On Vervaat and Vervaat-error type processes for partial sums and renewals

0.21 MB·English

by  Endre Csáki

On Vervaat and Vervaat-error type processes
for partial sums and renewals
4
0
0 Endre Cs´aki1
2
Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127,
n
a H-1364, Hungary. E-mail: [email protected]
J

6
Mikl´os Cs¨org˝o2

]
R School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6.
P E-Mail: [email protected]
.
h
t Zdzis law Rychlik3
a
m
Instytut Matematyki, Uniwersytet Marii Curie-Sk lodowskiej, pl. Marii Curie-Sk lodowskiej 1,
[
PL-20 031 Lublin, Poland. E-Mail: [email protected]
1
v
1 Josef Steinebach4
4
0 Universita¨t zu Ko¨ln, Mathematisches Institut, Weyertal 86-90, D-50 931 Ko¨ln, Germany.
1
E-Mail: [email protected]
0
4
0
/
h Abstract. We study the asymptotic behaviour of stochastic processes that are generated by sums
t
a
of partial sums of i.i.d. random variables and their renewals. We conclude that these processes
m
cannot converge weakly to any nondegenerate random element of the space D[0,1]. On the other
:
v
hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge
i
X
weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the
r
a deviations of these processes, the so-called Vervaat-error type processes.
Keywords. Partial sums, renewals, Vervaat and Vervaat-error type processes, Wiener process,
strong and weak approximations, weak convergence.
2000 Mathematics Subject Classification. Primary 60F17; secondary 60F05;60F15
1Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 037886 and T
043037.
2Research supported by an NSERCCanada Grant at Carleton University,Ottawa.
3Research supported by the Deutsche Forschungsgemeinschaft through the German–Polish project 436 POL
113/98/0-1 ”Probability measures”.
4ResearchsupportedbyaPolish-German ExchangeGrantNo. BWM-III/DAAD/801/JK99, andbyaspecialgrant
from Marie Curie-Skl odowska University,Lublin.
1
1 Introduction
Let X ,X ,... be an i.i.d. sequence with d.f. F(x) =P(X ≤ x), −∞ <x < ∞, and
1 2 1
(i) EX = µ > 0;
1
(ii) 0 < Var(X )= σ2 (< ∞);
1
(iii) EX4 < ∞.
1
Set S := X +···+X (n = 1,2,...), S := 0. For t ≥ 0 define pointwise the corresponding renewal
n 1 n 0
(or first passage time) process as N(t) := min{n ≥ 1 : S > t}. We further introduce the “average”
n
and “standardized” processes
(1.1) S (t) := S /(nµ) (t ≥ 0);
n [nt]
(1.2) s (t) := n1/2σ−1µ S (t)−t = n−1/2σ−1 S −nµt (t ≥0);
n n [nt]
(cid:16) (cid:17) (cid:16) (cid:17)
(1.3) N (t) := N(nµt)/n (t ≥ 0);
n
(1.4) r (t) := n1/2σ−1µ N (t)−t = n−1/2σ−1(µN(nµt)−nµt) (t ≥ 0),
n n
(cid:16) (cid:17)
where [x] denotes the integer part of x.
Throughout the paper k · k denotes the sup-norm on D[0,1], i.e., for f ∈ D[0,1], kfk :=
sup |f(t)|.
0≤t≤1
We first collect some well-known facts concerning the above processes.
Theorem A On a large enough probability space, for X ,X ,... satisfying (i), (ii) and (iii) above,
1 2
one can construct a standard Wiener process {W(t) : 0 ≤ t < ∞}, such that, the processes s and
n
r can be simultaneously approximated by W as follows:
n
(1.5) lim n1/4ks −W k= 0 a.s.
n n
n→∞
and
(1.6) limsupn1/4(logn)−1/2(loglogn)−1/4kr +W k = 21/4σ1/2µ−1/2 a.s.,
n n
n→∞
where W (t) := n−1/2W(nt).
n
The result (1.5) is due to Komlo´s et al. (1976), and that of (1.6) is due to Horva´th (1984). For
more details and further developments we refer to Chapter 2 of Cso¨rgo˝ and Horva´th (1993).
Consider now the stochastic process
(1.7) R∗(t) := s (t)+r (t), 0 ≤t ≤ 1,
n n n
whichcan beviewed as theremainderterm in theBahadur-Kiefer typerepresentation r = −s +R∗
n n n
of the renewal process r in terms of the partial sums process s . In particular, Bahadur (1966),
n n
Kiefer (1967, 1970) introduced and studied the stochastic process
R (t) := α (t)+β (t), 0 ≤ t ≤ 1,
n n n
2
that is known in the literature as the Bahadur–Kiefer process, where
α (t) := n1/2(F (t)−t), 0 ≤ t ≤ 1,
n n
β (t) := n1/2(F−1(t)−t), 0 ≤ t ≤ 1,
n n
the uniformempirical and quantile processes, with F−1 being theleft-continuous inverse of theright-
n
continuously defined empirical distribution function F of the independent uniform (0,1) random
n
variables U ,...,U , n ≥ 1.
1 n
In this regard, we summarize the most relevant results of Kiefer (1967, 1970) in the following
theorem.
Theorem B For every fixed t ∈ (0,1), we have
(1.8) n1/4R (t) →D (t(1−t))1/4N(|N|)1/2, n → ∞,
n
n1/4|R (t)| 25/4
(1.9) limsup n = (t(1−t))1/4 e a.s.,
n→∞ (loglogn)3/4 33/4
D
where N and N are independent standard normal variables and → denotes convergence in distribu-
tion. Also,
e kR k
(1.10) lim n1/4(logn)−1/2 n = 1 a.s.
n→∞ (kαnk)1/2
As to (1.10), Kiefer (1970) announced it butproved only convergence in probability (cf. Theorem
1A, and the two sentences right after, in Kiefer, 1970). The upper bound for the almost sure
convergence in (1.10) was proved by Shorack (1982), and the lower bound by Deheuvels and Mason
(1990).
Concerning similar known results for the process R∗, we summarize them in the next theorem.
n
Theorem C Under the assumptions (i), (ii), (iii) we have for every fixed t ∈ (0,1]
(1.11) n1/4R∗(t) →D t1/4σ1/2µ−1/2N(|N|)1/2, n → ∞.
n
where, as in Theorem B, N and N are independent stanedard normal random variables, and
(1.12) lim n1/4(elogn)−1/2kR∗k/kr k1/2 = σ1/2µ−1/2 a.s.
n n
n→∞
For the result in (1.11) we refer to Cso¨rgo˝ and Horva´th (1993, Theorem 2.1.5) and for that of
(1.12) to Deheuvels and Mason (1990, Theorem 1B).
As a consequence of (1.12), on account of
D
(1.13) kr k → kWk (cf., e.g., Vervaat (1972)),
n
as n → ∞, we have (cf. Deheuvels and Mason (1990))
(1.14) n1/4(logn)−1/2kR∗k →D σ1/2µ−1/2kWk.
n
3
Based on (1.11) and (1.14), the next conclusion is immediate.
Corollary 1.1 Given the assumptions (i), (ii), (iii), the statement
(1.15) a R∗ →D Y, n → ∞,
n n
cannot hold true in the space D[0,1] (endowed with the Skorokhod topology) for any non-degenerate
random element Y of D[0,1] with any normalizing sequence {a }.
n
The respective results of (1.11) and (1.12) are based on a strong invariance principle for the
Bahadur-Kiefer type process {R∗(t), 0 ≤ t ≤ 1; n = 1,2,...} that was explicitly stated in Cso¨rgo˝
n
and Horva´th (1993, p. 43) as follows.
Theorem D On assuming the conditions (i), (ii), (iii), on the probability space of Theorem A, as
n → ∞, we have
σ
(1.16) R∗(t) = n−1/2 W(nt)−W nt− W(nt) +o(n−1/4) a.s.,
n µ
(cid:18) (cid:18) (cid:19)(cid:19)
uniformly in t ∈ [0,1].
Corollary 1.1 and Theorem 3 of Vervaat (1972) serve as motivation for studying the following
Vervaat-type process for partial sums and renewals:
t
(1.17) V (t) := S (s)−s + N (s)−s ds
n n n
Z0 n(cid:16) (cid:17) (cid:16) (cid:17)o
t
= M (s)ds, 0 ≤ t ≤ 1,
n
Z0
where M (s):= n−1/2σµ−1R∗(s), s ≥ 0.
n n
So far we have been dealing with renewal processes of partial sums under the conditions (i),
(ii) and (iii), that is to say we had general renewal processes in mind. We will now see that the
just introduced Vervaat-type process V will be well-behaving asymptotically as n → ∞. However
n
the asymptotic behaviour of V will be different when it is based on general renewal processes, as
n
compared to it being based on ordinary renewal processes, i.e., when in addition to (i), (ii), (iii),
we also assume that X is positive. Consequently, in our Section 2 we will deal with Vervaat-type
1
processesforordinaryrenewals, whileSection 3willbedevoted tostudyingsuchprocessesforgeneral
renewals. Moreover, in Section 4 we will also be studying the asymptotic behaviour of Vervaat error-
type processes that, based on the respective results of Sections 2 and 3, will be defined there (cf.
(4.1)).
2 The Vervaat-type process for ordinary renewals
In addition to the conditions (i), (ii), (iii) of Section 1, in this Section we assume also the condition
(iv) P(X > 0) = 1.
1
4
Theorem 2.1 Assume the conditions (i), (ii), (iii) and (iv). Then, as n → ∞
n 1
(a) V − S¯2 → 0 a.s.
loglogn n 2 n
(cid:13) (cid:13)
(cid:13) (cid:13)
and (cid:13) (cid:13)
1
(b) n V − S¯2 →P 0,
n 2 n
(cid:13) (cid:13)
where S¯n(t) := Sn(t)−t. (cid:13)(cid:13) (cid:13)(cid:13)
Proof. Given condition (iv) it can be checked (cf. Figure 1 below) that V has the following
n
representation:
1
(2.1) V (t)= A (t)−S¯ (t)N¯ (t)− N¯2(t),
n n n n 2 n
where
t
(2.2) A (t) := S¯ (s)−S¯ (t) ds,
n n n
ZNn(t)(cid:16) (cid:17)
(2.3) S¯ (t) := S (t)−t = (nµ)−1 S −nµt ,
n n [nt]
(cid:16) (cid:17)
(2.4) N¯ (t):= N (t)−t = n−1 N(nµt)−nt ,
n n
(cid:16) (cid:17)
for all 0 ≤ t ≤ 1.
In order to check the above representation, we note that
S S
k−1 k
N(nµt)= k, for ≤ t < .
nµ nµ
Also, by calculations, we arrive at
t Nn(t)
S (s)+N (s) ds = t2+ (t−S (s))ds
n n n
Z0 (cid:16) (cid:17) Zt
t t
= (S (s)−s)−(S (t)−t) ds− (t−s)ds−(S (t)−t)(N (t)−t)+t2.
n n n n
ZNn(t)n o ZNn(t)
Consequently, we conclude (2.1) (cf. also Figure 1) as follows:
t 1
V (t) = (S (s)−s)+(N (s)−s) ds = A (t)− N¯2(t)−S¯ (t)N¯ (t)+t2−t2.
n n n n 2 n n n
Z0 (cid:16) (cid:17)
5
6
V (t)
n
SNn(t)
nµ t
SNn(t)−1
nµ
S[nt]
nµ
S2
nµ
S1
nµ
-
t
1 2 [nt] N(nµt) = N (t)
n n n n n
Figure 1
As an immediate consequence in our proof, we get
1 1 2
(2.5) V (t)− S¯2(t) = A (t)− S¯ (t)+N¯ (t)
n 2 n n 2 n n
1(cid:16) (cid:17)
= A (t)− M2(t).
n 2 n
Now, in view of (1.6), (1.12) and the law of the iterated logarithm for a Wiener process,
(2.6) kM2k= O n−3/2(logn)(loglogn)1/2 a.s.,
n
(cid:16) (cid:17)
and, as a consequence of (1.14),
(2.7) kM2k = O n−3/2logn .
n P
(cid:16) (cid:17)
Next, we show that A is the dominating process on the right-hand side of (2.5). In (2.2) put
n
s = t−u(t−N (t)) = t+uN¯ (t). Then we get
n n
1
(2.8) A (t) = −N¯ (t) S¯ (t+uN¯ (t))−S¯ (t) du.
n n n n n
Z0 (cid:16) (cid:17)
On account of (1.5),
nµ
(2.9) S¯ (t)= W(nt)+o(n1/4) a.s.,
n
σ
6
uniformly in t ∈ [0,1]. A combination of (2.8) and (2.9) yields
nµ 1
(2.10) A (t) = −N¯ (t) W(nt+unN¯ (t))−W(nt) du+o n−1/4(loglogn)1/2 a.s.,
n n n
σ
Z0 (cid:16) (cid:17) (cid:16) (cid:17)
uniformly in t ∈ [0,1], since
nN¯ (t) =O (nloglogn)1/2 a.s.,
n
(cid:16) (cid:17)
uniformly in t ∈ [0,1].
This rate in (2.10) can also be replaced by o (n−1/4), on account of
P
nN¯ (t) =O (n1/2),
n P
uniformly in t ∈ [0,1].
Moreover, by Theorem 1.2.1 of Cso¨rgo˝ and R´ev´esz (1981),
(2.11) W(nt+unN¯ (t))−W(nt)=O n1/4(logn)1/2(loglogn)1/4 a.s.
n
(cid:16) (cid:17)
as well as
(2.12) W(nt+unN¯ (t))−W(nt)= O n1/4(logn)1/2 ,
n P
(cid:16) (cid:17)
uniformly in u,t ∈ [0,1].
In view of (1.6), we also have
µ
(2.13) nN¯ (t) = −W(nt)+O n1/4(logn)1/2(loglogn)1/4 a.s.
n
σ
(cid:16) (cid:17)
As a consequence of (2.10)–(2.13), we arrive at
nµ σW(nt) 1
(2.14) A (t)= W(nt+unN¯ (t))−W(nt) du+o n−1/4(loglogn)1/2 a.s.,
n n
σ nµ
Z0 (cid:16) (cid:17) (cid:16) (cid:17)
uniformly in t ∈ [0,1], where this rate can also be replaced by o (n−1/4).
P
On making use of (2.13) in combination with Theorem 1.2.1 of Cso¨rgo˝ and R´ev´esz (1981), we
conclude the following strong and weak invariance principles for A (t) of (2.2):
n
σW(nt) 1 σ σ
(2.15)A (t) = W nt−u W(nt) −W(nt) du+o n−5/4(loglogn)1/2 a.s.
n
nµ nµ µ
= σ nσµWZ0(nt)(W(cid:16)(nt(cid:16)−nx)−W(nt)(cid:17))dx+o n−(cid:17)5/4(log(cid:16)logn)1/2 a.s. (cid:17)
nµ
Z0 (cid:16) (cid:17)
Yn(t)
= (Y (t−x)−Y (t))dx+o n−5/4(loglogn)1/2 a.s.,
n n
Z0 (cid:16) (cid:17)
uniformlyint ∈[0,1], whereY (t) := σ W(nt),andthis a.s.ratecan alsobereplaced byo n−5/4 .
n nµ P
Arguing similarly as in (2.11) and (2.12), we also have (cid:16) (cid:17)
(2.16) A (t) = O n−1/2(loglogn)1/2 n−3/4(logn)1/2(loglogn)1/4 a.s.
n
(cid:16) (cid:17)
= O n−5/4(logn)1/2(loglogn)3/4 a.s.,
(cid:16) (cid:17)
7
and
(2.17) A (t) = O n−5/4(logn)1/2 ,
n P
(cid:16) (cid:17)
uniformly in t ∈ [0,1].
Thus, in view of (2.5), (2.6) and (2.16) we conclude (a) of Theorem 2.1, while (2.5), (2.7) and
(2.17) result in (b) of Theorem 2.1. (cid:3)
As an immediate consequence of Theorem 2.1 and Theorem A, we get
Corollary 2.1 Assume the conditions (i), (ii), (iii) and (iv). Then, as n → ∞, on the probability
space of Theorem A with W as in (1.5), we have
n
2
1 1 σ
(a) nV − W → 0 a.s.
n n
loglogn 2 µ
(cid:13) (cid:18) (cid:19) (cid:13)
(cid:13) (cid:13)
and (cid:13) (cid:13)
2
1 σ
P
(b) nV − W → 0.
n n
2 µ
(cid:13) (cid:18) (cid:19) (cid:13)
(cid:13) (cid:13)
As a consequence of (b) of Coroll(cid:13)ary 2.1, one conclud(cid:13)es also
D
(2.18) nV → Z, n → ∞,
n
2
in the space C[0,1] (endowed with the uniform topology), where Z(t):= 1 σW(t) , t ∈ [0,1].
2 µ
The strong invariance principle (a) of Corollary 2.1 in turn implies Stra(cid:16)ssen (19(cid:17)64)-type laws of
the iterated logarithm for V as follows.
n
Corollary 2.2 Assume the conditions (i), (ii), (iii) and (iv). Then, as n → ∞, the set
µ2nV
n
(2.19) , n ≥ 3
σ2loglogn
(cid:26) (cid:27)
is relatively compact in C[0,1], equipped with the sup-norm k·k. Furthermore, the set of all limit
points of the functions in the set (2.19) almost surely coincides with the set {f2 : f ∈ S}, where S is
the Strassen class of all absolutely continuous functions f on [0,1] such that f(0) = 0 and kf′k ≤ 1,
2
where k·k denotes the L -norm, p ≥ 1.
p p
As examples of consequences of Corollary 2.2 `a la Strassen (1964), we mention the following
results.
Corollary 2.3 Under the conditions (i), (ii), (iii) and (iv) we have
nkV k σ2
n
limsup = a.s.
n→∞ loglogn µ2
and
nkV k 4σ2
n 1
limsup = a.s.
n→∞ loglogn µ2π2
8
3 The Vervaat process for general renewals
In this section we study the case of general renewals for which assumption (iv) of Section 2 may not
necessarily be true. In order to do this, we need to introduce some further notations and auxiliary
results:
Let ν denote the i-th (strong) ascending ladder index of the partial sum sequence {S } , i.e.,
i n n=0,1,...
ν = 0, and, recursively,
0
(3.1) ν := min n > ν : S −S > 0 .
i i−1 n νi−1
n o
We note that, under assumptions (i), (ii), (iii), {ν − ν } is an i.i.d. sequence of random
i i−1 i=1,2,...
variables with
(3.2) Eν4 < ∞.
1
Moreover, the sequence {S − S } of ladder heights is also an i.i.d. sequence of random
νi νi−1 i=1,2,...
variables with
(3.3) ES4 < ∞
ν1
(cf., e.g., Gut (1988, Sections III. 2-3)). By definition,
N(t) = ν , for S ≤ t < S (i = 1,2,...; t ≥ 0),
i νi−1 νi
i.e.,
S S
(3.4) N(nµs)= ν , for νi−1 ≤ s< νi (i = 1,2,...; s≥ 0).
i
nµ nµ
Let N(nµt)= ν , i.e.,
ℓ
(3.5) ℓ = min j : S > nµt ,
νj
n o
that is, ℓ = N (nµt), whereN denotes the (ordinary) renewal process correspondingto the (ladder
H H
height) sequence S ,S −S ,.... This will play a crucial role in the calculations below.
ν1 ν2 ν1
We first note that we continue to use the same definition for the Vervaat process V for general
n
renewals as in (1.17). Namely, we study the integral
t t
V (t)= M (s)ds = S¯ (s)+N¯ (s) ds,
n n n n
Z0 Z0 (cid:16) (cid:17)
where S¯ and N¯ are as in (2.3) and (2.4), respectively, i.e.,
n n
S¯ (t) = S (t)−t = (nµ)−1(S −nµt),
n n [nt]
N¯ (t) = N (t)−t = n−1(N(nµt)−nt),
n n
with S (t) and N (t) as in (1.1) and (1.3), respectively.
n n
Consider now the following figure.
9
6
Sνℓ
nµ
t
Sνℓ−1
nµ
A
ℓ
A
ℓ−1
Sν2
nµ
A
2
Sν1
nµ
1 ν1
n n
-
t
A 3 ν2 [nt] νℓ−1 N(nµt) = N (t) = νℓ
1 n n n n n n n
Figure 2
Note that, in contrast to the case of the ordinary renewal process (cf. Figure 1), the (random)
areas A ,A ,... (cf. Figure 2) are additional in the present case of the Vervaat process for general
1 2
renewals. Thus, in computing the above integral V , these areas are to be taken into consideration.
n
Our next lemma accomplishes this via establishing the following new representation for V in the
n
general case that will now replace that of (2.5).
Lemma 3.1 Under the conditions (i), (ii), (iii) we have
1 1
(3.6) V (t) = A (t)+B (t)− M2(t)+ S¯2(t),
n n n 2 n 2 n
with A as in (2.2), M = S¯ +N¯ and B is defined by
n n n n n
1 NH(nµt) νi−νi−1−1
(3.7) B (t) := (ν −ν −j)X ,
n n2µ  i i−1 νi−1+j
i=1 j=1
X X
 
with N (nµt):= ℓ of (3.5).
H
10