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

Endre Csáki^{1}

Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127,

H-1364, Hungary. E-mail: csaki@renyi.hu

Miklós Csörgő^{2}

School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6.

E-Mail: mcsorgo@math.carleton.ca

Zdzisław Rychlik^{3}

Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1,

PL-20 031 Lublin, Poland. E-Mail: rychlik@golem.umcs.lublin.pl

Josef Steinebach^{4}

Universität zu Köln, Mathematisches Institut, Weyertal 86-90, D-50 931 Köln, Germany.

E-Mail: jost@math.uni-koeln.de

Abstract. We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space D[0, 1]. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the 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 NSERC Canada Grant at Carleton University, Ottawa.

3Research supported by the Deutsche Forschungsgemeinschaft through the German–Polish project 436 POL 113/98/0-1 ”Probability measures”.

4Research supported by a Polish-German Exchange Grant No. BWM-III/DAAD/801/JK99, and by a special grant from Marie Curie-Skłodowska University, Lublin.

1 Introduction

Let X_{1}, X_{2}, … be an i.i.d. sequence with d.f. F (x) = P (X_{1} ≤ x), −∞ < x < ∞, and

(i) EX_{1} = μ > 0;

(ii) 0 < Var(X_{1}) = σ^{2} (< ∞);

(iii) EX_{1}^{4} < ∞.

Set S_{n} := X_{1} + ⋯ + X_{n} (n = 1, 2, …), S_{0} := 0. For t ≥ 0 define pointwise the corresponding renewal (or first passage time) process as N(t) := min{n ≥ 1 : S_{n} > t}. We further introduce the “average” and “standardized” processes S_{n}(t) := S_{[nt]}/(nμ) (t ≥ 0); (1.1)

s_{n}(t) := n^{1/2}σ^{−1}μ(S_{n}(t) − t) = n^{−1/2}σ^{−1}(S_{[nt]} − nμt) (t ≥ 0); (1.2)

N_{n}(t) := N(nμt)/n (t ≥ 0); (1.3)

r_{n}(t) := n^{1/2}σ^{−1}μ(N_{n}(t) − t) = n^{−1/2}σ^{−1}(μN(nμt) − nμt) (t ≥ 0), (1.4)

where [x] denotes the integer part of x.

Throughout the paper ∥ · ∥ denotes the sup-norm on D[0, 1], i.e., for f ∈ D[0, 1], ∥f∥ := sup_{0≤t≤1} |f(t)|.

We first collect some well-known facts concerning the above processes.

Theorem A On a large enough probability space, for X_{1}, X_{2}, … satisfying (i), (ii) and (iii) above, one can construct a standard Wiener process {W (t) : 0 ≤ t < ∞}, such that, the processes s_{n} and r_{n} can be simultaneously approximated by W as follows:

lim_{n→∞} n^{1/4}∥s_{n} − W_{n}∥ = 0 a.s. (1.5)

and

(1.6)  lim sup_{n→∞} n^{1/4}(log n)^{−1/2}(log log n)^{−1/4}∥r_{n} + W_{n}∥ = 2^{1/4}σ^{1/2}μ^{−1/2} a.s.,

where W_{n}(t) := n^{−1/2}W (nt).

The result (1.5) is due to Komlós et al. (1976), and that of (1.6) is due to Horváth (1984). For more details and further developments we refer to Chapter 2 of Csörgő and Horváth (1993).

Consider now the stochastic process

R_{n}^{∗}(t) := s_{n}(t) + r_{n}(t), 0 ≤ t ≤ 1, (1.7)

which can be viewed as the remainder term in the Bahadur-Kiefer type representation r_{n} = −s_{n} +R_{n}^{∗} of the renewal process r_{n} in terms of the partial sums process s_{n}. In particular, Bahadur (1966), Kiefer (1967, 1970) introduced and studied the stochastic process

R_{n}(t) := α_{n}(t) + β_{n}(t), 0 ≤ t ≤ 1,

that is known in the literature as the Bahadur–Kiefer process, where

α_{n}(t) := n^{1/2}(F_{n}(t) − t),  0 ≤ t ≤ 1,
β_{n}(t) := n^{1/2}(F_{n}^{−1}(t) − t),  0 ≤ t ≤ 1,

the uniform empirical and quantile processes, with F_{n}^{−1} being the left-continuous inverse of the rightcontinuously defined empirical distribution function F_{n} of the independent uniform (0,1) random variables U_{1}, …, U_{n}, n ≥ 1.

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)  n^{1/4}R_{n}(t) →^{D} (t(1 − t))^{1/4}N (|Ñ|)^{1/2}, n → ∞,
(1.9)  lim sup_{n→∞} \frac{n^{1/4}|R_{n}(t)|}{(log log n)^{3/4}} = (t(1 − t))^{1/4} \frac{2^{5/4}}{3^{3/4}}  a.s.,

where N and Ñ are independent standard normal variables and →^{D} denotes convergence in distribution. Also,

(1.10)  lim_{n→∞} n^{1/4}(log n)^{−1/2} \frac{∥R_{n}∥}{(∥α_{n}∥)^{1/2}} = 1  a.s.

As to (1.10), Kiefer (1970) announced it but proved 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_{n}^{∗}, we summarize them in the next theorem.

Theorem C Under the assumptions (i), (ii), (iii) we have for every fixed t ∈ (0, 1]

n^{1/4}R_{n}^{∗}(t) →^{D} t^{1/4}σ^{1/2}μ^{−1/2}N (|Ñ|)^{1/2}, n → ∞. (1.11)

where, as in Theorem B, N and Ñ are independent standard normal random variables, and

(1.12)  lim_{n→∞} n^{1/4}(log n)^{−1/2}∥R_{n}^{∗}∥/∥r_{n}∥^{1/2} = σ^{1/2}μ^{−1/2} a.s.

For the result in (1.11) we refer to Csörgő and Horváth (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

(1.13)  ∥r_{n}∥ →^{D} ∥W ∥  (cf., e.g., Vervaat (1972)),

as n → ∞, we have (cf. Deheuvels and Mason (1990))

n^{1/4}(log n)^{−1/2}∥R_{n}^{∗}∥ →^{D} σ^{1/2}μ^{−1/2}∥W ∥. (1.14)

Based on (1.11) and (1.14), the next conclusion is immediate.

Corollary 1.1 Given the assumptions (i), (ii), (iii), the statement

a_{n}R_{n}^{∗} →^{D} Y, n → ∞, (1.15)

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_{n}^{∗}(t), 0 ≤ t ≤ 1; n = 1, 2, …} that was explicitly stated in Csörgő and Horváth (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

R_{n}^{∗}(t) = n^{−1/2} (W (nt) − W (nt − \frac{σ}{μ}W (nt))) + o(n^{−1/4}) a.s., (1.16)

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:

(1.17)  V_{n}(t) := ∫_{0}^{t} {(S_{n}(s) − s) + (N_{n}(s) − s)}ds
= ∫_{0}^{t} M_{n}(s)ds,  0 ≤ t ≤ 1,

where M_{n}(s) := n^{−1/2}σμ^{−1}R_{n}^{∗}(s), s ≥ 0.

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_{n} will be well-behaving asymptotically as n → ∞. However the asymptotic behaviour of V_{n} will be different when it is based on general renewal processes, as compared to it being based on ordinary renewal processes, i.e., when in addition to (i), (ii), (iii), we also assume that X_{1} is positive. Consequently, in our Section 2 we will deal with Vervaat-type processes for ordinary renewals, while Section 3 will be devoted to studying such processes for general renewals. Moreover, in Section 4 we will also be studying the asymptotic behaviour of Vervaat errortype 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_{1} > 0) = 1.

Theorem 2.1 Assume the conditions (i), (ii), (iii) and (iv). Then, as n → ∞

(a)  \frac{n}{log log n}∥V_{n} − \frac{1}{2}S̄_{n}^{2}∥ → 0 a.s.

and

(b)  n∥V_{n} − \frac{1}{2}S̄_{n}^{2}∥ →^{P} 0,

where S̄_{n}(t) := S_{n}(t) − t.

Proof. Given condition (iv) it can be checked (cf. Figure 1 below) that V_{n} has the following representation:

V_{n}(t) = A_{n}(t) − S̄_{n}(t)N̄_{n}(t) − \frac{1}{2}N̄_{n}^{2}(t), (2.1)

where A_{n}(t) := ∫_{N_{n}(t)}^{t} (S̄_{n}(s) − S̄_{n}(t))ds, (2.2)

S̄_{n}(t) := S_{n}(t) − t = (nμ)^{−1}(S_{[nt]} − nμt), (2.3)

N̄_{n}(t) := N_{n}(t) − t = n^{−1}(N(nμt) − nt), (2.4)

for all 0 ≤ t ≤ 1.

In order to check the above representation, we note that

N(nμt) = k, for \frac{S_{k−1}}{nμ} ≤ t < \frac{S_{k}}{nμ}.

Also, by calculations, we arrive at

∫_{0}^{t} (S_{n}(s) + N_{n}(s))ds = t^{2} + ∫_{t}^{N_{n}(t)}(t − S_{n}(s))ds
= ∫_{N_{n}(t)}^{t} {(S_{n}(s) − s) − (S_{n}(t) − t)}ds − ∫_{N_{n}(t)}^{t}(t − s)ds − (S_{n}(t) − t)(N_{n}(t) − t) + t^{2}.

Consequently, we conclude (2.1) (cf. also Figure 1) as follows:

V_{n}(t) = ∫_{0}^{t} ((S_{n}(s) − s) + (N_{n}(s) − s))ds = A_{n}(t) − \frac{1}{2}N̄_{n}^{2}(t) − S̄_{n}(t)N̄_{n}(t) + t^{2} − t^{2}.

[figure: \frac{1}{n}; \frac{2}{n}; \frac{[nt]}{n}; \frac{N(nμt)}{n}; = N_{n}(t); t; \frac{S_{Nn(t)}}{nμ}; t; \frac{S_{Nn(t)−1}}{nμ}; \frac{S_{[nt]}}{nμ}; \frac{S_{2}}{nμ}; \frac{S_{1}}{nμ}; V_{n}(t)]

Figure 1

As an immediate consequence in our proof, we get

(2.5)  V_{n}(t) − \frac{1}{2}S̄_{n}^{2}(t) = A_{n}(t) − \frac{1}{2}(S̄_{n}(t) + N̄_{n}(t))^{2}
= A_{n}(t) − \frac{1}{2}M_{n}^{2}(t).

Now, in view of (1.6), (1.12) and the law of the iterated logarithm for a Wiener process,

(2.6) ∥M_{n}^{2}∥ = O(n^{−3/2}(log n)(log log n)^{1/2}) a.s.,

and, as a consequence of (1.14), ∥M_{n}^{2}∥ = O_{P} (n^{−3/2} log n). (2.7)

Next, we show that A_{n} is the dominating process on the right-hand side of (2.5). In (2.2) put s = t − u(t − N_{n}(t)) = t + uN̄_{n}(t). Then we get

A_{n}(t) = −N̄_{n}(t) ∫_{0}^{1} (S̄_{n}(t + uN̄_{n}(t)) − S̄_{n}(t))du. (2.8)

On account of (1.5),

\frac{nμ}{σ}S̄_{n}(t) = W (nt) + o(n^{1/4}) a.s., (2.9)

uniformly in t ∈ [0, 1]. A combination of (2.8) and (2.9) yields

(2.10) \frac{nμ}{σ}A_{n}(t) = −N̄_{n}(t) ∫_{0}^{1} (W (nt + unN̄_{n}(t)) − W (nt))du + o(n^{−1/4}(log log n)^{1/2}) a.s.,

uniformly in t ∈ [0, 1], since

nN̄_{n}(t) = O((n log log n)^{1/2}) a.s.,

uniformly in t ∈ [0, 1].

This rate in (2.10) can also be replaced by o_{P} (n^{−1/4}), on account of

nN̄_{n}(t) = O_{P} (n^{1/2}),

uniformly in t ∈ [0, 1].

Moreover, by Theorem 1.2.1 of Csörgő and Révész (1981),

(2.11) W (nt + unN̄_{n}(t)) − W (nt) = O(n^{1/4}(log n)^{1/2}(log log n)^{1/4}) a.s.

as well as W (nt + unN̄_{n}(t)) − W (nt) = O_{P} (n^{1/4}(log n)^{1/2}), (2.12)

uniformly in u, t ∈ [0, 1].

In view of (1.6), we also have

(2.13) \frac{μ}{σ}nN̄_{n}(t) = −W (nt) + O(n^{1/4}(log n)^{1/2}(log log n)^{1/4}) a.s.

As a consequence of (2.10)–(2.13), we arrive at

(2.14) \frac{nμ}{σ}A_{n}(t) = \frac{σW (nt)}{nμ} ∫_{0}^{1} (W (nt + unN̄_{n}(t)) − W (nt))du + o(n^{−1/4}(log log n)^{1/2}) a.s.,

uniformly in t ∈ [0, 1], where this rate can also be replaced by o_{P} (n^{−1/4}).

On making use of (2.13) in combination with Theorem 1.2.1 of Csörgő and Révész (1981), we conclude the following strong and weak invariance principles for A_{n}(t) of (2.2):

(2.15)A_{n}(t) = \frac{σW (nt)}{nμ} ∫_{0}^{1} \frac{σ}{nμ}(W (nt − u\frac{σ}{μ}W (nt)) − W (nt))du + o(n^{−5/4}(log log n)^{1/2}) a.s.
= \frac{σ}{nμ} ∫_{0}^{\frac{σ}{nμ} W (nt)}(W (nt − nx) − W (nt))dx + o(n^{−5/4}(log log n)^{1/2}) a.s.
= ∫_{0}^{Y_{n}(t)}(Y_{n}(t − x) − Y_{n}(t))dx + o(n^{−5/4}(log log n)^{1/2}) a.s.,

uniformly in t ∈ [0, 1], where Y_{n}(t) := \frac{σ}{nμ}W (nt), and this a.s. rate can also be replaced by o_{P} (n^{−5/4}). Arguing similarly as in (2.11) and (2.12), we also have

(2.16)  A_{n}(t) = O(n^{−1/2}(log log n)^{1/2} n^{−3/4}(log n)^{1/2}(log log n)^{1/4}) a.s.
= O(n^{−5/4}(log n)^{1/2}(log log n)^{3/4}) a.s.,

and A_{n}(t) = O_{P} (n^{−5/4}(log n)^{1/2}), (2.17)

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. □

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_{n} as in (1.5), we have

(a)  \frac{1}{log log n}∥nV_{n} − \frac{1}{2} (\frac{σ}{μ}W_{n})^{2} ∥ → 0 a.s.

and

(b)  ∥nV_{n} − \frac{1}{2} (\frac{σ}{μ}W_{n})^{2} ∥ →^{P} 0.

As a consequence of (b) of Corollary 2.1, one concludes also

nV_{n} →^{D} Z, n → ∞, (2.18)

in the space C[0, 1] (endowed with the uniform topology), where Z(t) := \frac{1}{2}(\frac{σ}{μ}W (t))^{2}, t ∈ [0, 1].

The strong invariance principle (a) of Corollary 2.1 in turn implies Strassen (1964)-type laws of the iterated logarithm for V_{n} as follows.

Corollary 2.2 Assume the conditions (i), (ii), (iii) and (iv). Then, as n → ∞, the set

\{\frac{μ^{2}nV_{n}}{σ^{2} log log n}, n ≥ 3\} (2.19)

is relatively compact in C[0, 1], equipped with the sup-norm ∥ · ∥. Furthermore, the set of all limit points of the functions in the set (2.19) almost surely coincides with the set {f^{2} : f ∈ S}, where S is the Strassen class of all absolutely continuous functions f on [0, 1] such that f(0) = 0 and ∥f′∥_{2} ≤ 1, where ∥ · ∥_{p} denotes the L_{p}-norm, p ≥ 1.

As examples of consequences of Corollary 2.2 à la Strassen (1964), we mention the following results.

Corollary 2.3 Under the conditions (i), (ii), (iii) and (iv) we have

lim sup_{n→∞} \frac{n∥V_{n}∥}{log log n} = \frac{σ^{2}}{μ^{2}} a.s.

and

lim sup_{n→∞} \frac{n∥V_{n}∥_{1}}{log log n} = \frac{4σ^{2}}{μ^{2}π^{2}} a.s.

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_{i} denote the i-th (strong) ascending ladder index of the partial sum sequence {S_{n}}_{n=0,1,...}, i.e., ν_{0} = 0, and, recursively, ν_{i} := min {n > ν_{i−1} : S_{n} − S_{ν_{i−1}} > 0}. (3.1)

We note that, under assumptions (i), (ii), (iii), {ν_{i} − ν_{i−1}}_{i=1,2,...} is an i.i.d. sequence of random variables with Eν_{1}^{4} < ∞. (3.2)

Moreover, the sequence {S_{ν_{i}} − S_{ν_{i−1}}}_{i=1,2,...} of ladder heights is also an i.i.d. sequence of random variables with ES_{ν_{1}}^{4} < ∞ (3.3)

(cf., e.g., Gut (1988, Sections III. 2-3)). By definition,

N(t) = ν_{i}, for S_{ν_{i−1}} ≤ t < S_{ν_{i}} (i = 1, 2, …; t ≥ 0),

i.e.,

N(nμs) = ν_{i}, for \frac{S_{ν_{i−1}}}{nμ} ≤ s < \frac{S_{ν_{i}}}{nμ} (i = 1, 2, …; s ≥ 0). (3.4)

Let N(nμt) = ν_{ℓ}, i.e., ℓ = min {j : S_{ν_{j}} > nμt}, (3.5)

that is, ℓ = N_{H}(nμt), where N_{H} denotes the (ordinary) renewal process corresponding to the (ladder height) sequence S_{ν_{1}}, S_{ν_{2}} − S_{ν_{1}}, . . .. This will play a crucial role in the calculations below.

We first note that we continue to use the same definition for the Vervaat process V_{n} for general renewals as in (1.17). Namely, we study the integral

V_{n}(t) = ∫_{0}^{t} M_{n}(s) ds = ∫_{0}^{t} (S̄_{n}(s) + N̄_{n}(s))ds,

where S̄_{n} and N̄_{n} are as in (2.3) and (2.4), respectively, i.e.,

S̄_{n}(t) = S_{n}(t) − t = (nμ)^{−1}(S_{[nt]} − nμt),
N̄_{n}(t) = N_{n}(t) − t = n^{−1}(N(nμt) − nt),

with S_{n}(t) and N_{n}(t) as in (1.1) and (1.3), respectively.

Consider now the following figure.

[figure: \frac{3}{n}; \frac{ν_{2}}{n}; \frac{[nt]}{n}; \frac{ν_{ℓ−1}}{n}; \frac{N(nμt)}{n}; \frac{1}{n}; \frac{ν_{1}}{n}; = N_{n}(t) = \frac{ν_{ℓ}}{n}; t; \frac{S_{ν_{ℓ}}}{nμ}; t; \frac{S_{ν_{ℓ−1}}}{nμ}; \frac{S_{ν_{2}}}{nμ}; \frac{S_{ν_{1}}}{nμ}; A_{1}; A_{2}; A_{ℓ−1}; A_{ℓ}]

Figure 2

Note that, in contrast to the case of the ordinary renewal process (cf. Figure 1), the (random) areas A_{1}, A_{2}, … (cf. Figure 2) are additional in the present case of the Vervaat process for general renewals. Thus, in computing the above integral V_{n}, these areas are to be taken into consideration. Our next lemma accomplishes this via establishing the following new representation for V_{n} in the general case that will now replace that of (2.5).

Lemma 3.1 Under the conditions (i), (ii), (iii) we have

V_{n}(t) = A_{n}(t) + B_{n}(t) − \frac{1}{2}M_{n}^{2}(t) + \frac{1}{2}S̄_{n}^{2}(t), (3.6)

with A_{n} as in (2.2), M_{n} = S̄_{n} + N̄_{n} and B_{n} is defined by

B_{n}(t) := \frac{1}{n^{2}μ} ∑_{i=1}^{N_{H}(nμt)} ( ∑_{j=1}^{ν_{i}−ν_{i−1}−1}(ν_{i} − ν_{i−1} − j)X_{ν_{i−1}+j}) , (3.7)

with N_{H}(nμt) := ℓ of (3.5).

Proof. We first show that

∫_{0}^{t} N̄_{n}(s)ds + ∫_{0}^{N_{n}(t)} S̄_{n}(s)ds = −\frac{1}{2}N̄_{n}^{2}(t) + B_{n}(t), (3.8)

We have

∫_{0}^{t}(N_{n}(s) − s)ds + ∫_{0}^{N_{n}(t)}(S_{n}(s) − s)ds = ∫_{0}^{t} N_{n}(s)ds + ∫_{0}^{N_{n}(t)} S_{n}(s)ds − \frac{t^{2}}{2} − \frac{N_{n}^{2}(t)}{2}. (3.9)

By (3.4) and the definition of N_{n} we arrive at

(3.10)  ∫_{0}^{t} N_{n}(s)ds = \frac{1}{n} ∑_{i=1}^{ℓ} ∫ \frac{Sν_{i}}{nμ} ; \frac{Sν_{i−1}}{nμ} ν_{i} ds − \frac{1}{n} ∫t^{\frac{Sν_{ℓ}}{nμ}} ν_{ℓ} ds
= \frac{1}{n^{2}μ} ∑_{i=1}^{ℓ} ν_{i}(S_{ν_{i}} − S_{ν_{i−1}}) − \frac{1}{n}ν_{ℓ}(\frac{S_{ν_{ℓ}}}{nμ} − t).

Also,

(3.11) ∫_{0}^{N_{n}(t)} S_{n}(s)ds = ∑_{i=1}^{ν_{ℓ}} ∫ \frac{i}{n} ; \frac{i−1}{n} S_{n}(s)ds = \frac{1}{n^{2}μ} ∑_{i=1}^{ν_{ℓ}} S_{i−1}
= \frac{1}{n^{2}μ} ∑_{i=1}^{ℓ} ∑_{j=1}^{ν_{i}−ν_{i−1}} (S_{ν_{i−1}+j−1} − S_{ν_{i−1}}) + \frac{1}{n^{2}μ} ∑_{i=1}^{ℓ}(ν_{i} − ν_{i−1})S_{ν_{i−1}}.

Now, for any real sequences {a_{i}}_{i=0,1,...} and {b_{i}}_{i=0,1,...}, it can easily be checked that

∑_{i=1}^{n} a_{i}(b_{i} − b_{i−1}) + ∑_{i=1}^{n} b_{i−1}(a_{i} − a_{i−1}) = a_{n}b_{n} − a_{0}b_{0}.

Hence, since ν_{0} = 0 = S_{ν_{0}},

∑_{i=1}^{ℓ} ν_{i}(S_{ν_{i}} − S_{ν_{i−1}}) = ν_{ℓ}S_{ν_{ℓ}} − ∑_{i=1}^{ℓ}(ν_{i} − ν_{i−1})S_{ν_{i−1}}. (3.12)

Thus, on combining (3.9) - (3.12), we obtain ∫_{0}^{t} N̄_{n}(s)ds + ∫_{0}^{N_{n}(t)} S̄_{n}(s)ds (3.13)

= \frac{1}{n^{2}μ} ∑_{i=1}^{ℓ} ∑_{j=1}^{ν_{i}−ν_{i−1}−1} (S_{ν_{i−1}+j} − S_{ν_{i−1}}) − \frac{t^{2}}{2} − \frac{N_{n}^{2}(t)}{2} + t\frac{ν_{ℓ}}{n}
= −\frac{1}{2}(N_{n}(t) − t)^{2} + \frac{1}{n^{2}μ} ∑_{i=1}^{ℓ} ( ∑_{j=1}^{ν_{i}−ν_{i−1}−1}(ν_{i} − ν_{i−1} − j)X_{ν_{i−1}+j}) .

This, in turn, proves (3.8).

As a consequence of (3.8) and the definition of A_{n}(t) we now conclude our claim in (3.6) as follows.

V_{n}(t) = ∫_{0}^{t} (S̄_{n}(s) + N̄_{n}(s))ds
= ∫_{0}^{t} N̄_{n}(s)ds + ∫_{0}^{N_{n}(t)} S̄_{n}(s)ds + ∫_{N_{n}(t)}^{t} S̄_{n}(s)ds
= −\frac{1}{2}N̄_{n}^{2}(t) + B_{n}(t) + A_{n}(t) − S̄_{n}(t)N̄_{n}(t)
= A_{n}(t) + B_{n}(t) − \frac{1}{2}M_{n}^{2}(t) + \frac{1}{2}S̄_{n}^{2}(t).

□

For the sake of stating and proving an analogue of Theorem 2.1 in case of the general renewal process, we need further notations.

Consider the summands

D_{i} := ∑_{j=2}^{ν_{i}−ν_{i−1}}(ν_{i} − ν_{i−1} − j)X_{ν_{i−1}+j−1} i = 1, 2, … (3.14)

in the definition of B_{n} (cf. 3.7). We note that the just introduced random variables D_{i} are i.i.d. with mean μ_{D} := ED_{1} and finite second moment ED_{1}^{2} < ∞. The latter is implied by (3.2) and (3.3), on applying the Cauchy-Schwarz inequality when computing ED_{1}^{2}. Put also μ_{H} := ES_{ν_{1}}, the so-called expected ladder height (cf., e.g., Gut (1988, Chap. IV, Theorem 2.7)).

The extension of our Theorem 2.1 to the general renewal situation now reads as follows.

Theorem 3.1 Assume the conditions (i), (ii), (iii). Then, as n → ∞,

(a)  \frac{1}{log log n}||nV_{n} − \frac{n}{2}S̄_{n}^{2} − (\frac{μ_{D}}{μ_{H}} )id|| → 0 a.s.

(b)  ||nV_{n} − \frac{n}{2}S̄_{n}^{2} − (\frac{μ_{D}}{μ_{H}} )id|| →^{P} 0 ,

where id(t) := t, t ∈ [0, 1].

Proof. For B_{n} of Lemma 3.1 we conclude the following asymptotic representation. As n → ∞, by the law of the iterated logarithm, we have almost surely

(3.15)  B_{n}(t) = \frac{1}{n^{2}μ} ∑_{i=1}^{N_{H}(nμt)} D_{i}
= \frac{1}{n^{2}μ}( ∑_{i=1}^{N_{H}(nμt)} D_{i} − nμt\frac{μ_{D}}{μ_{H}} ) + \frac{1}{n}(\frac{μ_{D}}{μ_{H}} )t
= \frac{1}{n}(\frac{μ_{D}}{μ_{H}} )t + O(\frac{(log log n)^{1/2}}{n^{3/2}} ),

uniformly in t ∈ [0, 1]. Consequently, via (2.6)/(2.7), (2.16)/(2.17), Lemma 3.1 and (3.15) we conclude the respective two statements of Theorem 3.1. □

Remark 3.1 Via the definition of the ladder index ν_{1}, we have

X_{1}, X_{1} + X_{2}, …, X_{1} + ⋯ + X_{ν_{1}−1} ≤ 0, X_{1} + ⋯ + X_{ν_{1}} > 0.

Hence (cf. (3.14)) we have also

D_{1} = ∑_{j=1}^{ν_{1}−1}(ν_{1} − j)X_{j} ≤ 0.

If, however,

(iv′)  P (X_{1} ≥ 0) = 1,

then P (D_{1} = 0) = 1. (3.16)

On the other hand, if

(v)  P (X_{1} < 0) > 0,

then we have P (D_{1} < 0) > 0, (3.17)

as well.

This shows that the results of Section 2 also remain true under assumption (iv’) replacing (iv), for then B_{n} = 0 a.s.

As an immediate consequence of Theorem 3.1 and Theorem A, we conclude

Corollary 3.1. Assume the conditions (i), (ii), (iii). Then, as n → ∞, on the probability space of Theorem A with W_{n} as in (1.5), we have

(a)  \frac{1}{log log n}||nV_{n} − \frac{1}{2} (\frac{σ}{μ}W_{n})^{2} − (\frac{μ_{D}}{μ_{H}} )id|| → 0 a.s.

and

(b)  ∥nV_{n} − \frac{1}{2}(\frac{σ}{μ}W_{n})^{2} − (\frac{μ_{D}}{μ_{H}} )id∥ →^{P} 0,

As a consequence of (b) of Corollary 3.1, we conclude also

nV_{n} →^{D} Z̃, n → ∞, (3.18)

in the space C[0, 1] (endowed with the uniform topology), where Z̃(t) := \frac{1}{2}(\frac{σ}{μ}W (t))^{2} + (\frac{μ_{D}}{μ_{H}} )t, t ∈ [0, 1].

The latter conclusion in turn also allows us a Vervaat (1972)-type proof of Corollary 1.1. Namely, if the statement (1.15) were true, then the sequence in there could be replaced by {n}, due to the convergence of types theorem (cf. Lemma 1 on p. 253 of Feller (1971)). But then one would also have the equality in distribution

∫_{0}^{t} Y (s)ds =^{D} \frac{1}{2}(\frac{σ}{μ}W (t))^{2} + (\frac{μ_{D}}{μ_{H}} ) t, t ∈ [0, 1],

which is impossible, since the Wiener process W is a.s. nowhere differentiable. Consequently, the relation (1.15) cannot hold true.

In concluding this section, it may be of interest to note that, while it is true that the asymptotic behaviour of the Bahadur-Kiefer-type process R_{n}^{∗} of (1.12) with ordinary and general renewals is the same pointwise and in sup-norm, their integrated form V_{n} behaves differently as n → ∞. Namely, in Corollary 3.1 and (3.18) the squared Wiener process appears with a drift term which is not present in Corollary 2.1 and (2.18).

4 Vervaat-error type processes for partial sums and renewals

Theorem 2.1 and Theorem 3.1 suggest the study of the asymptotic behaviour of the following Vervaaterror type process:

Q_{n}(t) := V_{n}(t) − \frac{1}{2}S̄_{n}^{2}(t) − (\frac{μ_{D}}{μ_{H}} ) t, 0 ≤ t ≤ 1. (4.1)

We note that the drift term (μ_{D}/μ_{H})t is identically zero in case of an ordinary renewal process (cf. Theorem 2.1).

Define the stochastic process Z_{n} by

Z_{n}(t) := ∫_{0}^{Y_{n}(t)} (Y_{n}(t − x) − Y_{n}(t))dx, 0 ≤ t ≤ 1, (4.2)

where Y_{n}(t) := \frac{σ}{nμ}W (nt).

We have the following representations for Q_{n} in terms of Z_{n}.

Proposition 4.1 Assume the conditions (i), (ii), (iii). Then, as n → ∞, on the probability space of Theorem A, we have (4.3) Q_{n}(t) = Z_{n}(t) + o(n^{−5/4}(log log n)^{1/2}) a.s.,

and Q_{n}(t) = Z_{n}(t) + o_{P} (n^{−5/4}), (4.4)

uniformly in t ∈ [0, 1].

Proof. In view of (3.6), via (2.15), (3.15), and (2.6), we conclude (4.3). Similarly, and again in view of (3.6), via (2.15) with the error term o_{P} (n^{−5/4}), (3.15), and (2.7), we arrive at (4.4) as well.

Theorem 4.1 Assume (i), (ii), (iii). Then, for 0 < t ≤ 1 fixed, as n → ∞

n^{5/4}Q_{n}(t) →^{D} (\frac{1}{3})^{1/2}(\frac{σ}{μ})^{5/2}t^{3/4}N |Ñ|^{3/2}, (4.5)

where N and Ñ are independent standard normal random variables.

We note in passing that the result of (4.5) parallels that of (2.5) of Theorem 2.1 of Csáki et al. (2002), which deals with a similar problem concerning the Vervaat-error process of the empirical and quantile processes. For further related results along these lines we refer to Csörgő and Zitikis (1999).

Proof. Given (4.4) of Proposition 4.1, it is clear that the pointwise asymptotic distribution of Q_{n}(t) for any fixed t ∈ (0, 1] can be established via that of Z_{n}(t). Clearly, from (4.2) we obtain

Z_{n}(t) = \frac{σW (nt)}{nμ} ∫_{0}^{1} \frac{σ}{nμ}(W (nt − s\frac{σ}{μ}W (nt)) − W (nt))ds. (4.6)

Let

W_{1}(u) := (nt)^{−1/2}(W (nt − unt) − (1 − u)W (nt)) + uN^{∗}, 0 ≤ u ≤ 1,
W_{2}(u) := (nt)^{−1/2}(W (nt + unt) − W (nt)), 0 ≤ u ≤ 1,

where N^{∗} is a standard normal random variable, independent of W (·). It is obvious that for any fixed n and t, the processes {W_{1}(u), 0 ≤ u ≤ 1} and {W_{2}(u), 0 ≤ u ≤ 1} are independent standard Wiener processes, which are also independent of W (nt).

Consider Z_{n}(t) of (4.6), and first assume that W (nt) > 0. Put unt = s(σ/μ)W (nt) in (4.6). Thus we obtain

Z_{n}(t) = \frac{tσ}{nμ} ∫0^{\frac{σW (nt)}{μnt}} (W (nt − unt) − W (nt)) du

= \frac{tσ}{μ} (\frac{t}{n})^{1/2} ∫0^{\frac{σW (nt)}{μnt}} W_{1}(u) du − \frac{tσ}{μn} ∫0^{\frac{σW (nt)}{μnt}} uW (nt) du − \frac{tσ}{μ} (\frac{t}{n})^{1/2} ∫0^{\frac{σW (nt)}{μnt}} uN^{∗} du

= \frac{tσ}{μ} (\frac{t}{n})^{1/2} ∫0^{\frac{σW (nt)}{μnt}} W_{1}(u) du + O_{P} (n^{−3/2}).

Assuming now that W (nt) < 0, on putting unt = −s(σ/μ)W (nt) in (4.6), we get

Z_{n}(t) = \frac{tσ}{μ} (\frac{t}{n})^{1/2} ∫0^{\frac{−σW (nt)}{μnt}} W_{2}(u) du.

Hence, for each fixed t ∈ (0, 1], Z_{n}(t) has the following representation

Z_{n}(t) = Z_{n}^{∗}(t) + O_{P} (n^{−3/2}), (4.7)

where

Z_{n}^{∗}(t) := \frac{tσ}{μ} (\frac{t}{n})^{1/2} ∫_{0}^{| \frac{σW (nt)}{μnt} |} W^{∗}(u) du,

and W^{∗}(u) := W_{1}(u)I{W (nt) > 0}+W_{2}(u)I{W (nt) < 0} is a standard Wiener process, independent of W (nt) for each fixed n and t.

It is well-known that for any fixed T > 0,

∫_{0}^{T} W^{∗}(u) du =^{D} N(0, \frac{T^{3}}{3} ) =^{D} \frac{T^{3/2}}{3^{1/2}} N(0, 1).

Consequently, by independence of W (nt) and {W^{∗}(u), 0 ≤ u ≤ 1} for each fixed n and t,

Z_{n}^{∗}(t) =^{D} (\frac{1}{3})^{1/2}(\frac{σ}{μ})^{5/2} \frac{1}{n^{2}} (\frac{|W (nt)|}{(nt)^{1/2}})^{3/2}(nt)^{3/4}N . (4.8)

Hence for each fixed n and t we conclude

Z_{n}^{∗}(t) =^{D} n^{−5/4}(\frac{1}{3})^{1/2}(\frac{σ}{μ})^{5/2}t^{3/4}|Ñ|^{3/2}N , (4.9)

where N and Ñ are independent standard normal random variables.

Consequently, on account of (4.7), as n → ∞, we obtain

n^{5/4}Z_{n}(t) →^{D} (\frac{1}{3})^{1/2}(\frac{σ}{μ})^{5/2}t^{3/4}N |Ñ|^{3/2}, (4.10)

for each fixed t ∈ (0, 1].

This in view of (4.4) of Proposition 4.1 also completes the proof of (4.5) of Theorem 4.1. □

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