Asymptotic sampling formulae for Lambda-coalescents

0.24 MB·English

by  Julien Berestycki

Asymptotic sampling formulae for Λ-coalescents
Julien Berestycki1, Nathana¨el Berestycki2, and Vlada Limic3
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0 Draft version — February 1, 2012
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J Abstract

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We present a robust method which translates information on the speed of coming down
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from infinity of a genealogicaltree into sampling formulae for the underlying population.

] We apply these results to population dynamics where the genealogy is given by a Λ-
R coalescent. This allows us to derive an exact formula for the asymptotic behavior of the
P site andallelefrequencyspectrumandthe number ofsegregatingsites,asthe samplesize
. tends to ∞. Some of our results hold in the case of a general Λ-coalescent that comes
h
t down from infinity, but we obtain more precise information under a regular variation
a
assumption. In this case, we obtain results of independent interest for the time at which
m
a mutation uniformly chosen at random was generated. This exhibits a phase transition
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at α=3/2, where α∈(1,2) is the exponent of regular variation.

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v AMS 2000 Subject Classification. 60J25, 60F99, 92D25
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1 Key words and phrases. Λ-coalescents, speed of coming down from infinity, exchangeable
5 coalescents, sampling formulae, infinite allele model, genetic variation.
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i
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1. Universit´ePierreetMarieCurie-ParisVI.SupportedinpartbyANRMAEVandANRMANEGE.
2. University ofCambridge. Supported in partby EPSRC grantEP/GO55068/1and EP/I03372X/1.
3. Universit´edeProvence. ResearchsupportedinpartbyNSERCDiscoveryGrant,byAlfredP.Sloan
Research Fellowship, and by ANR MAEV research grant.
1
1 Introduction and main results
Coalescents with multiple collisions, also known as Λ-coalescents, are a class of Markovian
coalescence models,introducedandfirststudiedbyPitman[26]andindependentlybySagitov
[28],andwerealreadyimplicitinacontemporaneousworkofDonnellyandKurtzin[11]. They
arise naturally as scaling limits for the genealogy of exchangeable population dynamics. This
connection to population genetics has motivated a large number of works around the study of
Λ-coalescents, an introduction to which may befoundin the recent surveys [9,8] for instance.
The following question is natural in the context of population genetics: assuming that the
genealogical tree of a sample is a Λ-coalescent (definitions will be given below), how much
genetic variation do we expect to see? A complete answer exists in the special case where
only pairwise collisions are possible, due to the celebrated Ewens sampling formula [14] for
the Kingman coalescent [21]. More recently, partial results have been obtained by Berestycki
et al. [7] in the particular case of Beta-coalescents (for a general overview of previous results
on the subject, we refer the reader to Durrett [13] or Berestycki [8].)
The main goal of this paper is to address this question in general, by providing a robust
method which translates information about the speed of coming down from infinity of the
genealogical tree into an explicit asymptotic formulae (as the sample size increases to ∞) for
quantifying the genetic variation. Since the speed of coming down from infinity was recently
analyzed by the authors in [4], this method in combination with results from [4] enables us
to obtain the following results:
(a) In Theorem 3, we obtain in complete generality (i.e., for arbitrary finite measures Λ
such that the corresponding Λ-coalescent comes down from infinity) a deterministic asymp-
toticrateofgrowthforthenumberofdistinctallelesinasampleandthenumberofsegregating
sites (thefamous SNPcount, or single nucleotide polymorphism). Theformulainvolves acer-
tain function ψ(q), which is the Laplace exponent of the subordinator whose L´evy measure is
precisely x−2Λ(dx). Furthermore, the above convergence in probability is strengthened to an
almost sureconvergence, providedthat themeasureΛsatisfies an additional regular variation
condition in the neighborhood of zero (precise assumptions will be given below).
(b) In Theorem 6 we derive explicit almost sure asymptotic formulae for the frequency
spectrum,inboththeinfinitesiteandtheinfiniteallelemodels,inthecasewhereΛisregularly
varying near zero.
This last result is a significant improvement and a generalization of previous work of
Berestycki et al. [6, 7] and of Schweinsberg [31], both in the sense that the result is valid for
more general measures Λ, and in the sense that the convergence holds almost surely rather
than in probability. Our methodology is completely different from that of [7] which relied on
an embeddinginto stable continuous randomtrees (CRT), allowing for explicit computations.
As explained above, the argument here is based on the recent work by the authors on the
speed of coming down from infinity [4], and a novel general method which translates such
results into results about sampling formulae. This method is more robust than previous
approaches to this problem, which explains why the results here are both stronger and more
general.
We note that the asymptotics in probability for the number of distinct alleles in a sample,
underthemodelwherethegenealogy isdrivenbythegeneral(regular)Ξ-coalescentdynamics,
was obtained in parallel by Limic [23] using an adaptation of the martingale method that led
to the results in [4] and [22] (see Remark 11).
2
d
In the sequel, we denote by ⇒ the convergence in distribution, and by = the equivalence
in distribution. We also use the standard Bachmann-Landau notation ∼,O(·),o(·),≍ for
comparing asymptotic behavior of deterministic and stochastic functions and sequences.
1.1 Mutation models
We now describe the underlying framework for the sampling results in more detail. Consider
a sample of n individuals, where n is a fixed number tending to infinity. Assume that the
genealogical relationship between these individuals is given by a Λ-coalescent, where Λ is
an arbitrary finite measure on [0,1]. That is, the genealogical tree is a Markov process
(Πn(t),t ≥ 0) on the space of partitions of {1,...,n} with the following transition rates:
whenever Πn has b blocks, any k-tupleof them merges at rateλ := 1xk−2(1−x)b−kΛ(dx).
b,k 0
In order to discuss genetic variation, we need to specify a mutation model. The two most
R
widelyusedandtractablemodelsaretheinfinite sites model andtheinfinite alleles model. To
familiarize oneself with these models, it is also useful to think in terms of the forward-in-time
evolution dynamics for the whole population (and not only in terms of the backward-in-time
coalescent dynamics).
In the classical infinite sites model, introduced by Kimura [19] in 1969, any individual
is affected by neutral mutations at constant rate θ > 0. Here it is also assumed that the
number of loci (the size of the genome) is large, so that each mutation occurs at a new locus.
In particular, if an individual is affected by a mutation, then all the descendants of this
individual carry this mutation (see Figure 1). Conversely, the genetic type of any individual
in the sample depends on the entire history of its ancestral lineage. We denote by S the
n
number of segregating sites, or the total number of distinct genetic types in the sense just
described. This is the same as the omnipresent SNP count (single nucleotide polymorphism)
from the biological literature.
The infinite alleles model is similar but with one difference. As above, any individual is
affected by a mutation at constant rate θ. However, it is now assumed that every mutation
changes the allelic type of the individual into something new, distinct from anything else
(already seen or yet unseen) in the population. Thus, the allelic type of an individual in
the sample is entirely determined by the most recent mutation affecting the corresponding
ancestrallineage. Theallelicpartitionisthepartitionofthesample(representedbyapartition
of {1,...,n}) obtained by grouping together the individuals that carry the same allelic type.
Denote by A the number of blocks in this partition, or equivalently, the total number of
n
allelic types expressed in the sample.
Remark 1. The two models differ only by the amount of information that is assumed to be
available in the sample. In the infinite sites model the assumption is that the precise allelic
type(e.g., theentire DNAsequence) is knownfor each individualin thesample. Ontheother
hand, in the infinite alleles model, the only available information is whether two individuals
carry the same type or not. Hence for different types we do not know how they differ.
Thus the infinite alleles model contains less information than the infinite sites model,
and is more appropriate in practice for situations where the only available information is,
for example, based on observed physiological differences. On the other hand the infinite
sites model is more natural when the full genetic information (the DNA sequences of each
individual in the sample) is available.
The above random variables can be realized in a natural way on a common probability
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space as follows. Consider a Λ-coalescent (Π ,t ≥ 0) that comes down from infinity, and let
t
T be the associated coalescent tree. Then T is a tree with infinitely many leaves 1,2,...,
and the root given by the most recent common ancestor among all the individuals. Each
branch of T is endowed with a positive number, its length or the size of the interval of time
that elapsed between the two defining coalescent events for this branch (the one that started
and the one that ended it) . Let P be a Poisson process of mutations on the branches of T,
where the intensity of mutations is constant and equal to θ per unit length. Restricting T
to the first n leaves produces a finite tree (even if the Λ-coalescent does not come down from
infinity), denoted by T , that has the law generated by the same Λ-coalescent started from
n
n particles. The restriction of P to T is identified as the mutation process on T , and it is
n n
a sufficient statistic for S and A . It is useful to note here that T and P alone determine,
n n
simultaneously for all n, the values of A and S , as well as various related quantities to be
n n
introduced in the sequel. Moreover, the coupling induced by this procedure between T and
n
T for m < n, is canonical from the sampling perspective, in that the mutations that arrive
m
onto T also arrive onto T . On the asymptotically unlikely event {P∩T = ∅}, we declare
n m n
A = S = 0.
n n
1 4 3 2 5 6
Figure1: ThegenealogicaltreeTn forasampleofsizen=6.Themutationsonthe(vertical)branches
of Tn areindicated as dots. The dot encircledin black correspondsto the mutation which is not seen
under the infinite alleles model. The dot encircled in gray will be referred to in Section 2. Thus we
have Sn =5 while An =4.
1.2 Sampling formulae
Let Λ be a finite measure on [0,1]. We will assume without further mention that Λ(1) = 0
(for reasons why this can be done without loss of full generality see any of [4, 26, 29]).
Definition 2. We say that Λ has (strong) α-regular variation at zero if Λ(dx) = f(x)dx
where f(x)∼ Ax1−α as x → 0 for some 1 <α < 2 and A> 0.
For any given finite measure Λ on [0,1], associate a function ψ = ψ defined by
Λ
ψ(q) := (e−qx −1+qx)x−2Λ(dx). (1)
Z[0,1]
Thefunction ψ is the Laplace exponent of a L´evy process, which is intimately connected with
the behavior of the Λ-coalescent. These links are discussed in a companion paper [5].
4
In this paper we will usually require that
∞ dq
<∞, (2)
ψ(q)
Z1
which is known as Grey’s condition. As was proved by Bertoin and Le Gall [10] (see also [5]
for a probabilistic proof), this is equivalent to the requirement that the Λ-coalescent comes
down from infinity. One can check (see e.g. [15] XIII.6) that in the case of strong α-regular
variation,
AΓ(2−α)
ψ(q) ∼ qα, as q → ∞, (3)
α(α−1)
where A is the constant from Definition 2, so in particular the Grey condition (2) holds if
α ∈ (1,2). Our first result concerns the asymptotic behavior of the number S of segregating
n
sites and the size A of the allelic partition.
n
Theorem 3. Assume (2) and let X denote either A or S . Then
n n n
X
n
−→ θ, (4)
n
qψ(q)−1dq
Z1
in probability as n → ∞. Moreover, if Λ has (strong) α-regular variation at zero, then the
above convergence holds almost surely, implying
nα−2X −→ θB, almost surely, (5)
n
where B = B(A,α) := α(α−1)/[AΓ(2−α)(2−α)].
Conjecture 4. Though our proofs rely on the fact that the Λ-coalescent comes down from
infinity, we observe that the above statement does not appeal to the function v(t) and hence
could hold in general. Due to the result of Basdevant and Goldschmidt [3] (see also Remark
8 in [23] for a short and more robust argument) on the number of allelic families in the
Bolthausen-Sznitman coalescent, it is easy to check that (4) holds in this special case, whereas
(2) does not hold. We conjecture that (4) holds for all the Λ-coalescents even in the L1
convergence sense.
Remark 5. Let ψ¯(q) := ((1−x)q −1+qx)/x2Λ(dx) be a close relative of ψ (note that
[0,1]
|ψ¯(q) − ψ(q)| = O(q) and moreover that ψ¯(q) ∼ ψ(q) as q → ∞). Applying the optional
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stopping formula to the following submartingale
n q t
dq− NΛ,n(u)du, t ≥ 0
ψ¯(q)
ZNΛ,n(t) Z0 !
gives an expectation upper bound in support of the above conjecture. We refer the reader to
(11) for the rest of the notation, and to the above mentioned remark in [22] as well as the
argument leading to (25) in [24] for applications of similar processes in the study of coming
down from infinity.
We also obtain precise results for the full frequency spectrum in both infinite sites and
infinite alleles models. For each n consider a sample of size n, and for each k ∈ {1,...,n},
let F be the number of families of size k in its allelic partition, and M be the number of
k,n n,k
mutations affecting precisely k of its individuals under the infinite sites model.
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Theorem 6. Suppose that Λ has (strong) α-regular variation at zero. Recall the constant
B ≡ B(α,A) from Theorem 3. Let X denote M or F , where 1 ≤ k ≤ n. As n → ∞,
k,n k,n k,n
X (2−α)Γ(k+α−2)
k,n
→ θB , a.s. (6)
n2−α k!Γ(α−1)
Moreover, if P ,P ,... are the ordered allele frequencies in the population, then
1 2
P ∼ Cj−1/(2−α), (7)
j
almost surely as j → ∞, and C = (θB/Γ(α−1))1/(2−α).
By the properties of the Gamma function, another expression for the constant on the
right-hand side of (6) is
(α−1)...(α+k−3)
θB(2−α) .
k!
As mentioned in the Introduction, the above results are improvements over previously known
results, since the convergence (6) was known to hold only in probability in the case of Beta-
coalescents (see [7]), while (7) was not known to hold even in this special case.
One key ingredient for our arguments, beside our earlier work on the speed of coming
down from infinity ([4]) is the asymptotic study of the time at which a randomly chosen
(uniformly) mutation was generated. More precisely, let the time run in the “coalescent
direction” (backward from the point of view of the population dynamics), so that the leaves
ofthetreearepresentattime0, andthenumberofbranchesdecreases intime. DenotebyM
n
the time-coordinate (age) of a point chosen at random from P ∩T . On the (asymptotically
n
unlikely) event {P∩T = ∅} = {S = A = 0}, we set M = 0 (although this value could be
n n n n
set to anything between 0 and the time of MRCA (the root of T ) and the next result would
n
still be true).
Define
n1−α, if 1< α < 3/2,
g(n) = n−1/2logn, if α= 3/2, (8)

nα−2, if 3/2 < α < 2.
Theorem 7. Suppose that Λ has (strong) α-regular variation at zero, for some α ∈(1,2).
(a) We have
Mn ⇒ α U−α2−−α1 −1, (9)
n1−α AΓ(2−α)
d
where U = Unif[0,1].
(b) If in addition Λ[1−η,1] = 0 for some η > 0, then there exists c ≡ c (α) ∈ (0,∞), such
1 1
that for g given by (8)
E(M )
n
lim = c . (10)
1
n→∞ g(n)
Remark 8. Note that M and E(M ) are only of the same order of magnitude if α < 3/2.
n n
Naturally, this is because the limit variable is integrable if and only if α < 3/2.
Interestingly, g(n) observed as a function of α decreases on (1,3/2) and increases on
(3/2,2), and moreover has a discontinuity on both sides at α = 3/2. It seems difficult to see
intuitively why this happens.
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Remark 9. We believe that the result (10) should hold without any further restriction on
Λ than strong regular variation. The techniques used in the proof of (10) can be used to
show, with some additional effort, that the sequence E(M )/g(n) is bounded away from 0
n
and infinity when no assumption is made on the support of Λ. However, in the interest of
brevity we decided to omit these arguments.
The link between Theorem 3 and Theorem 6 is provided by a remarkable Tauberian
theorem for random partitions of Gnedin, Hansen and Pitman [16]. The assumptions of this
theorem were recently extended in an independent but related work of Schweinsberg [31], to
deal with convergence in probability (to which the approach of [16] could not apply). This
allowed him to obtain the convergence in probability of Theorem 6 for the limiting behavior
of F (though not for that of M ). Gnedin, Hansen and Pitman [16] also derive a central
k,n k,n
limit theorem for F . It is natural to ask whether this result can be extended to our setting
k,n
with random frequencies. Kersting [18] has recently obtained precise fluctuation results for
the length of the genealogical tree in the regularly varying case. In particular, it follows from
hisTheorem1thatthesefluctuationsarenotGaussian. Inordertoresolvethejustmentioned
open problem, one would need to analyze the complex interplay between the fluctuations of
the tree length and the Poisson fluctuations of the mutations.
Organisation of the rest of the paper. Section 2 is devoted to proving the results
on the mutation frequency spectrum, announced in Section 1.1. More precisely, we prove
Theorem 7 in Section 2.1, Theorem 3 in Section 2.2, and Theorem 6 in Section 2.3. The final
section relaxes the technical condition on the support of Λ needed in the proofs of Theorems
3 and 6.
2 Proofs of the results
Fix some θ > 0. For each n ∈ N and t ≥ 0, let NΛ,n(t) denote the number of ancestral
lineages of the first n individuals remaining at time t. In particular, (NΛ,n(t), t ≥ 0) is a
continuous-time Markov jump process, starting from NΛ,n(0) = n. We assume throughout
this section that (2) holds, or that equivalently, the Λ-coalescent comes down from infinity:
NΛ(t) := lim NΛ,n(t) = supNΛ,n(t) < ∞, ∀t > 0. (11)
n→∞ n≥1
We will need some further notations. Define for k,n ∈ N,
τn = inf{t ≥ 0 :NΛ,n(t)≤ k}, and τ ≡ τ∞ = lim τn = inf{t ≥ 0 :NΛ(t) ≤ k}. (12)
k k k k
n→∞
Inparticularτn = inf{t ≥ 0: NΛ,n(t) = 1}isthetimeoftheMRCAforthesamplecontaining
1
the first n individuals, and L = τ1nNΛ,n(t)dt is the total length of the tree T .
n 0 n
The genealogical tree T is a path-connected set in R2. To any point x on the tree one can
R
associate a number t = t(x) called the time-coordinate or the age of x, which is defined as the
distance from that point to the set of leaves of T. Let Tˆ be the subtree of T consisting of
n
all the points in T having age in [τ ,τ ]. Then Lˆ = τ1NΛ(u)du is the length of Tˆ . The
n 1 n τn n
function
∞ Rdq
v(t) := inf{s ≥ 0 : < t} (13)
ψ(q)
Zs
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plays a central role in the analysis of asymptotic behavior of Tˆ . Finally, define t as
n n
∞ dq
t := ≡ v−1(n). (14)
n
ψ(q)
Zn
The following lemma gathers some asymptotic results which we will use in the rest of the
proof. Set
2−α
α¯ = ∈ (0,∞). (15)
α−1
Lemma 10. Assume (3) and define c = c(A,α) = α/(AΓ(2−α)) (compare with the constant
in (9)). Then, as n → ∞, we have almost surely
1. τ ∼ cn1−α, and
n
2. t ∼ cn1−α.
n
Furthermore, there exist c = c(Λ) > 0, and c = c(A,α) ∈ (0,∞), such that
1 2
3. P(τ > x)≤ e−c1·x, for all x ≥ 1,
1
1
4. Lˆ = τ1NΛ(u)du ∼ 1 v(u)du ∼ cα−1(τ )−α¯ ∼ cn2−α, a.s. as n→ ∞,
n τn tn α¯ n α¯
5. as x →R 0 R
c x−α¯+1, α¯ >1, c x−α¯+1, α¯ > 1,
τ1 2 1 2
uNΛ(u)du ∼ c log(1/x), α¯ =1, a.s. and uv(u)du ∼ c log(1/x), α¯ = 1,
2 2
 
Zx  Y, α¯ <1, Zx  c2, α¯ < 1,
(16)
where c = c (α) and Y := τ1uNΛ(u)du is a finite random variableif α¯ < 1.
2 2 0
Proof. Theorem 1 in [4] and (3) yRield
NΛ(t)∼ v(t) ∼ cα−11t−α−11, as t → 0, almost surely, (17)
where c = c(A,α) is as specified above. The asymptotic behavior (17) implies that NΛ(τ )=
n
n(1 + o(1)), almost surely, as n → ∞. Indeed, since τ → 0, we have NΛ(τ ) ∼ v(τ ) =
n n n
v(τ −) ∼ NΛ(τ −), and at the same time, P(NΛ(τ )≤ n < NΛ(τ −)) = 1. Since, again due
n n n n
to (17) NΛ(τn) ∼ cα−11(τn)−α−11, we obtain claim 1. Claim 2 is directly seen from (14) and
the asymptotic behavior of v in (17).
Claims 4 and 5 are derived similarly. One notes first that, due to P(τ > 0) = 1 and (17),
1
τ1 1 cα−11
NΛ(u)du ∼ v(u)du ∼ x−α¯ as x → 0,
α¯
Zx Zx
and then uses the facts that τ → 0, t → 0 as well as claims 1 and 2 to obtain claim 4. For
n n
1 1
claim 5, we use uNΛ(u) ∼ cα−1u−1/(α−1)+1 = cα−1u−α¯, and this uniquely determines c2. If
α¯ < 1, then both 1uv(u)du and τ1uNΛ(u)du are finite.
0 0
Itremains toverify claim 3. Dueto monotonicity of thecoalescent andthesimpleMarkov
R R
property, we have P(τn ≥ m+1|τn ≥ m)≤ P(τn ≥ 1). In turn, letting n → ∞,
1 1 1
P(τ ≥ m+1|τ ≥ m)≤ P(τ ≥ 1), for each m ≥ 1.
1 1 1
Hence, by induction, P(τ > m) ≤ e−c1m for all m ≥ 0, with c = logP(τ ≥ 1).
1 1 1
8
2.1 Proof of Theorem 7
Our first goal will be to prove Theorem 7. Let An = {S ≥ 1} = {A ≥ 1}. Since the length
1 n n
L of T diverges, we have that P(An) → 1, as n → ∞. Recall that on An, M is the age of
n n 1 1 n
a randomly chosen mutation in T , and that on the complement of An, M is set to 0. Due
n 1 n
to basic properties of Poisson point processes, on the event An (of overwhelming probability),
1
the random mutation is positioned as a point P∗ chosen uniformly at random from T . In
n n
symbols,
Mn = Mn∗·1An1 +0·1(An1)c, (18)
where M∗ is the time-coordinate of P∗, and P∗ is independent of An. Due to this indepen-
n n n 1
dence, and the fact P(An) → 1 as n → ∞, we have that E[M∗ ·1 ] = E(M∗)P((An)c) =
o(E(M∗)), and therefore1E(M )∼ E(M∗), as n → ∞. Similarnly (An1)c n 1
n n n
P(M ·nα−1 ≤ x)= P(M∗·nα−1 ≤ x)+O(P((An)c)), ∀x ≥ 0,
n n 1
therefore (9) is equivalent to
M∗
n ⇒ c(U−(α−1)/(2−α) −1), where U =d Unif[0,1], (19)
n1−α
Hence we proceed by studying M∗.
n
Now recall the subtree Tˆ of T. Consider a uniform random point on Tˆ and let Mˆ be
n n n
its age minus τ . Let E = {NΛ = n} be the event that Π ever attains a configuration with
n n τn
exactly n blocks. Due to the consistency property and the Markov property of (Π ,t ≥ 0),
t
the conditional law of Tˆ given F on the event E equals the law of T . (20)
n τn n n
This clearly induces the equivalence of the conditional law of Mˆ given F on E and the
n τn n
law of M∗, which will be used below. Recalling (12), we have
n
τ1 NΛ(u)du
P(Mˆ ≥ x|Tˆ ) = τn+x , x ∈[0,τ −τ ].
n n R Lˆn 1 n
So, recalling α¯ from (15), we have for a fixed y ∈ (0,1)
τ1 NΛ(u)du
P((Mˆ /τ +1)−α¯ ≤ y)= E τny−1/α¯∧τ1 . (21)
n n "R Lˆn #
By the argument used to show claim 4 of Lemma 10 and P(τ >0) = 1, we conclude
1
τ1
NΛ(u)du ∼ c′[τ y−1/α¯ ∧τ ]−1/(α−1)+1 ∼ c′[τ y−1/α¯]−1/(α−1)+1 = c′τ−α¯y, a.s.
n 1 n n
Zτny−1/α¯∧τ1
as n → ∞, where c′ = c1/(α−1)/α¯. Due to claim 4 of Lemma 10 we now see that the random
variable inside the expectation on the RHS of (21) converges to y, almost surely. Since
τ1 NΛ(u)du
|E[P((Mˆ /τ +1)−α¯ ≤ y|F )1 ]−yP(E )| ≤ E 1 τny−1/α¯ −y ,
n n τn En n " En(cid:12)R Lˆn (cid:12)#
(cid:12) (cid:12)
(cid:12) (cid:12)
and since the bounded (by 2) random variable inside the(cid:12)expectation converges(cid:12)to 0 al-
(cid:12) (cid:12)
most surely, the dominated convergence theorem implies that the left-hand side converges
9
to 0. Recalling (20), or its consequence for random points, this quantity can be rewritten
as |P((M∗/τ +1)−α¯ ≤ y)−y|P(E ). Theorem 1.8 in [6] gives P(E ) → α−1 as n → ∞,
n n n n
hence P((M∗/τ +1)−α¯ ≤ y)→ y, for all y ∈[0,1]. This, together with claim 1 of Lemma 10,
n n
implies (9).
The proof of (10) is analogous but technically more delicate. Due to P(M∗ ≥ x|T ) =
n n
τ1nNΛ,n(u)du/L and Fubini’s theorem, we have
x n
R ∞ τ1nNΛ,n(u)dudx τ1nuNΛ,n(u)du
E(M∗|T ) = 0 x = 0 . (22)
n n R R Ln R τ1nNΛ,n(u)du
0
Therefore, E(M∗)= E[Y ], where R
n n
τ1nuNΛ,n(u)du
Y = 0 . (23)
n R τ1nNΛ,n(u)du
0
R
Due to (20), the variable Y is equal in law to
n
τ1(u−τ )NΛ(u)du τ1uNΛ(u)du
Yˆ = τn n = τn −τ , given F , on the event E . (24)
n τ1NΛ(u)du τ1NΛ(u)du n τn n
R τn R τn
R R
UsingLemma10onecananalyzetheasymptoticbehaviorofYˆ /g(n)ineachofthethreecases
n
α¯ > 1, α¯ < 1 and α¯ = 1 (corresponding respectively to α < 3/2, α > 3/2 and α = 3/2). First
observe that τ /g(n) → 0 almost surely if α ≥ 3/2, and otherwise τ /g(n) → c almost surely,
n n
where c is the constant from claim 1 of Lemma 10. One can apply claims 4 and 5 (plugging
in τ as x) of Lemma 10 to the first term in (24). More precisely, if we let c = c α¯/cα¯ and
n 3 2
c = α¯/c, then
4
1. if α¯ > 1 then τ1uNΛ(u)du ∼ c τ−α¯+1 so Yˆ /g(n) ∼ c −c. almost surely.
τn 2 n n 3
2. if α¯ < 1 then Rτ1uNΛ(u)du ∼ Y so Yˆ /g(n) ∼ c Y, almost surely.
τn n 4
3. if α¯ = 1 thenR τ1uNΛ(u)du ∼ −c logτ ∼ c (α−1)logn so Yˆ /g(n) ∼ (α−1)c /c,
τn 2 n 2 n 2
almost surely.
R
With a slight abuseof notation, let Y := lim Yˆ /g(n), almost surely. Clearly P(Y ≥ 0) =
n→∞ n
1. Denote by D (Y) the set of points of continuity for the distribution function of Y. Then
c
for any x > 0 in D (Y) and any sequence (B ) of events, where B ∈ F , n ≥ 1, we have
c n n≥1 n τn
E(1 E(|1 −1 ||F ))= o(1), as n → ∞. (25)
Bn {Yˆn/g(n)≤x} {Y≤x} τn
We claim that
Y
n
⇒ Y, as n → ∞, (26)
g(n)
which can be verified as follows. Note that, due to (20), for each fixed x > 0
Y E[1 P(Yˆ /g(n) ≤ x|F )]
P n ≤ x = En n τn ,
g(n) P(E )
(cid:18) (cid:19) n
10