Laws of the iterated logarithm of Chover-type for operator stable Lévy processes

Statistics and Probability Letters 92 (2014) 17–25

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Laws of the iterated logarithm of Chover-type for operator stable Lévy processes Wensheng Wang Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

article

abstract

info

Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E, where E is an invertible linear operator on Rd . Integral tests for sample paths of operator stable Lévy process X are given. Laws of the iterated logarithm of Chover-type are derived from them as corollaries. Our results give information about the maximal growth rate of sample paths of X in terms of the real parts of the eigenvalues of E. © 2014 Published by Elsevier B.V.

Article history: Received 18 March 2014 Received in revised form 24 April 2014 Accepted 25 April 2014 Available online 2 May 2014 MSC: 60G51 60G17 60E07 60J30 Keywords: Operator stable law Lévy process Integral test Law of the iterated logarithm

1. Introduction Let X = {X (t ), t ∈ R+ } be a Lévy process in Rd , that is, X has stationary and independent increments, X (0) = 0 a.s. and such that t → X (t ) is continuous in probability. The finite-dimensional distributions of a Lévy process X are completely determined by the characteristic function of X (t ) given by

E[ei⟨ξ ,X (t )⟩ ] = e−t ψ(ξ ) , where, by the Lévy–Khintchine formula, 1

 

2

Rd

ψ(ξ ) = i⟨a, ξ ⟩ + ⟨ξ , Σ ξ ′ ⟩ +

ei⟨x,ξ ⟩ − 1 −

i⟨x, ξ ⟩  1 + ∥ x ∥2

φ(dx),

∀ξ ∈ Rd

(1.1)

and, a ∈ Rd is fixed, Σ is a non-negative definite, symmetric, (d × d) matrix, and φ is a Borel measure on Rd \ {0} that satisfies

 Rd

∥x∥2 φ(dx) < ∞. 1 + ∥ x ∥2

The function ψ is called the Lévy exponent of X , and φ is the corresponding Lévy measure. We refer to Bertoin (1996) and Sato (1999) for general theory of Lévy processes.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2014.04.025 0167-7152/© 2014 Published by Elsevier B.V.

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W. Wang / Statistics and Probability Letters 92 (2014) 17–25

There has been considerable interest in studying sample path properties of Lévy processes. Many authors have investigated fractal nature of various random sets generated by Lévy processes. See the survey papers of Taylor (1986) and Xiao (2004) and the references therein for more information. For a stable Lévy process X in Rd with index α ∈ (0, 2), many of the results on the sample paths of X can be formulated nicely in terms of α and d. For strictly α -stable processes X = {X (t ), t ∈ R+ } on R1 with 0 < α < 2, Chover (1966) showed that the LIL of Chover-type holds. Its integral test was first given by Khintchine (1938), and it implies that |X (t )| = 0 a.s. or = ∞ a.s. lim sup 1/α t →∞ (tht ) according as ∞



dt < ∞ or = ∞, tht if ht is increasing and limt →∞ ht = ∞. This fact was shown by Yamamuro’s result (2003) for Lévy processes in the case where φ((−∞, 0)) > 0 and φ((0, ∞)) > 0, and a similar result is reported by Fristedt (1974) for φ((0, ∞)) > 0. Watanabe (1996) investigated the integral test for self-similar processes with independent increments in R1 . Yamamuro (2005) investigated the integral test for a stable Lévy process X in Rd with index α ∈ (0, 2). The purpose of this paper is to prove integral tests for operator stable Lévy processes without Gaussian component. They will include the related results mentioned above as special cases. Laws of the iterated logarithm (LIL) of Chover-type are derived from them as corollaries. Our results give information about the maximal growth rate of sample paths of operator stable Lévy processes. We use the notations ft ∼ gt if lim ft /gt = 1 and ft ≍ gt if there exist constants c1 , c2 > 0 such that c1 ≤ lim inf ft /gt ≤ lim sup ft /gt ≤ c2 . All constants c appearing in this paper (with or without subscript) are positive and may not necessarily be the same in each occurrence. More specific constants will be denoted by c0 , c1 , c2 , . . . . 1

2. Main results A Lévy process X = {X (t ), t ∈ R+ } in Rd is called operator stable if the distribution ν of X (1) is full [i.e., not supported on any (d − 1)-dimensional hyperplane] and ν is strictly operator stable, i.e., there exists a linear operator E on Rd such that

ν t = t E ν for all t > 0, (2.1) where ν t denotes the t-fold convolution power of the infinitely divisible law ν and t E ν(dx) = ν(t −E dx) is the image measure of ν under the linear operator t E , which is defined by ∞  (log t )n n E , t > 0. tE = n! n =0 The linear operator E is called a stability exponent of X . The set of all possible exponents of an operator stable law is characterized in Theorem 7.2.11 of Meerschaert and Scheffler (2001). On the other hand, a stochastic process X = {X (t ), t ∈ R+ } with values in Rd is said to be operator self-similar if there exists a linear operator E on Rd such that for every λ > 0, d

{X (λt ), t ≥ 0} = {λE X (t ), t ≥ 0},

(2.2)

d

where X = Y denotes that the two processes X and Y have the same finite-dimensional distributions. Here the linear operator E is called a self-similarity exponent of X . Hudson and Mason (1982) proved that if X is a Lévy process in Rd such that the distribution of X (1) is full, then X is operator self-similar if and only if X (1) is strictly operator stable. In this case, every stability exponent E of X is also a self-similarity exponent of X . Hence, from now on, we will simply refer to E as an exponent of X . In this paper we restrict our study to an operator stable Lévy process without Gaussian component. To describe the characteristic function of X (1), we let SE = {x ∈ Rd : ∥x∥E = 1}, where ∞



∥e−tE x∥dt =

∥x ∥E = 0 d

1



∥sE x∥s−1 ds 0

is a norm on R , that is, SE is a unit sphere in Rd with respect to this norm. Then, the Lévy measure φ of X (1) in (1.1) (Jurek and Mason, 1993, Proposition 4.3.4) is

φ(A) =

∞

  SE

IA (sE x)s−2 dsm(dx) 0

for all A ⊂ Rd \ {0}, where m is the finite measure on B (SE ) the class of Borel subsets of SE , given by m(F ) = φ({sE x : x ∈ F , s ∈ [1, ∞)}).

(2.3)

W. Wang / Statistics and Probability Letters 92 (2014) 17–25

19

That is, the characteristic function of X (1) (Jurek and Mason, 1993, Theorem 4.3.7) is of the form ∞

 

E ei⟨ξ ,s x⟩ − 1 −

E[ei⟨ξ ,X (1)⟩ ] = exp

SE

0

 i⟨ξ , sE x⟩  −2 s dsm ( dx ) . 1 + ∥sE x∥2

(2.4)

Operator stable Lévy processes are scaling limits of random walks on Rd , normalized by linear operators; see Meerschaert and Scheffler (2001, Chapter 11). Clearly, all strictly stable Lévy processes in Rd of index α are operator stable with exponent E = α −1 I, where I is the identity operator in Rd . More generally, let X1 , . . . , Xd be independent stable Lévy processes in R1 with indices α1 , . . . , αd ∈ (0, 2), respectively, and define the Lévy process X = {X (t ), t ≥ 0} by X (t ) = (X1 (t ), . . . , Xd (t )). Then it is easy to verify that X is an operator stable Lévy process with exponent E which has α1−1 , . . . , αd−1 on the diagonal and 0 elsewhere. This class of Lévy processes was first studied by Pruitt and Taylor (1969). Examples of operator stable Lévy processes with dependent components can be found in Shieh (1998) and Becker-Kern et al. (2003). Meerschaert and Xiao (2005) investigated the dimensional results for sample paths of a large class of Lévy processes, i.e., the operator stable Lévy processes in Rd . For systematic information about operator stable laws and operator stable Lévy processes, we refer to Jurek and Mason (1993) and Meerschaert and Scheffler (2001). Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E. Factor the minimal polynomial 1 so of E into q1 (x), . . . , qp (x) for p ≤ d, where all roots of qi (x) have real part ai and ai < aj for i < j. Let αi = a− i that α1 > · · · > αp , and note that 0 < αi < 2 in view of Theorem 7.2.1 in Meerschaert and Scheffler (2001). Define Vi = Ker(qi (E )) and dim(Vi ) = di . Then d1 + · · · + dp = d and V1 ⊕ · · · ⊕ Vp is a direct sum decomposition of Rd into E-invariant subspaces. We may write E = E1 ⊕ · · · ⊕ Ep , where Ei : Vi → Vi and every eigenvalue of Ei has real part equal to ai . The matrix for E in an appropriate basis is then block-diagonal with p blocks, the ith block corresponding to the matrix for Ei . Write X (t ) = X (1) (t ) + · · · + X (p) (t ) with respect to this direct sum decomposition, and note that by Corollary 7.2.12 in Meerschaert and Scheffler (2001) we get the same decomposition for any exponent E. Since Vi is an E-invariant subspace it follows easily that {X (i) (t ), t ∈ R+ } is an operator stable Lévy process on the di -dimensional vector space Vi with exponent d

d

Ei . It follows from (2.1) that X (t ) = t E X (√ 1) and X (i) (t ) = t Ei X (i) (1) for all 1 ≤ i ≤ p. Choose an inner product ⟨·, ·⟩ on Rd such that Vi ⊥ Vj for i ̸= j, and let ∥x∥ = ⟨x, x⟩ be the associated Euclidean norm. Then

∥t E X (1)∥2 = ∥t E1 X (1) (1)∥2 + · · · + ∥t Ep X (p) (1)∥2 for all t > 0. Let α(θ ) : Rd \ {0} → {α1 , . . . , αp } be the spectral index function, that is, α(θ ) = αi = 1/ai for all θ ∈ Li \ Li−1 ,

(2.5)

(2.6)

where L0 = {0}, Li = V1 ⊕ · · · ⊕ Vi and Rd = V1 ⊕ · · · ⊕ Vp is the spectral decomposition with respect to E. For any unit vector θ , define β : (0, ∞) → (0, ∞) by

β(r )=β( ˆ r , θ ) = ∥(r E )∗ θ∥−1 , where (r E )∗ is the transpose of r E . Our main results read as follows. Theorem 2.1. Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E. Let ht be an increasing positive function on [c , ∞) with some c > 0, such that limt →∞ ht = ∞ and lim infk→∞ h2k /h2k+1 ≥ c0 with some c0 > 0. Then, for all unit vectors θ ∈ Rd , lim sup β(tht )|⟨X (t ), θ⟩| = 0 a.s. or = ∞ a.s. t →∞

according as ∞

 c

dt tht

< ∞ or = ∞.

By using Theorem 2.1 we obtain the LIL of Chover-type for operator stable Lévy processes as follows. Corollary 2.1. For every unit vector θ ∈ Rd , we have 1

1

lim sup |β(t )⟨X (t ), θ⟩| log log n = e α(θ) t →∞

a.s.,

where β(t ) and α(θ ) are defined as above. Theorem 2.2. Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E, and let Ei and X (i) (t ) be defined as above. Let ht be an increasing positive function on [c , ∞) with some c > 0, such that limt →∞ ht = ∞ and lim infk→∞ h2k /h2k+1 ≥ c0 with some c0 > 0. Then, for any 1 ≤ i ≤ p, lim sup ∥(tht )−Ei X (i) (t )∥ = 0 a.s. or = ∞ a.s. t →∞

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W. Wang / Statistics and Probability Letters 92 (2014) 17–25

according as ∞



dt tht

c

< ∞ or = ∞.

The following result is an easy consequence of the above theorem. Corollary 2.2. Let ht be as in Theorem 2.2. Then lim sup ∥(tht )−E X (t )∥ = 0 a.s. or = ∞ a.s. t →∞

according as ∞

 c

dt tht

< ∞ or = ∞.

By using Theorem 2.2 and Corollary 2.2 we obtain the LILs of Chover-type for operator stable Lévy processes as follows. Corollary 2.3. We have 1

lim sup ∥t −Ei X (i) (t )∥ log log n = eai

a.s. for all 1 ≤ i ≤ p

(2.7)

t →∞

and 1

lim sup ∥t −E X (t )∥ log log n = eap

a.s.,

t →∞

(2.8)

where ai is the real part of the eigenvalue of Ei , 1 ≤ i ≤ p. Remark 2.1. The assertions of Corollaries 2.1 and 2.3 give information about the maximal growth rate of sample paths of X in terms of the real parts of the eigenvalues of E. For example, by Lemma 3.1, we have that for any δ > 0 there is some −

1

−δ

−

1

+δ

≤ β(t ) ≤ t α(θ) for all t ≥ t0 . It follows from Corollary 2.1 that the maximal growth constant t0 > 0 such that t α(θ) rate of X (t ) projected onto the direction θ is of the order (t log t )ai when θ ∈ Li \ Li−1 . 3. Preliminaries In this section we investigate some technical results necessary for our argumentation. Let L(Rd ) denote the set of linear operators A : Rd → Rd . The operator norm of the linear operator A on L(Rd ) is defined by

∥A∥ = sup{∥Ax∥ : ∥x∥ = 1}. The following properties on the operator norm are easy (Meerschaert and Scheffler, 2001; Wang, 2014) and will be used to our proofs. Lemma 3.1. The operator norm of the linear operator A on L(Rd ) satisfies (i) ∥Ax∥ ≤ ∥A∥ · ∥x∥ for all A ∈ L(Rd ) and all x ∈ Rd (ii) If A ∈ L(Rd ) then ∥Ax∥ ≥ ∥x∥/∥A−1 ∥ for all x ∈ Rd (iii) For any 0 < δ < a1 , we have ct (a1 −δ) ≤ ∥t E ∥ ≤ ct ap +δ for all t ≥ 1, where a1 and ap are the smallest and largest real parts of the eigenvalues of E, respectively (iv) For any 0 < δ < a1 , we have ct ap +δ ≤ ∥t E ∥ ≤ ct (a1 −δ) for all 0 < t < 1. Lemma 3.2. Let ht be an increasing positive function on [c , ∞) with some c > 0, and let limt →∞ ht = ∞. Then, for any unit vector θ ∈ Rd and ε > 0, there exists t0 > 0 such that 1 −ε − α(θ)

β(t )ht

1 − α(θ) +ε

≤ β(tht ) ≤ β(t )ht

(3.1)

for all t ≥ t0 . Proof. For any unit vector θ ∈ Rd , there exists a unique 1 ≤ i ≤ p such that θ ∈ Li \ Li−1 , where Li = V1 ⊕ · · · ⊕ Vi and V1 , . . . , Vp is the spectral decomposition Rd = V1 ⊕ · · · ⊕ Vp with respect to E. Moreover, for any θ ∈ Li \ Li−1 , there exist θ (j) ∈ Vj , θ (j) ̸= 0, such that θ = θ (1) + · · · + θ (i) and (t E )∗ θ = (t E1 )∗ θ (1) + · · · + (t Ei )∗ θ (i) , where E1 , . . . , Ep is the spectral decomposition of E. Then, i −1  ∥(r E )∗ θ∥2 ∥(r Ej )∗ θ (j) ∥2 β 2 (r , θ (i) ) = = 1 + . β 2 (r , θ ) ∥(r Ei )∗ θ (i) ∥2 ∥(r Ei )∗ θ (i) ∥2 j=1

W. Wang / Statistics and Probability Letters 92 (2014) 17–25

21

Noting that every eigenvalue of Ej has a real part equal to aj , by Lemma 3.1, we have easily that for any 0 < τ < ai there exist constants c3 , c4 > 0 such that

∥(r Ej )∗ θ (j) ∥ ≤ c3 r aj +τ for all 1 ≤ j ≤ i − 1, ∥(r Ei )∗ θ (i) ∥ ≥ c4 r ai −τ . Since τ > 0 is arbitrary and aj < ai for all 1 ≤ j ≤ i − 1 it follows that β(r , θ ) ∼ β(r , θ (i) ). Thus, by using Lemma 3.1(i) and (iv), for any ε > 0 there exists t0 > 0 such that

β(tht ) ∥(ht i )∗ ((tht )Ei )∗ θ (i) ∥ ∥(t E )∗ θ ∥ ∥(t Ei )∗ θ (i) ∥ −E −a +ε = ≤ ∥ht i ∥ ≤ ht i = ∼ ∗ ( i ) E ∗ E β(t ) ∥((tht ) ) θ∥ ∥((tht ) i ) θ ∥ ∥((tht )Ei )∗ θ (i) ∥ −E

for all t ≥ t0 . Similarly to the above inequality, we have

β(t ) E a +ε ≤ ∥ht i ∥ ≤ ht i . β(tht ) Therefore, (3.1) is proved.



Lemma 3.3. Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E. Then, for any unit vector θ ∈ Rd ,

P(|⟨X (1), θ⟩| > x) ∼ H (x)

(3.2)

as x → ∞, where H : (0, ∞) → [0, ∞] is defined by H (x) = φ({v ∈ Rd : |⟨v, θ⟩| > x}). Here φ is the Lévy measure as in (2.3). Proof. Let F (x) = 1 − min{H (x), 1} be a distribution on [0, ∞). By (2.3), for any λ > 0 and A ⊂ Rd \ {0},

φ(λ−E A) =

∞

  SE

=λ

IA ((sλ)E x)s−2 dsm(dx)

0

∞

  SE

IA (ξ E x)ξ −2 dξ m(dx) = λφ(A).

(3.3)

0

It follows that λ · φ = λE φ for all λ > 0. Thus, from Theorem 6.4.15 in Meerschaert and Scheffler (2001), it follows that F belongs to the subexponential class (Embrechts et al., 1979). Hence, (3.2) follows from Theorem 2.1 of Rosinski and Samorodnitsky (1993).  Lemma 3.4. Let X = {X (t ), t ∈ R+ } be an operator stable Lévy process in Rd with exponent E. Let n and m be arbitrary positive integers. Then, for any unit vector θ ∈ Rd and a > 0,

 P



sup |⟨X (t ), θ⟩| > 3a ≤ 3 sup P(|⟨X (t ), θ⟩| > a).

n≤t ≤m

(3.4)

n ≤t ≤m

Proof. See Yamamuro (2005, Lemma 2.4).



4. Proofs of main results Proof of Theorem 2.1. First, we suppose that

∞ c

(tht )−1 dt < ∞. Noting that for any t , r > 0 and unit vector θ

⟨X (t ), θ⟩ = ⟨r E r −E X (t ), θ⟩ = ⟨r −E X (t ), (r E )∗ θ ⟩ = β −1 (r )⟨r −E X (t ), θr ⟩, where θr = β(r )(r E )∗ θ is a unit vector in Rd , it follows that

β(r )⟨X (t ), θ⟩ = ⟨r −E X (t ), θr ⟩.

(4.1) d

Thus, since X is operator self-similar, X (t ) = t E X (1), by (4.1) and Lemma 3.3, for any δ > 0 and sufficiently large t, we have

P(|β(tht )⟨X (t ), θ⟩| > δ) = P(|⟨(tht )−E X (t ), θ¯t ⟩| > δ) E ¯ = P(|⟨h− t X (1), θt ⟩| > δ) ˜ = P(|⟨X (1), θt ⟩| > δβ(ht , θ¯t ))

≍ φ({v ∈ Rd : |⟨v, θ˜t ⟩| > δβ(ht , θ¯t )}) E ¯ ≍ φ({v ∈ Rd : |⟨h− t v, θt ⟩| > δ}),

where θ¯t = β(tht )((tht ) ) θ and θ˜t = β(ht , θ¯t )( E ∗

(4.2)

) θ¯ are two unit vectors in R .

hEt ∗ t

d

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W. Wang / Statistics and Probability Letters 92 (2014) 17–25

Fix D = {v ∈ Rd : |⟨v, θ¯t ⟩| > δ}. By (3.3), we have 1 φ(hEt D) = h− t φ(D).

(4.3)

By Lemma 6.3.1 in Meerschaert and Scheffler (2001), for any given δ > 0, there exist constants c5 = c5 (δ), c6 = c6 (δ) > 0 such that c5 ≤ φ({v ∈ Rd : |⟨v, θ⟩| > δ}) ≤ c6 for all unit vectors θ ∈ Rd . Thus, by (4.2) and (4.3), we obtain 1 d −1 ¯ P(|β(tht )⟨X (t ), θ⟩| > δ) ≍ h− t φ({v ∈ R : |⟨v, θt ⟩| > δ}) ≍ ht .

(4.4)

Now, let tk = 2k for any positive integer k. Let δ > 0. Put Gk = {|⟨X (t ), θ⟩| > δβ −1 (tk htk ) for some t with tk ≤ t < tk+1 }. Set l(t ) = δβ −1 (tht ). By virtue of Lemma 3.4 and (4.4) we have, for any sufficiently large integer k,

P(Gk ) = P

≤3



sup

tk ≤t
sup

tk ≤t
|⟨X (t ), θ⟩| > l(tk )



1 P(|⟨X (t ), θ⟩| > l(tk )) ≤ ch− tk .

Thus, for any sufficiently large integer m, ∞ 

P(Gk ) ≤ c

∞ 

du

m−1 h2u

k=m

k=m

∞



1 h− ≤c 2k



∞

du

2m−1

uhu

≤c

< ∞.

Hence, by virtue of the Borel–Cantelli lemma, we obtain that P(lim supk→∞ Gk ) = 0. Furthermore, since h is increasing, by Lemma 3.1, we have for any 0 < τ < a1 , where a1 is the smallest real parts of the eigenvalues of E, sup

tk ≤t
β(tht ) = β(tk htk )

sup

tk ≤t
≤c

∥((tk htk )E )∗ θ ∥ 1 E ≤ sup ∥(tk htk t −1 h− t ) ∥ ∥((tht )E )∗ θ∥ tk ≤t
sup

tk ≤t
1 a1 −τ (tk htk t −1 h− ≤ c. t )

Thus, ∞  ∞ 

1 = P

{|⟨X (t ), θ⟩| ≤ δβ −1 (tk htk ) for all t with tk ≤ t < tk+1 }



m=1 k=m

∞  ∞ 

≤P

 {|⟨X (t ), θ⟩| ≤ c δβ −1 (tht ) for all t with tk ≤ t < tk+1 } .

m=1 k=m

Therefore we have lim sup β(tht )|⟨X (t ), θ⟩| ≤ c δ

a.s.

t →∞

As δ → 0, we have lim sup β(tht )|⟨X (t ), θ⟩| = 0

a.s.

t →∞

d

∞

Secondly, we suppose that c (tht )−1 dt = ∞. Let M > 0 and set ζk = X (tk+1 ) − X (tk ). Then, by noting ζk = X (tk ) and the virtue of (4.4), we obtain that, for any sufficiently large integer k,









1 P |⟨ζk , θ⟩| > M β −1 (tk htk ) = P |⟨X (tk ), θ⟩| > M β −1 (tk htk ) ≥ ch− tk .

Hence we obtain that, for any sufficiently large integer m,

 ∞ ∞     1 P |⟨ζk , θ⟩| > M β −1 (tk htk ) ≥ c h− ≥ c 2k k=m

k=m

∞

du

2m+1

uhu

By virtue of the Borel–Cantelli lemma, we almost surely have

|⟨ζk , θ⟩| > M β −1 (tk htk ) for infinitely many k. Thus we almost surely have either

|⟨X (tk+1 ), θ⟩| > M β −1 (tk htk ) for infinitely many k, or

|⟨X (tk ), θ⟩| > M β −1 (tk htk )

= ∞.

W. Wang / Statistics and Probability Letters 92 (2014) 17–25

23

for infinitely many k. Note that, since h is increasing and lim infk→∞ h2k /h2k+1 ≥ c0 with some c0 > 0, by Lemma 3.1(ii) and (iii), we have for any τ > 0,

β(tk+1 htk+1 ) ∥((tk htk )E )∗ θ ∥ 1 −E −1 ∥ = ≥ ∥(tk htk tk−+11 h− tk+1 ) β(tk htk ) ∥((tk+1 htk+1 )E )∗ θ∥ 1 ap +τ ≥ (tk htk tk−+11 h− ≥ 2−ap −τ (htk ht−k+1 1 )ap +τ ≥ c , tk+1 )

where ap is the largest real part of the eigenvalues of E. Therefore, we almost surely have

|⟨X (tk ), θ⟩| > cM β −1 (tk htk ) for infinitely many k. From the inequality above, we obtain that lim sup β(tht )|⟨X (t ), θ⟩| ≥ cM

a.s.

t →∞

As M → ∞, we have lim sup β(tht )|⟨X (t ), θ⟩| = ∞ a.s. t →∞

The proof has been completed.



Proof of Corollary 2.1. Let δ > 0. Take ht = (log t )1+δ . Since ∞



dt t (log t )1+δ

e

< ∞,

it follows that lim sup(log t )

1 +ε) −(1+δ)( α(θ)

t →∞

β(t )|⟨X (t ), θ⟩| = 0 a.s.

from Theorem 2.1 and Lemma 3.2. Then the limsup above implies that there is a positive constant c such that

  1   β(t )|⟨X (t ), θ⟩| ≤ c exp (1 + δ) + ε log log t a.s. α(θ ) for any sufficiently large t. Therefore we have 1 (1+δ)( α(θ) +ε)

1

lim sup |β(t )⟨X (t ), θ⟩| log log n ≤ e

a.s.

t →∞

Letting δ ↓ 0 and ε ↓ 0, we have 1

1

lim sup |β(t )⟨X (t ), θ⟩| log log n ≤ e α(θ)

a.s.

t →∞

We prove the reverse inequality of the above one. Take ht = log t. Since



∞ e2

dt t log t

= ∞,

it follows that 1 −( α(θ) −ε)

lim sup(log t ) t →∞

β(t )|⟨X (t ), θ⟩| = ∞ a.s.

from Theorem 2.1 and Lemma 3.2. Hence we have 1

1

lim sup |β(t )⟨X (t ), θ⟩| log log n ≥ e α(θ)

a.s.

t →∞

The proof has been completed.



Proof of Theorem 2.2. Fix 1 ≤ i ≤ p. Let {θ (1) , . . . , θ (di ) } be an orthonormal basis of Vi , where di = dim(Vi ). Then it is easy to see that for any random vector with values in Vi we have

∥Z ∥2 = |⟨Z , θ (1) ⟩|2 + · · · + |⟨Z , θ (di ) ⟩|2 . On the other hand, following the same lines as the proof of Theorem 2.1, we conclude that for each 1 ≤ j ≤ di , lim sup |⟨(tht )−Ei X (i) (t ), θ (j) ⟩| = 0 t →∞

a.s. or = ∞ a.s.

(4.5)

24

W. Wang / Statistics and Probability Letters 92 (2014) 17–25

according as ∞

 c

dt tht

< ∞ or = ∞.

This, together with (4.5), yields the desired result.



Proof of Corollary 2.2. Since, similarly to (2.5),

∥(tht )−E X (t )∥2 = ∥(tht )−E1 X (1) (t )∥2 + · · · + ∥(tht )−Ep X (p) (t )∥2 , then the corollary follows easily from Theorem 2.2.

(4.6)



Proof of Corollary 2.3. Note that, by Lemma 3.1, we have that for all ε > 0 and sufficiently large t −ai −ε

ht

∥t −Ei X (i) (t )∥ ≤ ∥t −Ei X (i) (t )∥ · ∥ht i ∥−1 ≤ ∥ht i t −Ei X (i) (t )∥ −E

E

≤ ∥ht i ∥ · ∥t −Ei X (i) (t )∥ ≤ ht

−ai +ε

−E

Let δ > 0 and take ht = (log t )

1+δ

−Ei −Ei

lim sup ∥ht

t

t →∞

(i)

X (t )∥ = 0

∥t −Ei X (i) (t )∥.

(4.7)

. Then, it follows from Theorem 2.2 that a.s.

Then, by (4.7), the limsup above implies that there is a positive constant c such that

  ∥t −Ei X (i) (t )∥ ≤ c exp (1 + δ)(ai − ε) log log t a.s.

(4.8)

for any sufficiently large t. Therefore we have 1 lim sup ∥t −Ei X (i) (t )∥ log log n ≤ e(1+δ)(ai −ε)

a.s.

t →∞

Letting δ ↓ 0 and ε ↓ 0, we have 1

lim sup ∥t −Ei X (i) (t )∥ log log n ≤ eai

a.s.

(4.9)

t →∞

We prove the reverse inequality of the above one. Take ht = log t. It follows from Theorem 2.2 again that −Ei −Ei

lim sup ∥ht t →∞

t

X (i) (t )∥ = ∞ a.s.

Hence, by (4.7) we have 1

lim sup ∥t −Ei X (i) (t )∥ log log n ≥ eai

a.s.

(4.10)

t →∞

This yields (2.7). Now we show (2.8). By (4.6), (4.9) and (4.10) we obtain (2.8) immediately. We have completed the proof.



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