Hitting times of threshold exceedances and their distributions

0.1 MB·English

by  Natalia Markovich

Lecture / Ponencia XX
Hitting times of threshold exceedances and their
distributions
N. Markovich1
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1 1 Institute of Control Sciences, Russian Academy of Sciences, Russia,
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[email protected],[email protected]
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n
a Abstract
J

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We investigate exceedances of the process over a sufficiently high threshold. The

] exceedances determine the risk of hazardous events like climate catastrophes, huge
T
insurance claims, the loss and delay in telecommunication networks. Due to de-
S
. pendence such exceedances tend to occur in clusters. Cluster structure of so-
h
t cial networks is caused by dependence (social relationships and interests) between
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m nodes and possibly heavy-tailed distributions of the node degrees. A minimal time
[ to reach a large node determines the first hitting time. We derive asymptotically

equivalent distribution and a limit expectation of the first hitting time to exceed
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v the threshold u as sample size n tends to infinity. The results can be extended
n
1
to the second and, generally, to kth (k > 2) hitting times.
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1. Introduction
0
.
1
0 Let {Xn}n≥1 be a stationary process with marginal distribution function F(x) and
5 M = max{X ,...,X }. We investigate the exceedances of the process over a
1 n 1 n
: sufficiently high threshold u. Due to dependence such exceedances tend to occur
v
in clusters. Let us consider the inter-cluster size
i
X
r T (u) = min{j ≥ 1 : M ≤ u,X > u|X > u}, (1)
a 1 1,j j+1 1
i.e. the number of inter-arrivals of observations running under the threshold be-
tween two consecutive exceedances, where M = max{X ,...,X }, M = −∞.
1,j 2 j 1,1
Let T∗(u ) be a first hitting time to exceed the threshold u . We get
n n
P{T∗(u ) = j +1} = P{M ≤ u ,X > u }, (2)
n j n j+1 n
j = 0,1,2,..., M = −∞.
0
The necessity to evaluate quantiles of the first hitting time and its mean is arisen
in many applications. In social networks it is important to compare sampling
strategies (Avrachenkov et al. (2012), Avrachenkov et al. (2014a), Avrachenkov
et al. (2014b), Chul-Ho Lee et al. (2012)) like random walks, Metropolis-Hastings
Markov chains, Page Ranks and other with regard to how quickly they allow to
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reach a node with a large degree that is the number of links with other nodes. It
is important to investigate the first hitting time of significant nodes since it allows
us to disseminate advertisement or collect opinions more effectively within the
clusters surrounded such nodes. Similar problem is required in telecommunication
peer-to-peer networks to find a node with a large number of peers, Dán and Fodor
(2009), Markovich (2013).
Definition1.1. (Leadbetteretal. (1983),p.53)Thestationarysequence{X }
n n≥1
is said to have extremal index θ ∈ [0,1] if for each 0 < τ < ∞ there is a sequence
of real numbers u = u (τ) such that
n n
lim n(1−F(u )) = τ and (3)
n
n→∞
lim P{M ≤ u } = e−τθ (4)
n n
n→∞
hold.
Theextremalindexθ of{X }relatestothefirsthittingtimeT∗(u ),Robertsetal.
n n
(2006). Really, since u is selected according to (3) it follows that P{X > u } is
n n n
asymptotically equivalent to 1/n. Notice, that P{M ≤ u } = P{T∗(u ) > k}.
k n n
Hence, substituting τ by (3) we get from (4)
P{T∗(u )/n > k/n} ∼ e−θkP{Xn>un} ∼ e−θk/n
n
and
lim P(T∗(u )/n > x) = e−θx
n
n→∞
for positive x. It follows
lim E(T∗(u )/n) = 1/θ. (5)
n
n→∞
This implies, the smaller θ, the longer it takes to reach an observation with a large
value.
Using achievements regarding the asymptotically equivalent geometric distribution
of T (x ) derived in (Theorem 1.1, Markovich (2014)), where the (1 − ρ )th
1 ρn n
quantile x of {X } is taken as u , we derive an asymptotically equivalent dis-
ρn n n
tribution of the first hitting time and its limiting expectation that specifies (5) in
Section 2.
We use the mixing condition proposed in Ferro and Segers (2003)
α (u) = max sup|P(B|A)−P(B)| = o(1), n → ∞, (6)
n,q
1≤k≤n−q
where for real u and integers 1 ≤ k ≤ l, F (u) is the σ-field generated by the
k,l
events {X > u}, k ≤ i ≤ l and the supremum is taken over all A ∈ F (u) with
i 1,k
P(A) > 0 and B ∈ F (u) and k, q are positive integers.
k+q,n
We use the following partition
1 ≤ k∗ ≤ k∗ ≤ k∗ ≤ k∗ ≤ j, j = 2,3,..., (7)
n,1 n,2 n,3 n,4
k∗ = 1, k∗ = j, k∗ = [jk /n] + 1,i = {1,2}, k∗ = j −[jk /n], k∗ =
n,0 n,5 n,i n,i n,3 n,4 n,4
2
1
j − [jk /n], of the interval [1,j] for a fixed j, where positive integers {k }
n,3 n,i
are such that
{k = o(k ),i ∈ {2,3,4}}, k = o(n). (8)
n,i−1 n,i n,4
Theorem 1.1. (Markovich (2014)) Let {X } be a stationary process with the
n n≥1
extremal index θ. Let {xρn} and {xρ∗n} be sequences of quantiles of X1 of the
levels {1−ρ } and {1−ρ∗}, respectively,2 those satisfy the conditions (3) and (4)
n n
if un is replaced by xρn or by xρ∗n and, qn = 1−ρn, qn∗ = 1−ρ∗n, ρ∗n = (1−qnθ)1/θ.
Let positive integers {k∗ }, i = 0,5, and {k }, i = 1,4, be respectively as in (7)
n,i n,i
and (8), p∗ = o(∆ ), ∆ = k∗ −k∗ , q∗ = o(p∗ ), i ∈ {1,2,...,5} and
n,i n,i n,i n,i n,i−1 n,i n,i
{p∗ } be an increasing sequence, such that
n,3
αn∗(xρn) = max{αkn∗,4,qn∗,1;αkn∗,3,qn∗,2;α∆n,3,qn∗,3;αj+1−kn∗,2,qn∗,4; (9)
αj+1−kn∗,1,qn∗,5;αj+1,kn∗,4−kn∗,1} = o(1)
and
αj+1,kn∗,4−kn∗,1 = o(ρn) (10)
hold as n → ∞, where α = α (x ) is determined by (6). Then it holds for
n,q n,q ρn
j ≥ 2
lim P{T (x ) = j}/(ρ (1−ρ )(j−1)θ) = 1, (11)
1 ρn n n
n→∞
The achievements can be similarly extended to the second and, generally, to kth
(k > 2)hitting times. Thepaperisorganizedasfollows. InSection 2we derivethe
asymptotically equivalent distribution and the limit expectation of the first hitting
time to exceed sufficiently high threshold. Proofs are given in Section 3.
2. Distribution and expectation of the first hitting time
Theorem 2.1. Let all conditions of Theorem 1.1 are satisfied. We assume
supρ E(T (x )) < ∞. (12)
n 1 ρn
n
Then we get
lim nP{T∗(u ) = n} = e−θτ/θ, (13)
n
n→∞
ρ
P{T∗(x ) = j} ∼ n(1−θρ )j−1. (14)
ρn θ n
as n → ∞.
1[x] represents the integer part of the real number x.
2F(xρn)=P{X1 >xρn}=ρn.
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Expression (14) implies that P{T∗(x ) = j} ∼ P{T = j}, where
ρn θ
χ, with probability 1/θ2;
T =
θ (cid:26) 0, with probability 1−1/θ2
holds and χ has the geometric distribution with probability ρ θ.
n
Remark 2.1. The condition (12) provides a uniform convergence of the range
∞ P{T (x ) = j} by n. The latter condition is fulfilled for a geometrically
j=1 1 ρn
Pdistributed T (x ), i.e. P{T (x ) = j} = ρ (1−ρ )j−1.
1 ρn 1 ρn n n
Lemma 2.1. Let conditions of Theorem 2.1 be satisfied. Then it follows
lim ρ ET∗(x ) = 1/θ3. (15)
n ρn
n→∞
Since ρ ∼ τ/n according to (3), the expression (15) specifies (5).
n
3. Proofs
3.1 Proof of Theorem 2.1
It follows from (2) that
P{T∗(u ) = n} = P{M ≤ u ,X > u }
n n−1 n n n
= P{M ≤ u }−P{M ≤ u }. (16)
n−1 n n n
We obtain due to (1) and the stationarity of {X } that it holds
n
P{T (u ) = n} = P{M ≤ u ,X > u |X > u }
1 n 1,n n n+1 n 1 n
= (P{M ≤ u ,X > u }−P{M ≤ u ,X > u })/P{X > u }
1,n n n+1 n n n n+1 n 1 n
= (P{M ≤ u ,X > u }−P{M ≤ u ,X > u })/P{X > u }
n−1 n n+1 n n n n+1 n 1 n
= (P{T∗(u ) = n}−P{T∗(u ) = n+1})/P{X > u }. (17)
n n 1 n
Following Ferro and Segers (2003) we get alternatively for n ≥ 1
P{T (u ) > n} = P{M ≤ u |X > u }
1 n 1,n+1 n 1 n
= (P{M ≤ u }−P{M ≤ u })/P{X > u }
1,n+1 n n+1 n 1 n
= (P{M ≤ u }−P{M ≤ u })/P{X > u }
n n n+1 n 1 n
= P{T∗(u ) = n+1}/P{X > u }.
n 1 n
Thus, we get
P{T∗(x ) = n+1} = P{X > x }·P{T (x ) > n}
ρn 1 ρn 1 ρn
∞
= ρ P{T (x ) = i}, (18)
n 1 ρn
i=Xn+1
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where ρ = P{X > x }. Now we use the fact followed from (11) in Theorem
n 1 ρn
1.1, i.e. it holds
c P{T (x ) = i} ∼ η (1−η )i−1 (19)
n 1 ρn n n
as n → ∞, where 1−η = (1−ρ )θ, c = η /(1−(1−η )1/θ), 0 < η < 1.
n n n n n n
Furthermore, we obtain
∞ ∞
ρ ρ
P{T∗(x ) = n+1} = n c P{T (u ) = i} ∼ n η (1−η )i−1
ρn c n 1 n c n n
n i=Xn+1 n i=Xn+1
ρ ρ
= n(1−η )n = n 1−(1−η )1/θ (1−η )n
n n n
c η
n n (cid:0) (cid:1)
ρ2(1−θρ )n ρ ρ
∼ n n = n(1−θρ )n ∼ ne−θτ, (20)
n
1−(1−ρ θ) θ θ
n
since (1−ρ )θ = 1−θρ +o(ρ ) and lim (1−θρ )n = e−θτ as ρ ∼ τ/n,
n n n n→∞ n n
n → ∞. Formula (20) implies both (13) rewriting it with regard to P{T∗(x ) =
ρn
n+1}/ρ and (14) replacing n+1 by j.
n
Hence, we have from (20)
1
P{T∗(x ) = j} ∼ ρ θ(1−ρ θ)j−1, n → ∞. (21)
ρn θ2 n n
The similarity in the first string of (20) follows from (12) and since
∞
supr (n) = sup P{T (x ) = i}
k 1 ρn
n n X
i=⌊k/ρn⌋
∞
i ρ
n
= sup P{T (x ) = i} ≤ sup E(T (x )) → 0
i 1 ρn k 1 ρn
n X n
i=⌊k/ρn⌋
holds as k → ∞.
3.2 Proof of Lemma 2.1
Using (18) and (19) we find the expectation of the first hitting time
∞ ∞ ∞
ET∗(x ) = jP{T∗(x ) = j} = jρ P{T (x ) = i}
ρn ρn n 1 ρn
Xj=1 Xj=1 Xi=j
∞ ∞ ∞
ρ ρ
∼ j n η (1−η )i−1 = n jη (1−η )j−1
n n n n
c c η
Xj=1 n Xi=j n n Xj=1
ρ2 ρ2
= n = n .
η3 (1−(1−ρ )θ)3
n n
The similarity follows from the same arguments as before. Then (15) follows.
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Acknowledgments
Special thanks to the partial financial support by the Russian Foundation for Basic
Research, grant 13-08-00744 A.
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