The extremal index for a Markov chain

1992 ◽  
Vol 29 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Extreme Values ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extremal Index ◽  
Scaling Properties ◽  
Continuous State Space ◽  
Continuous State ◽  
Tail Behaviour

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.

The extremal index for a Markov chain

1992 ◽  
Vol 29 (01) ◽  
pp. 37-45 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Extreme Values ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extremal Index ◽  
Scaling Properties ◽  
Continuous State Space ◽  
Continuous State ◽  
Tail Behaviour

The paper presents a method of computing the extremal index for a discrete-time stationary Markov chain in continuous state space. The method is based on the assumption that bivariate margins of the process are in the domain of attraction of a bivariate extreme value distribution. Scaling properties of bivariate extremes then lead to a random walk representation for the tail behaviour of the process, and hence to computation of the extremal index in terms of the fluctuation properties of that random walk. The result may then be used to determine the asymptotic distribution of extreme values from the Markov chain.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

Tree-dependent extreme values: the exponential case

1990 ◽  
Vol 27 (01) ◽  
pp. 124-133 ◽  
Author(s):  
Vijay K. Gupta ◽  
Oscar J. Mesa ◽  
E. Waymire
Keyword(s):  
Branching Process ◽  
Extreme Values ◽  
Rooted Tree ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Extinction Time ◽  
Brownian Excursion ◽  
Weighted Trees

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2 L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

scholarly journals Extreme values of an infinite mixture of normally distributed variables

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 909-916
Author(s):  
Ehfayed Shneina ◽  
Vladimir Bozin
Keyword(s):  
Distribution Function ◽  
Extreme Values ◽  
Infinite Sequence ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Value Distribution ◽  
Common Distribution ◽  
Common Distribution Function ◽  
The Common ◽  
Normally Distributed

We study distribution of extreme values of a mixture of an infinite sequence of independent normally distributed variables with the same mean and an increasing sequence of standard deviations, and prove that the common distribution function belongs to the domain of attraction of Gumbel extreme value distribution. The norming constants for the maximum also are given.

scholarly journals Autoregressive Modeling of Forest Dynamics

Forests ◽  
2019 ◽  
Vol 10 (12) ◽  
pp. 1074
Author(s):  
Olga Rumyantseva ◽  
Andrey Sarantsev ◽  
Nikolay Strigul
Keyword(s):  
Random Walk ◽  
Forest Dynamics ◽  
Forest Biomass ◽  
Autoregressive Process ◽  
Autoregressive Models ◽  
Forest Inventories ◽  
Continuous State Space ◽  
Continuous State ◽  
Yearly Average ◽  
Geometric Random Walk

In this work, we employ autoregressive models developed in financial engineering for modeling of forest dynamics. Autoregressive models have some theoretical advantage over currently employed forest modeling approaches such as Markov chains and individual-based models, as autoregressive models are both analytically tractable and operate with continuous state space. We performed a time series statistical analysis of forest biomass and basal areas recorded in Quebec provincial forest inventories from 1970 to 2007. The geometric random walk model adequately describes the yearly average dynamics. For individual patches, we fit an autoregressive process (AR) of order 1 capable to model negative feedback (mean-reversion). Overall, the best fit also turned out to be geometric random walk; however, the normality tests for residuals failed. In contrast, yearly means were adequately described by normal fluctuations, with annual growth on average of 2.3%, but with a standard deviation of order of 40%. We used a Bayesian analysis to account for the uneven number of observations per year. This work demonstrates that autoregressive models represent a valuable tool for the modeling of forest dynamics. In particular, they quantify the stochastic effects of environmental disturbances and develop predictive empirical models on short and intermediate temporal scales.

Tail equivalence and its applications

1971 ◽  
Vol 8 (01) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 &lt;A &lt;∞ , as x → ∞, then for normalizing constants an &gt; 0, bn, n &gt; 1, Fn (anx + bn ) → φ(x), φ(x) non-degenerate, iff Gn (anx + bn )→ φ A−1(x). Conversely, if Fn (anx+bn )→ φ(x), Gn (anx + bn ) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C &gt;0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (02) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(J n , X n ), n ≧ 0} with X 0 = - ∞ a.s. The marginal sequence {J n } is an irreducible, aperiodic, m-state M.C., m &lt; ∞, and the r.v.'s X n are conditionally independent given {J n }. Furthermore P{J n = j, X n ≦ x | J n − 1 = i} = p ij H i (x) = Q ij (x), where H 1(·), · · ·, H m (·) are c.d.f.'s. Setting M n = max {X 1, · · ·, X n }, we obtain P{J n = j, M n ≦ x | J 0 = i} = [Q n (x)] i, j , where Q(x) = {Q ij (x)}. The limiting behavior of this probability and the possible limit laws for M n are characterized. Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then: (a)ρ(x) is a c.d.f.; (b) if for a suitable normalization {Q ij n (a ijn x + b ijn )} converges completely to a matrix {U ij (x)} whose entries are non-degenerate distributions then U ij (x) = π j ρ U (x), where π j = lim n → ∞ p ij n and ρ U (x) is an extreme value distribution; (c) the normalizing constants need not depend on i, j; (d) ρ n (a n x + b n ) converges completely to ρ U (x); (e) the maximum M n has a non-trivial limit law ρ U (x) iff Q n (x) has a non-trivial limit matrix U(x) = {U ij (x)} = {π j ρ U (x)} or equivalently iff ρ(x) or the c.d.f. π i = 1 m H i π i(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {M n } are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

scholarly journals The extremes of random walks in random sceneries

2009 ◽  
Vol 41 (2) ◽  
pp. 452-468 ◽  
Author(s):  
Brice Franke ◽  
Tatsuhiko Saigo
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Stationary Process ◽  
Random Sequence ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Extreme Value Distributions ◽  
Stable Law ◽  
Random Scenery

In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law, and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit distribution is not in the class of classical extreme value distributions.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (01) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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