The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models

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Model 3G

pp. 1–8 (col. fig: nil)

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The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models✩ Jinghong Xiao a , Zhongquan Tan b , a b

∗

School of Economics and Management, Tongji University, Shanghai 200092, PR China College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, PR China

article

info

a b s t r a c t

Article history: Received 5 September 2018 Received in revised form 12 January 2019 Accepted 17 January 2019 Available online xxxx

Let {X (t), t ≥ 0} be a nonstationary strongly dependent Gaussian process. Under some conditions related to the correlation function of the Gaussian process, the point processes formed by the upcrossings of level u by {X (t), t ≥ 0} converge weakly to a Poisson process N with random intensity, as u → ∞. © 2019 Elsevier B.V. All rights reserved.

MSC: 60G70 60G15 Keywords: Gaussian process Strongly dependent Upcrossings Poisson process

1. Introduction

1

Let {X (t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. Many authors studied the tail asymptotics of extremes of Gaussian process X under the following condition α

α

r(t) = 1 − C |t | + o(|t | ) as t → 0 and r(t) < 1 for t > 0

(1)

for some positive constant C and α ∈ (0, 2], see e.g., Leadbetter et al. (1983) and Piterbarg (1996). If further, the Berman condition (see Berman, 1966 or Berman, 1992) lim r(T ) ln T = 0

2 3 4

5

6 7

(2)

8

holds, then from Theorem 12.3.5 of Leadbetter et al. (1983) we know that the maximum M(T ) = maxt ∈[0,T ] X (t), obeys the Gumbel law as T → ∞, namely

10

T →∞

⏐ ⏐

⏐ ⏐

lim sup ⏐⏐P {aT (M(T ) − bT ) ≤ x} − Λ(x)⏐⏐ = 0 T →∞

(3)

9

11

x∈R

holds with Λ(x) = exp(− exp(−x)), x ∈ R, and some normalizing constants aT > 0, bT ∈ R. ✩ Tan’s work was supported by Natural Science Foundation of Zhejiang Province of China (No. LY18A010020), National Science Foundation of China (No. 11501250). ∗ Corresponding author. E-mail address: [email protected] (Z. Tan). https://doi.org/10.1016/j.spl.2019.01.028 0167-7152/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

12

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1 2 3

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The Gaussian process {X (t), t ≥ 0} is called weakly dependent if the Berman condition (2) holds. A natural generalization of Berman condition is the following assumption lim r(T ) ln T = r ∈ [0, ∞]

T →∞ 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

28 29 30 31

32

33

(4)

In analogy, the stationary Gaussian process {X (t), t ≥ 0} with correlation function satisfying assumption (4) with r > 0 is called strongly dependent. For related studies on extremes for strongly dependent Gaussian process, we refer to Mittal and Ylvisaker (1975), Piterbarg (1996), Stamatovic and Stamatovic (2010), Tan et al. (2012), Tan and Hashorva (2013a), Hashorva et al. (2013) and Tan and Tang (2017). In general, it is very difficult to derive a Gumbel limit law for the maximum as in (3) for nonstationary Gaussian processes. Exceptions are some special cases including locally stationary Gaussian processes, see Berman (1974) and Hüsler (1990, 1995), cyclostationary Gaussian processes, see Konstant and Piterbarg (1993), Konstant et al. (2004) and Tan and Hashorva (2013b). Recently, Azaïs and Mercadier (2003) derived a Gumbel limit law for some general nonstationary Gaussian process by studying its upcrossings point process. Since our main results are closely related to that of the last paper, we state first the key finding taken from it. In the following part, let {X (t), t ≥ 0} be a nonstationary Gaussian process with covariance function r(·, ·). Let m(t) = E(X (t)) be the expectation function and assume that the variance of X is a constant, without loss of generality we can take it i+j equal to one: for all t in R, r(t , t) = Var(X (t)) = 1. The partial derivatives of r(·, ·) are given for all (i, j), rij (s, t) = ∂∂si ∂ t j r(s, t). Suppose that X satisfies the following assumptions. (A1) ∀t ∈ R, r11 (t , t) = 1, (A2) r(s, t) ln(|s − t |) → 0, |s − t | → ∞, (A3) ∀ε > 0, sup|s−t |>ε |r(s, t)| < 1, (A4) r√is of class C 4 ; ∀γ > 0, r01 and r04 are bounded on {(s, t) ∈ R2 , s > γ , t > γ }, (A5) ln(t)m(t) → 0, t → ∞, (A6) m′ (t) → 0, t → ∞. For a positive real number u and a process X with C 1 − sample paths, we define the point processes formed by the upcrossings of level u by {X (t), t ≥ 0}:

∀B ⊂ R, Nu (B) = ♯{t ∈ B, X (t) = u, X ′ (t) > 0}. Under the above conditions, Azaïs and Mercadier (2003) obtained the following result. Theorem 1.1. Let {X (t), t ≥ 0} be a Gaussian process with unit variance and suppose that the assumptions (A1)–(A6) are satisfied. Define the family of point processes Nu∗ by

∀B ⊂ R, Nu∗ (B) = Nu (Cu−1 B), where Cu is the asymptotic mean number of upcrossings on any interval of size one, i.e., Cu =

1 2π

( exp −

u2 2

)

.

Then as u → ∞, Nu∗ converges weakly on (0, ∞) to a standard Poisson process.

36

The above result can also be extended to more general cases in some degree by the transformation of unit-time, see Azaïs and Mercadier (2003) for details. In this paper, we extend Theorem 1.1 to the strongly dependent case. Brief organization of the paper: In Section 2 we present the main result and in Section 3, we prove the main result.

37

2. Main result

34 35

38 39 40 41 42 43 44 45

46 47 48 49

In order to extend the result of Theorem 1.1 to strongly dependent nonstationary Gaussian processes we replace the weak dependence condition (A2) by the following one (B2) lims→∞ supt ≥0 |r(t , t + s) ln(s) − r | = 0 with r ≥ 0. It is known that the slow decay of correlations of strongly dependent stationary Gaussian processes destroys properties of asymptotic independence of maxima in disjoint intervals, thus the limiting distribution of the extremes is no longer the exp(−e−x ), but a convolution of exp(−e−x ) and a normal distribution function. For more details, we refer to pp. 133–136 of Leadbetter et al. (1983). Our main result is the following theorem. Theorem 2.1. Let {X (t), t ≥ 0} be a Gaussian process with unit variance and suppose that the assumptions (A1),(B2) and (A3)–(A6) √ are satisfied. Then as u → ∞, Nu∗ converges weakly on (0, ∞) to a Poisson process N with random intensity exp(−r + 2r ξ ), where ξ is an N(0, 1) random variable. The following corollary can be obtained from Theorem 2.1 directly. Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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3

Corollary 2.2. Let {X (t), t ≥ 0} be a Gaussian process with unit variance and suppose that the assumptions (A1),(B2) and (A3)–(A5) are satisfied. Then

⏐ ⏐ ∫ +∞ √ ⏐ ⏐ exp(−e−x−r + 2rz )φ (z)dz ⏐⏐ = 0, lim sup ⏐⏐P {αT (MX (T ) − βT ) ≤ x} − (5) T →∞ x∈R −∞ √ where MX (T ) = maxt ∈[0,T ] X (t), αT = 2 ln T , βT = αT + αT−1 ln(2π )−1 and φ (z) denotes the density function of a standard Gaussian random variable.

1 2

3

4 5

Remark 2.3. (1) Similar to Theorem 2.3 of Azaïs and Mercadier (2003), Theorem 2.1 can also be extended to more general case by some transformation of time. (2) Similar to Corollary of 2.1 of Azaïs and Mercadier (2003), assumption (A6) is not needed for Corollary 2.2. (3) Assumption (B2) has been used in Tan and Hashorva (2013b) for deriving the limit distribution of the maximum of a strongly dependent cyclo-stationary χ -process. (4) It is worth comparing the main result with Theorem 15.2 of Piterbarg (1996). The result in Piterbarg (1996) considered the limit Poisson character of extremes for the strongly dependent Gaussian homogeneous field. For the one dimensional setup, that result corresponds to the stationary case. In this paper we consider the more general case of nonstationary Gaussian processes. (5) For the case r = ∞, whether the point processes formed by the upcrossings of level u by Gaussian process converge is not clear, even if the Gaussian process is stationary. 3. Proof

6 7 8 9 10 11 12 13 14 15 16

17

We will use Kallenberg’s Theorem (see Kallenberg, 1976) to prove Theorem 2.1, so we need to consider a set of the form p B = ∪i=1 (ci , di ], where 0 < c = c1 < d1 < c2 < d2 < · · · < cp < dp = d ≤ 1. Let Du,i = (Cu−1 ci , Cu−1 di ], where Cu is defined in Theorem 1.1. Let 0 < a < b < 1 be constants whose value will be chosen in Lemma 3.1. We shall denote throughout in the sequel S = Cu ,

S = Cu .

−a

−b

′

1/2

ξu (t) = (1 − ρ (u))

−1

η(t) + ρ

′

1/2

(u)η, Cu c ≤ t ≤ Cu d, −1

−1

where η is an N(0, 1) random variable independent of {η(t), t ≥ 0}. Note in passing that {ξu (t), t ∈ [Cu c , Cu d]} is a nonstationary Gaussian process with covariance function ϱ(s, t) given by r(s, t) + (1 − r(s, t))ρ (u), ρ (u),

−1

s ∈ Ju,i , t ∈ Ju,j , i = j; s ∈ Ju,i , t ∈ Ju,j , i ̸ = j,

{

Lemma 3.1. Let q = q(u) = u−1 δ with some δ > 0. Under assumptions (B2) and (A3)–(A6), we have as u → ∞,

kq∈Iu,i ,lq∈Iu,j ,kq̸ =lq,

|r(kq, lq) − ϱ(kq, lq)|

1

∫ 0

(

1

√

1−

, lq)

r (h) (kq

23 24 25 26 27 28 29

31 32

33

which is very close to X in some sense. p In the next lemma we shall work with subintervals of [Cu−1 c , Cu−1 d]. Let Iu,i = Su,i ∩ ∪j=1 Du,j , i = 1, 2, . . . , n. Note that Iu,i may be an empty set for some i ∈ {1, 2, . . . , n}. Let K denote some positive constant whose values may vary from place to place.

∑

21

30

−1

ϱ(s, t) =

20

22

We divide the interval [Cu c , Cu d] into intervals of length S alternating with shorter intervals of length S . Denote the long intervals by Su,l , l = 1, 2, . . . , n = ⌊Cu−1 (d − c)/(S + S ′ )⌋, and the short intervals by Su′ ,l , l = 1, 2, . . . , n, where ⌊x⌋ denotes the integer part of x. It will be seen from the proofs that, a possible remaining interval with length different than S or S ′ plays no role in our asymptotic considerations; we call also this interval a short interval. We will construct a nonstationary Gaussian process to approximate the original process X . Let {Yi (t), t > 0}, i = 0, 1, 2, . . . be independent copies of X and {η(t), t > 0} be such that η(t) = Yi (t) for t ∈ Ju,i = Su,i ∪ Su′ ,i . Define ρ (u) = r /ln(Cu−1 d) and −1

18 19

exp −

2

(u − m(kq)) + (u − m(lq)) 2(1 + r (h) (kq, lq))

2

)

dh → 0,

34 35 36 37

38

(6)

39

i,j∈{1,2,...,n}

where r

(h)

(kq, lq) = hr(kq, lq) + (1 − h)ϱ(kq, lq).

Proof. Let ϑ (t) = supkq−lq>t {ϖ (kq, lq)}, where ϖ (kq, lq) = max{r(kq, lq), ϱ(kq, lq)}. From assumption (A3), it is easy to see that for any ε ∈ (0, 2−1/2 ), ϑ (ε ) < 1 for all sufficiently large u. Consequently, we may choose some positive constant β such 1−ϑ (ε ) that β < 1+ϑ (ε) for all sufficiently large u. Let also 0 < a < b < β .

Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

40

41 42 43

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In the first step, we consider the case that kq, lq are in the same interval Iu,i , which implies ϱ(kq, lq) = r(kq, lq) + (1 − r(kq, lq))ρ (u) ∼ r(kq, lq) for sufficiently large u uniformly for all kq, lq ∈ Iu,i . Split the left-hand-side of (6) into two parts as

∑

3

kq,lq∈Iu,i i,j∈{1,2,...,n},|lq−kq|<ε

4

5

∑

+

( ) (u − m(kq))2 + (u − m(lq))2 |r(kq, lq) − ϱ(kq, lq)| √ exp − 2(1 + r(kq, lq)) 1 − r(kq, lq) 1

∑

Tu,1 ≤

1 − r(kq, lq)

∑

= ρ (u)

√

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

7

∑

= ρ (u)

√

1 − r(kq, lq)

9

10

11

12 13 14

15 16

17

(

1 − r(kq, lq) exp −

1 + r(kq, lq)

u2 2

)

( exp −

1

r(s, t) = 1 −

2

23 24

25

2(1 + r(kq, lq))

+

u(m(kq) + m(lq)) 1 + r(kq, lq)

)

.

(8)

(9)

Let µ(t) = sups>t |m(s)|. From assumption (A5), we have for any lq ∈ Iu,i , i = 1, 2, . . . , n um(lq) ≤ uµ(cCu−1 ) = o(1). Consequently, since further q = u

(10)

∑

Tu,1 ≤ K ρ (u)

−1

δ we obtain ( 2) ( ) √ u (1 − r(kq, lq))u2 1 − r(kq, lq) exp − exp − exp(2uµ(cCu−1 )) 2 2(1 + r(kq, lq))

−1

∑

−1

≤ K (ln(Cu )) Cu

√

(

1 − r(kq, lq) exp −

≤ K (ln(Cu−1 ))−1 Cu Cu−1 q−1 u−1 ≤ K (ln(Cu−1 ))−1

∞ ∑

(1 − r(kq, lq))u2

)

2(1 + r(kq, lq))

exp(2uµ(cCu−1 ))

( ) 1 |kq − lq| exp − |kq − lq|2 u2 exp(2uµ(cCu−1 ))

∑

≤ K (ln(Cu−1 ))−1 Cu

8

∑

) ( 1 |kq|u exp − |kq|2 u2 exp(2uµ(cCu−1 )) 8

( ) 1 k exp − k2 exp(2uµ(cCu−1 )) 8

≤ K (ln(Cu−1 ))−1 exp(2uµ(cCu−1 )).

(11)

Thus, limu→∞ Tu,1 = 0. Using the fact that u ∼ (−2 ln Cu )1/2 and (10), we have further Tu,2 ≤

∑

(

|r(kq, lq) − ϱ(kq, lq)| exp −

(u − m(kq))2 + (u − m(lq))2

≤

∑ kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|≥ε

( |r(kq, lq) − ϱ(kq, lq)| exp −

)

2(1 + ϑ (ε ))

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|≥ε

26

(1 − r(kq, lq))u2

u2 = −2 ln Cu + O(1).

k=1 22

1 + r(kq, lq)

1 |s − t |2 ≤ 1 − r(s, t) ≤ 2|s − t |2 . 2 From the definition of Cu , we have

0
21

)

for any s, t ∈ (0, ε ) and |s − t | → 0. Thus, for all |t − s| ≤ ε < 2−1/2

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

20

u(m(kq) + m(lq))

|t − s|2 + o(|t − s|2 )

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

19

+

By Taylor expansions and assumption (A4) we have

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

18

u2

( exp −

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

8

(7)

We deal with Tu,1 . By the fact that ϱ(kq, lq) = r(kq, lq) + (1 − r(kq, lq))ρ (u), we have

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|<ε

6

=: Tu,1 + Tu,2 .

kq,lq∈Iu,i i,j∈{1,2,...,n},|lq−kq|≥ε

u2 1 + ϑ (ε )

)

exp(2uµ(cCu−1 ))

Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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( ≤ exp −

u2

)

1 + ϑ (ε )

1−ϑ (ε )

1

1

kq,lq∈Iu,i i∈{1,2,...,n},|lq−kq|≥ε

( ≤ KCu−(1+b) q−2 exp − ≤ KCu1+ϑ (ε)

∑

exp(2uµ(cCu−1 ))

5

u2

)

exp(2uµ(cCu−1 ))

1 + ϑ (ε )

2

−b 2

u exp(2uµ(cCu−1 )).

(12) 1−ϑ (ε ) 1+ϑ (ε )

Consequently, limu→∞ Tu,2 = 0 since b < β < and Cu → 0 as u → ∞. In the second step, we deal with the case that kq ∈ Iu,i and lq ∈ Iu,j , i ̸ = j. Note that in this case, ϱ(kq, lq) = ρ (u) and the distance between any two intervals Iu,i and Iu,j is larger than Cu−b . We split the left-hand-side of (6) into two parts as

∑

∑

+

kq∈Iu,i ,lq∈Iu,j

=: Tu,3 + Tu,4 .

kq∈Iu,i ,lq∈Iu,j

−β

i̸ =j∈{1,2,...,n},|lq−kq|
(13)

4 5 6

7

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

Similarly to the derivation of (12), we have

8

(

∑

Tu,3 ≤

3

|r(kq, lq) − ϱ(kq, lq)| exp −

kq∈Iu,i ,lq∈Iu,j

) 2

(u − m(kq))2 + (u − m(lq))

9

2(1 + ϑ (Cu−b ))

−β

i̸ =j∈{1,2,...,n},|lq−kq|
( |r(kq, lq) − ϱ(kq, lq)| exp −

∑

≤

kq∈Iu,i ,lq∈Iu,j

u2

)

1 + ϑ (Cu−b )

exp(2uµ(cCu−1 ))

10

−β

i̸ =j∈{1,2,...,n},|lq−kq|
( ≤ exp −

u

2

)

1 + ϑ (ε )

∑

exp(2uµ(cCu−1 ))

kq∈Iu,i ,lq∈Iu,j

1

11

−β

i̸ =j∈{1,2,...,n},|lq−kq|
(

≤ KCu−(1+β ) q−2 exp − 1−ϑ (ε )

≤ KCu1+ϑ (ε)

u

2

)

1 + ϑ (ε )

12

−β 2

u exp(2uµ(cCu−1 )).

Consequently, limu→∞ Tu,3 = 0 since β < For Tu,4 , we have

(14)

1−ϑ (ε ) 1+ϑ (ε )

and Cu → 0 as u → ∞.

kq∈Iu,i ,lq∈Iu,j

13

14 15

) ( (u − m(kq))2 + (u − m(lq))2 |r(kq, lq) − ϱ(kq, lq)| exp − 2(1 + ϑ (ε ))

∑

Tu,4 ≤

exp(2uµ(cCu−1 ))

16

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

u2

( ≤ exp −

) −β

1 + ϑ (Cu )

∑

exp(2uµ(cCu−1 ))

kq∈Iu,i ,lq∈Iu,j

|r(kq, lq) − ρ (u)|

17

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

=

Cu−2 q2

×

u2

( −1

ln(Cu )

exp −

q2 ln(Cu−1 )

) −β

1 + ϑ (Cu )

∑

−2

Cu

kq∈Iu,i ,lq∈Iu,j

exp(2uµ(cCu−1 ))

18

|r(kq, lq) − ρ (u)|.

19

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

−β

−β

By assumption (B2), we have ϑ (t) ln t ≤ K for all large t and some constant K . Thus, ϖ (kq, lq) ≤ ϑ (Cu ) ≤ K /ln(Cu ) for −β |kq − lq| > Cu . Now using (9) again, we obtain Cu−2 −1

q2 ln(Cu )

≤

u2

( exp −

Cu−2 q2 ln(Cu−1 )

) −β

1 + ϑ (Cu )

( exp −

exp(2uµ(cCu−1 ))

u2

) −β

1 + K /ln(Cu )

exp(2uµ(cCu−1 ))

Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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22

23

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Cu−2

1

∼

2

= KCu

( 2) Cu

q2 ln(Cu−1 )

−β

1+K /ln(Cu

−β

3

exp(2uµ(cCu−1 ))

)

−β

−(2K /ln(Cu ))/(1+K /ln(Cu ))

exp(2uµ(cCu−1 )) = O(1).

Thus, we have Tu,4 ≤ K

4

q2 ln(Cu−1 )

∑

|r(kq, lq) − ρ (u)|

−2

Cu

kq∈Iu,i ,lq∈Iu,j

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

≤K

5

q2 ln(Cu−1 )

⏐ ⏐ ⏐ ⏐ r ⏐ ⏐r(kq, lq) − ⏐ ln(|lq − kq|) ⏐

∑

Cu−2

kq∈Iu,i ,lq∈Iu,j

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

+K

6

q2 ln(Cu−1 )

⏐ ⏐ ⏐ ⏐ r r ⏐ ⏐ − ⏐ ln(|lq − kq|) ln(dC −1 ) ⏐ =: Tu,41 + Tu,42 . u

∑

Cu−2

kq∈Iu,i ,lq∈Iu,j

(15)

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

7

By assumption (B2) again, we have −β

Tu,41 ≤ K

8

Cu−1 − Cu

|ln(|lq − kq|)r(kq, lq) − r | = o(1)

max

Cu−1

kq∈Iu,i ,lq∈Iu,j

−β

i̸ =j∈{1,2,...,n},|lq−kq|≥Cu

9

as u → ∞. Furthermore, by an estimate as in the proof of Lemma 6.4.1 of Leadbetter et al. (1983), we have −1

Tu,42 ≤ K

10

dCu q ln(Cu−1 ) ∑ ⏐

⏐ ⏐ ⏐ 1 ⏐ 1 − ⏐ ⏐ ln(kq) ln(dC −1 ) ⏐ = o(1), u −β

Cu−1

kq=Cu

11

as u → ∞. The proof follows now from (7)–(15).

12

Lemma 3.2. Under assumptions (B2) and (A3)–(A5), we have for any λ ∈ (0, 1)

( P

13

) max

t ∈(cCu−λ ,dCu−λ ]

X (t) > u

= (d − c)Cu1−λ + o(Cu1−λ )

14

with o(Cu1−λ ) uniform for all intervals (cCu−λ , dCu−λ ] such that c ≥ γ > 0.

15

Proof. It can be proved with the same arguments as those used in Lemma 4.1 and (1) of Azaïs and Mercadier (2003).

16

Lemma 3.3. For any set E ⊂ R, let MX (E) = maxt ∈E X (t). Under assumptions (B2) and (A3)–(A6), we have as u → ∞,

( P

17

p ⋂

)

∫

{MX (Du,i ) ≤ u} →

−∞

i=1

18

p +∞ ∏

√

exp(−(di − ci )e−r +

2rz

)φ (z)dz .

i=1

Proof. In the first step, we prove that

) ( p ) n ⋂ ⋂ 0≤P {MX (Iu,i ) ≤ u} − P {MX (Du,i ) ≤ u} → 0. (

19

i=1 20

21

22

23

(16)

i=1

Let p

∆ = {k : Su,k ∩ (∪i=1 Du,i ) ̸= ∅} and l1 = ♯∆, we have (denote by |E | the length of some interval E) l1 ∼

p ∑

1 −a

−b

Cu + Cu

i=1

|Du,i | =

p ∑

Cu−1 −a

(di − ci ) ≤

−b

Cu + Cu

i=1

Cu−1 −a

Cu + Cu−b

(d − c) ≤ Cu−1+b (d − c).

Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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7

Using Lemma 3.2, we have 0≤P

( n ⋂

1

)

)

⋂

{MX (Iu,i ) ≤ u} − P

i=1

{MX (Du,i ) ≤ u}

2

i=1

∑ (

P MX (Su,k ) > u

≤

p

(

)

3

k∈∆

∑

≤K

Cu1−a

4

k∈∆

≤ KCu−1+b (d − c)Cu1−a = K (d − c)Cub−a → 0,

5

since a < b and Cu → 0, as u → ∞. Thus, (16) is proved. In the second step, let q = q(u) = u−1 δ and δ ↓ 0, by repeating the proof of (5) and (7) of Azaïs and Mercadier (2003) we can prove that 0≤P

( n ⋂

)

(

{X (jq) ≤ u, jq ∈ Iu,i } − P

i=1

P

⋂ {MX (Iu,i ) ≤ u} → 0,

(17)

(

10 11

)

n n ⋂ ⋂ {X (jq) ≤ u, jq ∈ Iu,i } − P {ξu (jq) ≤ u, jq ∈ Iu,i } → 0, i=1

(18)

13 14

)

n

⋂ {ξu (jq) ≤ u, jq ∈ Iu,i } →

∫

+∞

p

∏

−∞

i=1

√

exp(−(di − ci )e−r +

2rz

)φ (z)dz .

(19)

P

15

i=1

First, by the definitions of ξu (t) and η(t), we have

(

12

i=1

as u → ∞. In the last step, we prove that P

9

i=1

)

(

8

)

n

as u → ∞. In the third step, by Normal comparison Lemma (see e.g. Leadbetter et al., 1983) and Lemma 3.1, we can prove that

(

6 7

16

)

n ⋂ {ξu (jq) ≤ u, jq ∈ Iu,i }

17

i=1

=P

( n ⋂

) 1/2

{(1 − ρ (u))

η(jq) + ρ

1/2

(u)η ≤ u, jq ∈ Iu,i }

18

i=1

+∞

∫ =

−∞

}) n { ⋂ u − ρ 1/2 (u)z P η(jq) ≤ , jq ∈ Iu,i φ (z)dz . (1 − ρ (u))1/2 (

(20)

A direct calculation leads to uz :=

19

i=1

u−ρ

1/2

20

√ (u)z

(1 − ρ (u))1/2

=u+

− 2rz + r u

1

+ o( ). u

(21)

By the same arguments as those used in the proofs of (16) and (17), we have

⏐ ( n ( p )⏐ ) ⏐ ⏐ ⋂{ ⋂ } ⏐ ⏐ η(jq) ≤ uz , jq ∈ Iu,i − P {Mη (Du,i ) ≤ uz } ⏐ ⏐P ⏐ ⏐ i=1 i=1 ⏐ ( n ) ( n )⏐ ⏐ ⋂{ ⏐ ⋂ } ⏐ ⏐ ≤ ⏐P η(jq) ≤ uz , jq ∈ Iu,i − P {Mη (Iu,i ) ≤ uz } ⏐ ⏐ ⏐ i=1 i=1 ⏐ ( n ) ( p )⏐ ⏐ ⏐ ⋂ ⋂ ⏐ ⏐ + ⏐P {Mη (Iu,i ) ≤ uz } − P {Mη (Du,i ) ≤ uz } ⏐ → 0, ⏐ ⏐ i=1

21

22

23

24

(22)

25

i=1

as u → ∞. Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

26

STAPRO: 8434

8

1 2

J. Xiao and Z. Tan / Statistics and Probability Letters xxx (xxxx) xxx

From the definition of η(t) we know that η(s) and η(t) are independent for |s − t | ≥ Cu−a + Cu−b . Note that mini∈{1,2,...,p} (ci+1 − di )Cu−1 ≥ Cu−a + Cu−b for large u, we have by Eq. (21) and Lemma 4.4 of Azaïs and Mercadier (2003) P

3

( p ⋂

) {Mη (Du,i ) ≤ uz }

=

i=1

p ∏ (

P Mη (Du,i ) ≤ uz

)

i=1

=

4

p ∏

√

exp(−(di − ci )e−r +

2rz

) + o(1).

i=1 5 6

7

Now combining the last relation with (20), (22) and applying the dominated convergence theorem completes the proof of (19). The lemma follows now from (16)–(19). Proof of Theorem 2.1. From Kallenberg’s Theorem, to prove the convergence of Nu∗ , it suffices to show as u → ∞ E(Nu∗ ((c , d])) → E(N((c , d])), for any 0 < c < d ≤ 1

8 9

and

( 10

P

p ⋂

) {Nu ((ci , di ]) = 0} ∗

) p ⋂ →P {N((ci , di ]) = 0} (

i=1

i=1 p

∞

∫

∏

=

11

√

exp(−(di − ci )e−r +

2rz

)φ (z)dz ,

−∞ i=1 12 13 14 15 16 17 18

for 0 < c1 < d1 < c2 < d2 < · · · < cp < dp ≤ 1, where p ≥ 1 is an arbitrary integer. By the same arguments as those used in the proof of Theorem 2.2 of Azaïs and Mercadier (2003), we have as u → ∞ E(Nu∗ ((c , d])) = (d − c)(1 + o(1)), so since E(N((c , d])) = E((d − c)e−r +

2r ξ

) = (d − c)e−r e(

0≤P

p ⋂

) p ⋂ {MX (Du,i ) ≤ u} → 0, {Nu ((ci , di ]) = 0} − P )

(

i=1

as u → ∞. Furthermore, applying Lemma 3.3, we have

( 21

= (d − c),

∗

i=1 20

2r)2 /2

the first assertion follows immediately. From the arguments in the proof of Theorem 2.2 of Azaïs and Mercadier (2003), we have

( 19

√

√

P

p ⋂

) {Nu ((ci , di ]) = 0} → ∗

i=1

∫

p +∞ ∏

−∞

√

exp(−(di − ci )e−r +

2rz

)φ (z)dz ,

i=1

22

which completes the proof of the second assertion.

23

Acknowledgments

25

The authors would like to thank the two referees for several corrections and important suggestions which significantly improved this paper.

26

References

24

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Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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Please cite this article as: J. Xiao and Z. Tan, The limit properties of point processes of upcrossings in nonstationary strongly dependent Gaussian models. Statistics and Probability Letters (2019), https://doi.org/10.1016/j.spl.2019.01.028.

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