Poster at University of Cincinnati

Invariance principle for tempered fractional time series
models
Farzad Sabzikar
Michigan State University
Abstract
2. Tempered Hermite Process of Order One
4. Weak Convergence Results
Autoregressive tempered fractionally integrated moving average (ARTFIMA) time series is a useful model for velocity
data in turbulence flows. In this paper, we obtain an invariance principle for the partial sum of an ARTFIMA process.
The limiting process is called tempered Hermite process
of order one, T HP 1, which is well-defined for any H > 12 .
When 12 < H < 1, we develop the Wiener integral with
respect to T HP 1 to provide the sufficient condition for the
convergence
Definition: Given an independently scattered
Gaussian random measure B(dx) on R with control measure σ 2dx, H > 21 , λ > 0, the stochastic
integral
∫ ∫ t(
)
3
H− 2 −λ(s−y)+
1
ZH,λ(t) :=
(s − y)+ e
ds B(dy)
Assume H = 12 + α for α > 0 and let {Zj }j∈Z be
a sequence of independent and identically distributed random variables mean zero and variance one . Define the random variables
∑ λ
λ
Ykn :=
Cjn Zk−j , k = 1, 2, . . .
(14)
∫
+∞ ( ) λ
∑
k
−H
1 (du)
f
n
Xkn →
f (u)ZH,λ
n
R
k=0
1. Tempered Fractional Time Series
The tempered fractional difference operator is defined by:
∆α,λ
h f (x) =
wj e−λjhf (x − jh),
(1)
j=0
where
( )
j
α
(−1)
Γ(1 + α)
j
wj := (−1)
=
j
j!Γ(1 + α − j)
∆α,λ
1 Xt =
wj e−λjhXt−j ,
(5)
where (x)+ = xI(x > 0), will be called a tempered
Hermit process of order one (T HP 1).
j∈Z
where
{
Cj =
Proposition:
1
• The process ZH,λ
given by (5) has the covariance function
∫ t∫ s
R(t, s) = CH,λ
|u−v|H−1K1−H (λ|u−v|)dv du,
0
0
2Γ(H− 12 )
where CH,λ = √π(2λ)H−1 .
1
• The process ZH,λ
given
(6)
(2)
j=0
follows an ARMA(p, q) model (see [MSPA]).
Theorem: Let {Xt} be an ARTFIMA (0, α, λ, 0)
times series.
• {Xt} has the spectral density
2
−2α
σ −(λ+iω)
h(ω) = 1 − e
(3)
,
2π
for −π ≤ ω ≤ π.
• The covariance function of {Xt} is
σ e Γ(α + k)
−2λ
γk =
F
(α;
k
+
α;
k
+
1;
e
),
2 1
2π Γ(α)k!
(4)
∑∞ Γ(a+j)Γ(b+j)Γ(c)z j
where 2F1(a; b; c; z) = j=0 Γ(a)Γ(b)Γ(c+j)Γ(j+1) is the
hypergeometric function.
Remark:
• The ARTFIMA (0, α, λ, 0) times series with the
covariance function γk given by (4) has the following asymptotic behavior:
by (2) has stationary in-
σ 2 1 −λk α−1
e k (1 − e−2λ)−α
γk ∼
2π Γ(α)
as |k| → ∞.
• The ARTFIMA (0, α, λ, 0) is short memory process,∑since by the previous part one can show
∞
that k=0 γk < ∞.
0
if j ≤ 0.
(15)
For t ≥ 0, we define S (t) as the partial sum of
λ
{Xkn },
λ
n
S (t) :=
[t]
∑
λ
n
λ
n
Yk + (t − [t])Y[t]+1,
t ≥ 0,
(16)
where∑
[t] is the largest integer less than or equals
0
t and k=1 = 0. We also define
(7)
Theorem: Given 12 < H < 1 and λ > 0, the class
of functions
}
{
∫ (
)
2
1
H−
,λ
A1 := f ∈ L2(R) :
I− 2 f (x) dx < ∞ ,
∑−m
j=1−m
= 0.
Lemma: For any θ1, θ2, . . . , θp, t1, t2, . . . , tp ≥ 0, we
have
p
2
∑ ∑
λ
−2H
n
θr ξm(nt) →
n
m∈Z r=1
∫ t
p
+∞ ∑
∫
−∞
θr
H− 32
(s − y)+ e−λ(s−y)+
2
ds dy
0
r=1
as n → ∞.
Theorem: Let {Zj }j∈Z be a sequence of i.i.d
random variables with mean zero and finite vari−H nλ
ance. Then n S (nt) converges weakly to
1
ZH,λ
(t), given by (5), in C[0, 1] ,as n → ∞ (C[0, 1]
is the space of all continuous functions defined on
[0, 1]). That is
n
−H
λ
n
1
S (nt) ⇒ ZH,λ
(t),
(18)
(10)
(11)
m
(j )
∑
+
=
fn,m
f
1[ j , (j+1) ],
n
n
n
j=0
+
,
fn− = fn,∞
+
,
fn+ = fn,∞
(12)
The set of elementary functions E is dense in the
space A1. The space A1 is not complete.
1
2
(17)
where ⇒ means weak convergence in C[0, 1].
For m ∈ N ∪ {∞}, we define the approximation
is a linear space with inner product
1 ( H− 12 ,λ )
F (x) = Γ(H − ) I−
f (x)
2(
1 H− 12 ,λ )
G(x) = Γ(H − ) I−
g (x).
2
Cj + (t − [t])Ct+1−m,
for m ∈ Z and t ≥ 0, where
is called tempered fractional integral of the function f (see [MS]).
⟨f, g⟩A1 := ⟨F, G⟩L2(R)
λ
n
λ
n
j=1−m
Lemma: For a tempered Hermite process of order one given by (5), T HP 1, with λ > 0 and H > 21 ,
we have:
∫ +∞ (
)
1
1
H− 2 ,λ
1
ZH,λ(t) = Γ(H − )
I−
1[0,t] (x) B(dx),
2 −∞
(8)
where
∫ ∞
1
α,λ
−λ(u−t)+
f (u)(u − t)α−1
I− f (t) =
e
du (9)
+
Γ(α) −∞
R
∑
[t]−m
λ
n
ξm(t) :=
3. Wiener integrals with respect to tempered
Hermite process of order one
where
if j ≥ 1
k=1
crements, such that
{ 1
}
{ H 1
}
ZH,λ(ct) t∈R , c ZH,cλ(t) t∈R
2 −λk
1 α−1 − nλ j
e
Γ(α) j
λ
n
for any scale factor c > 0.
for α > 0, λ > 0, and Γ(·) is the Euler gamma
function. If λ = 0 and α is a positive integer, then
equation (1) reduces to the usual definition of the
fractional difference operator.
Definition: The discrete time stochastic process {Xt} is called an autoregressive tempered fractional integrated moving average ,
ART F IM A (p, λ, α, q), if
∞
∑
0
λ
n
in distribution, as n → ∞, where Xk is an ARTFIMA time
1 is T HP 1.
series and ZH,λ
∞
∑
R
Definition: For any < H < 1 and λ > 0, we
define
∫
∫ (
)
1
1
H− 2 ,λ
1
f (x)ZH,λ(dx) := Γ(H − )
I−
f (x) B(dx)
2 R
R
(13)
for any f ∈ A1.
−
=
fn,m
−1
∑
f
j=−m
fm = fn+ +
(j )
n
1[ j , (j+1) ],
n
n
fn−
Theorem: Let α > 0 and Xjλ be the ARTFIMA
(0, α, λ, 0) times series. Suppose also that the following, condition A, is satisfied:
±
∥A1 → 0
f, fn± ∈ A1, ∥fn± − fn,m
∥f − fn∥A1 → 0 as n → ∞.
as m → ∞,
Then,
∫
+∞ ( )
∑
λ
k
−H
1
n
f
n
Xk →
f (u)ZH,λ
(du)
n
R
k=0
in distribution as n → ∞. We refer the reader to
see [S] for more details.
References
[S]
F. Sabzikar (Submitted): Invariance principle for tempered fractional time series models.
[MS] M. Meerschaert and F. sabzikar (2014): Stochastic integration for tempered fractional Brownian motion. Stochastic Processes and their Applications, Vol. 124 (2014) pp. 2363–2387.
[MSPA] M. Meerschaert, F. Sabzikar, M. Phanikumar, A. Zeleki (2014): Tempered fractional time series model for turbulence in geophysical flows. Journal of Statistical Mechanics: Theory and
Experiment, to appear in the Special Issue on Fractional Dynamics: Theory and Applications
Cincinnati Symposium on Probability Theory and Applications, September 19th - 21th 2014, University of Cincinnati, Cincinnati, Ohio

Download Report