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Tsitsiashvili G.Sh., Osipova M.A. - DIRECT CALCULATIONS OF A REACHING MOMENT DISTRIBUTION FOR AN AUTOREGRESSIVE RANDOM SEQUENCE R&RATA # 1 (12)
BY RECCU RENT INTEGRAL EQUALITIES (Vol.2) 2009, March
DIRECT CALCULATIONS OF A REACHING MOMENT DISTRIBUTION FOR AN AUTOREGRESSIVE RANDOM SEQUENCE BY RECCURENT
INTEGRAL EQUALITIES
Tsitsiashvili G.Sh., Osipova M.A.
Institute for Applied Mathematics, Far Eastern Branch of RAS 690041, Vladivostok, Radio str. 7, e-mail: [email protected], [email protected]
In this paper we consider an autoregressive random sequence
Xk = RXk-l +%-!, X = 0, withO < R < 1 and assume that rjk has exponentials mixture distribution with
P (n > t )=ZPr exp (-Xrt).
r=1
Our problem is to calculate a distribution of a reaching moment t = inf (k: Xk > X). This problem origins in the risk theory, in the financial mathematics, in the statistics of random processes and in the reliability theory. Interest to mixtures of exponentials as approximations of distributions with heavy tails is initiated by papers [1], [2]. At first look this problem may be solved by martingale technique. But in different applications when R > 1 or R,X depend on k or we need a distribution of a jump over X it is too complicated. In this paper we apply some recurrent integral equalities to get over these difficulties. We solve the considered problem in symbols and illustrate obtained solutions by numerical calculations.
Denote the deriving moment for this sequence reaching some boundary and designate
Xj = XRj, k > 1, 0 < j < k -1, Xk = 0. Theorem 1. The following formula are true for k > 1:
Tk (x) = P(Xk > x, t> k) = £ Takk-SJr expf ^'
xk-s+1 < x < xk-s:
s = 1,...,k, and
r=1 j=0
p ( = k )=2 ak 00r exp (~AX ) • r=1
Rj
Here for 1 < r ,q < l
s-1 k-t i pqArakk t jr 2k+1 k+1-s o q = I(s * 1)2 2 Z7 rkt1jrA(k,t,q,r, j)
t=1 j=0 r=1 Àr - Rj+1Aq
B ( k s q r
1 r=1 A - Rj+1Aq
+I (s * k + 1)ï 2 PAakk+-7B (k, s, q, r, j), 1 < s < k +1,
j 11 1 , r 'q
^ PqKakk-s j—1 r q
ak+1 k+1-s j r 2 - ni'i
q=1 Ar - RJA,
0 < j < k +1 - s , 1 < s < k,
(1) (2)
a100r = pr ,
(3)
(4)
(5)
with
B(k,s, q, r, j) = exp{-X*^- (R+1 )) , A (k, t, q, r, j ) = B (k, t, q, r, j )-exp X^-t ( -Aq
Tsitsiashvili G.Sh., Osipova M.A - DIRECT CALCULATIONS OF A REACHING MOMENT DISTRIBUTION FOR AN AUTOREGRESSIVE RANDOM SEQUENCE BY RECCU RENT INTEGRAL EQUALITIES
Proof. As T1 ( x )=2 pr exp (-Arx ), 0 < x < X, so we have an equality
r=1
P(t = 1)= 2 pr exp(-ArX) and (1)-(3) are true.
r=1
Denote Qk (x) = P (RXk > x, t> k +1) and calculate
Then we have
So
with
Qi (x) = 2 Pr exp[-R I , 0 < x < XR .
min(x,XR) i
T2 ( x ) = - I 2 Pq eXP (-Aq ( x-U ))dQ1 (U ) =
0 q=1
l l Ar exP (-Aqx) ) [ . , mjA.
=2=12 pqpr A, - R*
- exp J - min (x, XR )[ -A
R
Arx
iipp exp (-Aqx)- exp (-R II, 0 < x < XR,
a, exp (-Aqx )
i i
q=1 r=1 q r Ar - RAq
2 2 pqpr ' ^\„q 7 (1 - exp (-X (a, - RAq ))), XR < x < X.
T2 (x)=2 2a22_S]r exp {-j, X2-s < x < X22-s, s = 1,2, r=1 j=0 V RJ
= i ar = i ar
a210 q = 2 pqpr~i T^T , a211q = - 2 pqpr~j ,
r=1 ar - Raq q=1 ar - Raq
= i ar
a200q = 2 pqpr j^T
r=1 ar - Raq
A (1,1, q, r,0), A (1,1, q, r, 0) = (1 - exp (-X (Ar - RAq ))).,
So
Then we have P(t = 2) = 2 a200r exp(-xrX). So for k = 1 the formulas (4), (5) are true also.
r=1
Assume that the formulas (1), (2) are true for fixed k then
-dQk (u)=2,2ak
r=1 j=0
a,
k k-S j r Rj +1exp I Rj + 1 I dU , Xk +1
au
x£?~s < u < Xk+ts, s = 1,...,k .
min(x,X1+1 ) i
Tk+1 ( x ) = - I 2 pq exp (-Aq ( x-u ))dQk (u ) =
0 q=1
= 2 mmT'%£ 2 2 aja1 -] exp i- +uaq ) du =
S =1min(x, Xkk++2-s )j=0 q=1 r=1 RJ+1 V RJ+1 V
kk2S 2 2 pqAr exp (-Aqx ) akk-sjr a (k . ) 0 . <X
= 2 2 2 2—-. v Dj+/,--A1 (k,s,q,r,jx<X,
s =1 j=0 q=1 r=1 ar - RJ aq
q
with
Ar -A
Rj+1 q
A (k,s,q,r, j) = expmin(x,s))XJ)-expf-min(x,X^) From another side we search Tk+1 (x) as follows:
l k+1-s f x X ^
Tk+1 (x)=2 2 ak+1 k+i-S]r exp Rj j, Xkk+12-S < x < Xkk++1-s, s = 1,..., k +1.
Tsitsiashvili G.Sh., Osipova M.A - DIRECT CALCULATIONS OF A REACHING MOMENT DISTRIBUTION FOR AN AUTOREGRESSIVE RANDOM SEQUENCE BY RECCU RENT INTEGRAL EQUALITIES
So for s = 1 we have
£ £ * - kjr exp ( j .1 £ £ (exp ^) ex> (- j
and obtain (4), (5). For 2 < s < k +1 we have
£k£ ( V^ £ £ £ iPgk exp(-Vx)akk-tjr A(, .)
£ £ ak+1 k+1-sjrexpI I = £ £ £ £ ——. Dj+1o--A(k,s,q,r, j)
r=1 j=0 V RJ J t=1 j=0 q=1 r=1 vr - RJ +1aq
W , ,)k~SLLPqVr exp (-Vqx) akk-sjr (D(, . ( ( Vr V
+I(s *k +1) £ £ £ —--—^-—I B(k,s,q,r, j)-expl -xl —-V
V j0q=1 r=1 Vr-Rj+Vq f V ' ,J) PV VR+1 '
As a result obtain (2), (4), (5).
Denominators in (4), (5) may be small or even zero. This circumstance creates difficulties in calculations. These difficulties may be got over by a following statement.
Lemma 1. Suppose that V1,...,Vl are different positive numbers and R = m / n where m, n are integers and mutually simple. Then for any s> 0 satisfying inequalities v -V. > 2s, 1 < i * j < l, there are rational numbers A1,..., Al so that
V - A< s, v * Rkvj, 1 < i * j < l, k > 0 . (6)
Proof. Denote L = mn for any s > 0 there are integers V,k1,...,k so that
1 £
— <-,
NL 2
A, - k <£, 1 < / < l, i N 2
then
|a,-Aj <£, with A, = k,L+1, 1 <i < l. I ' M ' nl
As A, -A j > 2s so A *Aj , 1 < i * j < l. Each pair of integers ( ktL +1, L ), 1 < i < l, has not joint divisors larger 1 so (L + 1)nk *(kjL +1) mk, 1 < i * j < l, k > 0.
Remark 1. Suppose that X = 1, R = 0.9, l = 13 and a, , pi,1< i < l, are described by Table 1 then in an accordance with Theorem 1 we obtain Table 2.
i A, P,
1 4.491 0.193963
2 1.422 0.651199
3 0.371 0.147817
4 0.076 0.006832
5 0.014 1.88 x10-4
6 0.03 4.61x10-6
7 5 x 10-4 1.11x10-7
8 8.8 x 10-5 2.65 x10-9
9 1.6 x 10-5 6.35 x10-11
10 2.9 x10-6 1.52 x10-12
11 5.4 x10-7 3.63 x 10-14
12 9.7 x 10-8 8.61x 10-16
13 1.5 x 10-8 1.72 x 10-17
Table 1.
k P(t = k )
3 0.267786
6 0.214032
9 0.001387
12 0.000051
15 1.621 x 10-6
18 4.747 x 10-8
21 1.345 x 10-9
24 3.755 x10-11
27 1.042 x10-12
30 2.9 x 10-14
Table 2.
Tsitsiashvili G.Sh., Osipova M.A - DIRECT CALCULATIONS OF A REACHING MOMENT DISTRIBUTION FOR AN AUTOREGRESSIVE RANDOM SEQUENCE BY RECCU RENT INTEGRAL EQUALITIES
Remark 2. The results of Theorem 1 remain valid for variable boundary
t = inf (k: Xk > X (k)) with the replacement of the designations
Xk = min(R,X(k)), j = 1,...,k-1, Xl = 0, Xk = X(k) , k > 1.
Remark 3. Obtained formulas are true for variable R:
Xk = Rk-1Xk -1 -1, 0 < Rk-1 < 1 , with the replacement of the designations
Xj = XRj, Rk = Rk:X-1, 0 < j <k-1 , R0 = 1, Xk = 0.
Remark 4. In an accordance with (2) we have that a jump of Xk, k > 1, over a level X may be characterized by the following formula:
P (X > Y + / k) 2r=1 ak 00r exp (-Xr (X + y)) > 0
P(Xk > X + y/T = = -( X y) ,y > 0 .
2r=1 ak00r exp(-XrX)
The authors thank A.A. Novikov for useful discussions.
References
[1] Asmussen S. Ruin Probabilities. Singapore: World Scientific, 2000.
[2] Feldmann A., Whitt W. Fitting mixtures of exponentials to long-tailed distributions to analyze networkperfomance models //Perfomance Evaluation. 1998. Vol. 31. Pp. 245-279.
