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Y = (. . . , Y−1 , Y0 , Y1 , . . .),
Yt ∈ {0, 1, ...}, t ∈ Z = {. . . , −1, 0, 1, . . .} M K &0 -# 8
5 * * 0* ** * t $% 0,
)I -&*# 0 & 4 - '
)%0* 0% ωs # τs θs (i) ∈ Z + 0& & - 08
s ∈ Z + 0& -# 5 0* s ** ωs + i − 1
/ θs (i) > 0 &, &H)5 i ∈ {1, . . . τs } θs (i) = 0 - i ≤ 0, i ≥ τs + 1
% ξt ∈ Z+ )%0* 0& 0# -.5, ** t
** 7 / --%', -# *5 %&0*
* ** t 0*
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s∈Z
-&4*# 0
;" &0 &0 τs , s ∈ Z+ ƒ
C" &0 &0 ξt , t ∈ Z + # ξt * -& # / P {ξt = k} = e k!λ , 0 < λ < ∞, k ∈ Z + ƒ
?" &0 &0 τs %* %* ξt ωs ƒ
& i ∈ {1, ..., τs},
A" 0 *H -,H /@ θs (i) = 1,
0, & i ≤ 0, i ≥ τs + 1.
−λ
k
-& Yt &, 4 t ,&,, -*# *
- DYt = λEτ +/ ξ τ *H 4 -&# ξt τs
H *&/ 5,I - 0 %H M/G/1 *&/H# *8
& M 0 % - ξt # G / -& %*5
5 * )&4, τs
& & 5 - τs &, 5 s ∈ Z + 5 l ∈ N *H 8
-& # /
EYt = λEτ
P (τs = l) = c0 l
−α−1
, c0 =
∞
−1
l
−α−1
, 1 < α < 2,
!;"
l=1
- Yt ) *-0 *-)* -'* -** 6
3−α
H ∈ ( 12 , 1) - 1 < α < 2# {Yt }, t ≥ 0 K -' &
2
-*,/H * - 1 < α < 2 τs *H 0 )0H
-H
H =
DE
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V
Eτ < ∞ P (ξ = 0) < 1 Y /D/1/h . ' 6
. /&@ D(h) &+ " )'E
Ploss ≥
∞
1
P (τ ≥ n),
λEτ (Eτ + Eκ)2 n=n
!C"
1
)
n1
=
a =
h+b
+ 2,
a
Eκ = [1 − P (ξ = 0)]−1 − 1,
(Eτ + Eκ)−1 ≤ 1,
b=a+1
Ploss ≤
∞
1
P (τ ≥ n),
(1 − λEτ )Eτ
n=h
λEτ < 1
!?"
D<
& &0, &0 τ * -& # 48
,, 5,, ' &, , - - - h → ∞ ) */
Ploss ≥
c0
α(α − 1)λEτ (Eτ + Eκ)2
Ploss ≤
−α+1
h+b
,
+2
a
c0
h−α+1 ,
α(α − 1)Eτ (1 − λEτ )
!A"
!D"
λE(τ ) < 1
&, )&/.5 h * -&0*
Ploss ≥ clow h−α+1
!E"
Ploss ≤ cupper h−α+1 .
!<"
c0
α(α − 1)λEτ (Eτ + Eκ)α+1
!B"
K* clow =
c0
α(α − 1)Eτ (1 − λEτ )
!G"
c0
h−α+1 = q1 h−α+1
α(α − 1)(Eτ + Eκ)α+1
!;="
c0 λ
h−α+1 = q2 h−α+1 .
α(α − 1)(1 − λEτ )
!;;"
cupper =
%, λ, α, Eτ h
* )%*# ' !E" !<" 0/H *4&,# %,I h# H / ), , - - * %* )7
h **# 0 &0 '5 *& &7 ,/
Ploss ) * h K-'&/# -* )%*
,/ - - ,% ,/H --&, )78
0# - &5 . --&4,5 &/ 7 Yt
&, Pover -& &HI *-0 '@
Pover ≥
Pover ≤
,/ Pover 4 ) * h K-'&/# -*
)%*
6 *4% i 5*45*$*45(: !*4#d: T*"*;*:
'# -&0 :)* $ # 0/ ) -05 -&8
4,5# -&/ %, %* )7 h ".
DB
# h .
6&/ ) -&0/ &0 %&/
&, 1 * 5,I* -*
-&4*# 0 * ,% 0 * )/ 2 ≤ r < ∞ %5
- !t;u" * &* 4 0 - k, k = 1, . . . , r 5%,
* & - τ (k) # , * -&
-** α(k) # 1 ≤ k ≤ r@
(k)
P (τ (k) = n) = c0 n−α
(k)
−1
,
!;C"
1 < α(1) < . . . α(r) < 2.
K* &0 ξt = ξt(1) + ... + ξt(r) # ξt(i) / &0 0 - i#
-.5, ** t ξt(i) * -& -**
(i)
λ(i) , 1 ≤ i ≤ r )*# 0) &0 ξt , 1 ≤ i ≤ r )& %* ξt 4 * -& -** λ = λ(1) + ... + λ(r) -8
&4* 4# 0 &0 λ = const# &0 λ(i) , 1 ≤ i ≤ r * %*,,
* )%*# * *4* %*,/ &# 0* 4 -
)I* -# &,, - K* %** / )I -
4 0 %* - R = 1 - *,
- %* )7 h < ∞# --, -)/
& C ; - - ,, - --& )7# %
'-& d ∈ D(h) &/. 4, &,
** 5,I -' Y ∗ = Y (1) + . . . + Y (r) # /
--%', %*5 -' Y (k) , k = 1, . . . , r -&5 .
-&* &0 d(k) = λλ -&, 5 ξ ∗ τ ∗ )
*/H -& ξ (k) τ (k) # %,5 * d(k) # /
(k)
P {ξ ∗ = m} =
r
d(k) P {ξ (k) = m} =
k=1
P {τ ∗ = x} =
r
r
k=1
d(k) P {τ (k) = x} =
k=1
r
d(k)
(λ(k) )m −λ(k)
e
m!
(k)
d(k) c0 x−α
(k)
−1
.
!;?"
!;A"
k=1
75 λ(k) λ - &* F * x ) *&8
0* &/ )0 P {τ ∗ = x} -) *-
cx−α −1 # / *- P {τ ∗ = x} - *- * ,8
4& -&, % P
-0 0 %, ',# ,4& %,
-5, # H % -4H &/ )& &
%,
)* d(k) &HI* )%*
(1)
d(k)
d(r)
(k)
(r)
= hα −α , k = 1, . . . , r − 1
= 1 − d(1) − . . . − d(r−1) .
( /# 0 d(r) → 1, d(k) → 0, k = 1, . . . , r − 1 - h → ∞
!;D"
DG
& λ 7# -&, ξ ∗ τ ∗ ) %/ h
-&/%, !C" !?" 7# 0 Eτ ∗ = d(k) Eτ (k) → Eτ r - h → ∞# *
-&0*# 0
Ploss ≥
∞ r
1
d(k) P (τ (k) ≥ n).
λEτ ∗ (Eτ ∗ + Eκ∗ )2 n=n
!;E"
r
∞ 1
≤
d(k) P (τ (k) ≥ n).
Eτ ∗ (1 − λEτ ∗ )
!;<"
1
Ploss
k=1
n=h k=1
&/
(k)
(k)
(k)
(k)
c0
c
n−α ≤ P (τ (k) ≥ n) ≤ 0(k) (n − 1)−α
α(k)
α
∞
(k)
(k)
1
1
−α(k) +1
≤
n−α ≤ (k)
n
(n1 − 2)−α +1 , n1 > 2, 1 < α(k) < 2,
1
(k)
(α − 1)
(α − 1)
n=n
1
% !;C" !;D" &# 0
∞
d(k)
(k)
P (τ (k) ≥ n) ≥
n=n∗
1
d(k)
∞
d(k) c0
α(k)
n=h
d(k) c0
α(k)
(k)
(k)
≥
(k)
d(k) c0
(n∗ )−α +1
α(k) (α(k) − 1) 1
(k)
≤
(k)
d(k) c0
(h − 2)−α +1 .
(k)
α (α(k) − 1)
n−α
n=n∗
1
∞
(k)
P (τ (k) ≥ n) ≤
∞
(n − 1)−α
(k)
n=h
)&/.5 h
(k)
(k)
d(k) c0
(n∗1 )−α +1
(k)
(k)
α (α − 1)
c̃1 h−α
∼
c̃2 h−α
(k)
(k)
(k)
d(k) c0
(h − 2)−α +1
(k)
(k)
α (α − 1)
(r)
∼
(r)
(k)
+1
+1
.
0&/ -&0*#
(r)
c̃1 h−α
+1
(r)
≤ Ploss ≤ c̃2 h−α
+1
,
c̃1 =
r
(k)
c0
1
λ(r) Eτ (r) (Eτ (r) + Eκ(r) )2 k=1 α(k) (α(k) − 1)
!;B"
!;G"
E=
c̃2 =
r
(k)
c0
1
.
Eτ (r) (1 − λ(r) Eτ (r) ) k=1 α(k) (α(k) − 1)
!C="
* )%*# /, & 0 %5 - )I* -# *48
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,/ - -
) t;u &, *& :) * 0* )& -&8
0 4,, *-0, ' &, , --&, )7 '* 4*@
(r)
Pover ≥ q̃1 h−α
+1
,
q̃1 =
r
(k)
c0
1
.
(Eτ (r) + Eκ(r) )2
α(k) (α(k) − 1)
!C;"
!CC"
k=1
+**# 0 &, , --&, )7 '* 4*
-& 4 *-0, 5,, '@
(r)
Pover ≤ q̃2 h−α
q̃2 =
+1
,
r
(k)
λ(r)
c0
.
(1 − λ(r) Eτ (r) ) k=1 α(k) (α(k) − 1)
!C?"
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