aleatoires, Gauthier-Villars,
Paris, 1937.
[14] A. Rényi, Representations for real numbers and their ergodic properties, Acta
Math. Acad. Sci. Hungar. 8 (1957), 477–493.
[15] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc.
Nat. Acad. Sci. USA 42 (1956), 43–47.
[16] R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad.
Sci. USA 33 (1947), 333–338.
[17] P. Schatte, On strong versions of the central limit theorem, Math. Nachr. 137(1988), 249–256.
[18] S. Takahashi, A gap sequence with gaps bigger than the Hadamards, Tôhoku
Math. J. 13 (1961), 105–111.
[19] S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite
gaps, Tôhoku Math. J. 15 (1963), 281–288.
[20] A. Walfisz, Ein metrischer Satz über diophantische Approximationen, Fund.
Math. 16 (1930), 361–385.
[21] H. Weyl, Über die Gleichverteilung von Zahlen mod eins, Math. Ann. 77(1916), 313–352.

The mathematical work of Walter Philipp

15

Walter Philipp’s scientific publications
1. Ein metrischer Satz über die Gleichverteilung mod 1, Arch. Math. 12 (1961), 429–433.
2. Über die Einfachheit von Funktionenalgebren, Monatsh. Math. 66 (1962), 441–451(with Wilfried Nöbauer).
3. Über die Einfachheit von Funktionenalgebren über Verbänden, Monatsh. Math. 67(1963), 259–268.
4. Die Einfachheit der mehrdimensionalen Funktionenalgebren, Arch. Math. 15 (1964),
1–5 (with Wilfried Nöbauer).
5. Über einen Satz von Davenport–Erdős–LeVeque, Monatsh. Math. 68 (1964), 52–58.
6. An n-dimensional analogue of a theorem of H. Weyl, Compositio Math. 16 (1964),
161–163.
7. A note on a problem of Littlewood about Diophantine approximation, Mathematika
11 (1964), 137–141.
8. Some metrical theorems in number theory, Pacific J. Math. 20 (1967), 109–127.
9. Ein zentraler Grenzwertsatz mit Anwendungen auf die Zahlentheorie, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 185–203.
10. Das Gesetz vom iterierten Logarithmus für stark mischende stationäre Prozesse, Z.
Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 204–209.
11. Mischungseigenschaften gewisser auf dem Torus definierter Endomorphismen, Math.
Z. 101 (1967), 369–374.
12. Das Gesetz vom iterierten Logarithmus mit Anwendungen auf die Zahlentheorie, Math.
Ann. 180 (1969), 75–94. (Corrigendum: Math. Ann. 190 (1970), 338).
13. The remainder in the central limit theorem for mixing stochastic processes, Ann. Math.
Statist. 40 (1969), 601–609.
14. Zwei Grenzwertsätze für Kettenbrüche, Math. Ann. 181 (1969), 152–156 (with Olaf
P. Stackelberg).
15. The central limit problem for mixing sequences of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 12 (1969), 155–171.
16. The law of the iterated logarithm for mixing stochastic processes, Ann. Math. Statist.
40 (1969), 1985–1991.
17. Some metrical theorems in number theory II, Duke Math. J. 37 (1970), 447–458 (Errata: Duke Math. J. 37 (1970), 788).
18. Mixing sequences of random variables and probabilistic number theory, Memoirs of
the American Mathematical Society 114 (1971).
19. An invariance principle for mixing sequences of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972), 223–237 (with Geoffrey R. Webb).
20. On a theorem of Erdős and Turán on uniform distribution, Proceedings of the Number
Theory Conference (University of Colorado, Boulder 1972) (1972), 180–182 (withHarald Niederreiter).
21. Berry–Esseen bounds and a theorem of Erdős and Turán on uniform distribution mod
1, Duke Math. J. 40 (1973), 633–649 (with Harald Niederreiter).
22. Empirical distribution funcitons and uniform distribution mod 1, in: Diophantine Approximation and its applications (Proc. Conf. Washington, D.C. 1972), 211–234. Academic Press, New York, 1973.
23. Arithmetic functions and Brownian motion. Analytic methods in number theory, Proc.
Sympos. Pure Math. XXIV, 233–245, American Mathematical Society, Providence
R.I., 1973.

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24. Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith. 26(1974), 241–251.
25. Distances of probability measures and uniform distribution mod 1, Math. Z. 142 (1975),
195–202 (with Rudolf Mück).
26. A conjecture of Erdős on continued fractions, Acta Arith. 28 (1975), 379–386.
27. Asymptotic fluctuation behaviour of sums of weakly dependent random variables, in:Limit theorems of probability theory (Colloq. Keszthely 1974), Colloquia Math. Soc.
János Bolyai 11, 273–296, North-Holland, Amsterdam, 1975 (with William F. Stout).
28. Almost sure invariance principles for partial sums of weakly dependent random variables, Memoirs of the American Mathematical Society 161 (1975) (with William F.
Stout).
29. Almost sure invariance principles for empirical distribution functions of weakly dependent random variables, Empirical distribution and processes, Lecture Notes inMathematics 566, 83–105, Springer, Berlin, 1976.
30. A functional law of the iterated logarithm for empirical distribution functions of weakly
dependent random variables, Ann. Probability 5 (1977), 319–350.
31. An almost sure invariance principle for the empirical distribution function of mixing
random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977), 115–
137 (with István Berkes).
32. A uniform law of the iterated logarithm for classes of functions, Ann. Probability 6(1978), 930–952 (with Robert Kaufman).
33. Approximation theorems for independent and weakly dependent random vectors. Ann.
Probability 7 (1979), 29–54 (with István Berkes).
34. An a.s. invariance principle for lacunary series f (nk x), Acta Math. Acad. Sci. Hungar.
34 (1979), 141–155 (with István Berkes).
35. Almost sure invariance principles for sums of B-valued random variables, in: Probability in Banach spaces II (Proc. Second Internat. Conference, Oberwolfach 1978),Lecture Notes in Math. 709, 171–193, Springer, Berlin 1979.
36. A new method to prove limit theorems for mixing random variables and its applications, in: Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Prague 1978), Vol. C, 39–46, Reidel,
Dordrecht–Boston, 1979 (with István Berkes).
37. Weak and L p -invariance principles for sums of B-valued random variables, Ann. Probability 8 (1980), 1003–1036 (Correction: Ann. Probability 14 (1986), 1095–1101).
38. Almost sure approximation theorems for the multivariate empirical process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 54 (1980), 1–13 (with Laurence Pinzur).
39. Almost sure invariance principles for partial sums of mixing B-valued random variables, Ann. Probability 8 (1980), 1003–1036 (with James Kuelbs).
40. Almost sure invariance principles for sums of B-valued random variables with applications to random Fourier series and the empirical characteristic process, Trans. Amer.
Math. Soc. 269 (1982), 67–90 (with Michael B. Marcus).
41. Almost sure invariance principles for weakly dependent vector-valued random variables, Ann. Probability 10 (1982), 689–701 (with Herold Dehling).
42. An almost sure invariance principle for Hilbert space valued martingales, Trans. Amer.
Math. Soc. 273 (1982), 231–251 (with Gregory Morrow).
43. Invariance Principles for sums of Banach space valued random elements and empirical
processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 62 (1983), 509–552 (withRichard M. Dudley).

The mathematical work of Walter Philipp

17

44. An almost sure invariance principle for triangular arrays of Banach space valued random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 65 (1984), 483–491(with André Dabrowski and Herold Dehling).
45. Versik processes and very weak Bernoulli processes with summable rates are independent, Proc. Amer. Math. Soc. 91 (1984), 618–624 (with Herold Dehling and Manfred
Denker).
46. Invariance principles for von Mises and U-statistics, Z. Wahrscheinlichkeitstheorie und
Verw. Gebiete 67 (1984), 439–167 (with Herold Dehling and Manfred Denker).
47. Approximation by Brownian motion of Gibbs measures and flows under a function,
Ergodic Theory Dynam. Systems 4 (1984), 541–552 (with Manfred Denker).
48. Invariance principles for sums of mixing random elements and the multivariate empirical process, in: Limit theorems in probability and statistics, Veszprém 1982, Colloq.
Math. Soc. János Bolyai 36, 843–873, North-Holland, Amsterdam, 1984.
49. A bounded law of the iterated logarithm for Hilbert space valued random variables
and its application to U-statistics, Probab. Theory Rel. Fields 72 (1986), 111–131(with Herold Dehling and Manfred Denker).
50. Invariance principles for martingales and sums of independent random variables, Math.
Z. 192 (1986), 253–264 (with William F. Stout).
51. Invariance principles for partial sum processes and empirical processes indexed bysets, Probab. Theory Rel. Fields 73 (1986), 11–42 (with Gregory Morrow).
52. A strong approximation theorem for sums of random vectors in the domain of attraction of a stable law, Acta Math. Hungar. 48 (1986), 161–172 (with István Berkes,
André Dabrowski and Herold Dehling).
53. A note on the almost sure approximation of weakly dependent random variables,
Monatsh. Math. 102 (1986), 227–236.
54. Central limit theorems for mixing sequences under minmal conditions, Ann. Probability 14 (1986), 1359–1370 (with Herold Dehling and Manfred Denker).
55. Invariance principles for independent and weakly dependent random variables, Dependence in Probability and Statistics (Oberwolfach 1985), Progress in Probability
and Statistics 11, 225–268, Birkhäuser, Boston, 1986.
56. The almost sure invariance principle for the empirical process of U-statistic structure,
Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), 121–134 (with Herold Dehling and
Manfred Denker).
57. Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math.
105 (1988), 195–206.
58. A note on the almost sure central limit theorem, Statist. Probab. Letters 9 (1990),
201–205 (with Michael Lacey).
59. The problem of embedding vector-valued martingales in a Gaussian process, Theory
Probab. Appl. 35 (1990), 374–377 (with Ditlev Monrad).
60. Nearby variables with nearby conditional laws and a strong approximation theorem
for Hilbert space valued martingales, Probab. Theory Rel. Fields 88 (1991), 381–404(with Ditlev Monrad).
61. Conditional versions of the Strassen–Dudley theorem, Probability in Banach spaces
8 (Brunswick, Maine 1991), Progress in Probability 30, 116–127, Birkhäuser Boston
1992 (with Ditlev Monrad).
62. Empirical distribution functions and strong approximation theorems for dependent
random variables. A problem of Baker in probabilistic number theory, Trans. Amer.
Math. Soc. 345 (1994), 705–727.

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63. The size of trigonometric and Walsh series and uniform distribution mod 1, J. London
Math. Soc. 50 (1994), 454–464 (with István Berkes).
64. Trigonometric series and uniform distribution mod 1, Studia Sci. Math. Hungar. 31(1996), 15–25 (with István Berkes).
65. Limit theorems for mixing sequences without rate assumptions, Ann. Probability 26(1998), 805–831 (with István Berkes).
66. A limit theorem for lacunary series ∑ f (nk x), Studia Sci. Math. Hungar. 34 (1998),
1–13 (with István Berkes).
67. Pair correlations and U-statistics for independent and weakly dependent random variables, Illinois J. Math. 45 (2001), 559–580 (with István Berkes and Robert Tichy).
68. Bell’s theorem and the problem of decidability between the views of Einstein andBohr, Proc. Nat. Acad. Sci. USA 98 (2001), 14228–14233 (with K. Hess).
69. A possible loophole in the theorem of Bell, Proc. Nat. Acad. Sci. USA 98 (2001),
14224–14227 (with K. Hess).
70. Metric theorems for distribution measures of pseudorandom sequences, Dedicated to
Edmund Hlawka on the occasion of his 85th birthday. Monatsh. Math. 135 (2002),
321–326 (with Robert Tichy).
71. The profound impact of Paul Erdős on probability limit theory – a personal account,
in: Paul Erdős and his mathematics, Budapest 1999, Bolyai Soc. Math. Stud. 11, 549–
566, Budapest, 2002.
72. Empirical process techniques for dependent data, in: Empirical process techniques for
dependent data, 3–113, Birkhäuser, Boston, 2002 (with Herold Dehling).
73. Quantum computing and the theorem of Bell, Microelectr. Eng. 63 (2002), 11–15 (withK. Hess).
74. Exclusion of time in the theorem of Bell, Europhys. Lett. 57 (2002), 775–781 (with K.
Hess).
75. A local mathematical model for EPR-experiments, in: Foundations of probability andphysics, 2 (Växjö, 2002), 469–488, Math. Model. Phys. Eng. Cogn. Sci., 5, Växjö Univ.
Press, 2003 (with K. Hess).
76. Reply to the comment by R. D. Gill et al. on “Exclusion in time in the theorem ofBell”, Europhys. Letters 61 (2003), 284–285 (with K. Hess).
77. Exclusion of time in Mermin’s proof of Bell-type inequalities, in: Quantum theory:reconsideration of foundations—2, 243–253, Math. Model. Phys. Eng. Cogn. Sci., 10,Växjö Univ. Press, Växjö, 2004 (with K. Hess).
78. Breakdown of Bell’s theorem for certain objective local parameter spaces, Proc. Nat.
Acad. Sci. USA 101 (2004), 1799–1805 (with K. Hess).
79. The Bell theorem as a special case of a theorem of Bass, Found. Phys. 35 (2005),
1749–1767 (with K. Hess).
80. Bell’s theorem: critique of proofs with and without inequalities, in: Foundations ofprobability and physics—3, 150–157, AIP Conf. Proc., 750, Amer. Inst. Phys., Melville,
NY, 2005 (with K. Hess).
81. Empirical processes in probabilistic number theory: the LIL for the discrepancy of nk ω
mod 1, Illinois J. Math. 50 (2006), 107–145 (with István Berkes and Robert Tichy).
82. What is quantum information? Intern. J. Quant. Inf. 4 (2006), 585–625 (with K. Hessand M. Aschwanden).
83. Bell’s theorem and the consistency problem of common probability distributions: amathematical perspective on the Einstein–Bohr debate (in German), Math. Semesterber. 53 (2006), 152–183 (with K. Hess).

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19

84. Local time dependent instruction-set model for the experiment of Pan et al., in: Quantum theory: reconsideration of foundations—3, pp. 437–446, AIP Conf. Proc., 810,
Amer. Inst. Phys., Melville, NY, 2006 (with M. Aschwanden, K. Hess, S. BarrazaLopez, G. Adenier).
85. Pseudorandom numbers and entropy conditions, J. Complexity 23 (2007), 516–527(with István Berkes and Robert Tichy).
86. Metric discrepancy results for sequences {nk x} and Diophantine equations, in: Diophantine Approximations (eds. H. P. Schlickewei, K. Schmidt and R. Tichy), 95–105.
Springer, 2008 (with István Berkes and Robert Tichy).
87. Entropy conditions for subsequences of random variables with applications to empirical processes, Monatsh. Math. 153 (2008), 183–204 (with István Berkes and Robert
Tichy).

Walter Philipp’s Ph.D. students
1. Lawrence Pinzur (1978)
2. Gregory Morrow (1979)
3. Sompong Dhompongsa (1981)
4. Herold Dehling (1981, University of Göttingen)
5. André Dabrowski (1982)
6. Wolfram Strittmatter (1985, University of Freiburg)
7. Michael Lacey (1987)