O. Kallenberg, \Random Measures", Akademie-Verlag, Berlin, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....chosen independently according to . Let L(R; be the straight line passing through the points X and Y. The constant (2.1) 2 exists and satis es 0 1. Furthermore, 2.2) 2 We are ready to state our main theorem. For accounts of vague and weak convergence, see [8] [11]. Theorem 2.2. Assume that (A1) and (A2) hold. a) The set n 2 ; g converges vaguely to a Poisson process with constant intensity = and the convergence is weak when restricted to any bounded interval of R. In particular, the random variable n 1 converges weakly, as n 1, ....

....Poisson progress with constant intensity c, and, for all T 0, the set n [0; T ] converges weakly to the process restricted to [0; T ] In particular, n 1 converges weakly to the exponential distribution with parameter c. Proof. We apply Theorem 3.2 and Lemma 2. 2 of [8] see also [11]) with I the semiring of half open intervals of the form (a; b] a; b 0. Note that, in the notation of [8] if U 2 I is such that U [0; T ] for some T , then P(X j 2 U) P(X j T ) and similarly for P(X j 2 U; X k 2 U ) Lemma 3.1 is used to bound 1 by 2 . In Sections 4 and 5, we ....

Kallenberg, O., Random Measures, 3rd edn, Akademie-Verlag, Berlin, 1983.

....with i = 1 holds true. Thus, P (1; u ln ;N (x) o( and the assertion (2.2) follows as in (2.23) This completes the proof of the lemma. Proof of Theorem 1.1. The necessity of (1.6) is immediate from the assertion (2.2) of Lemma 2.1. The suciency follows from Kallenberg s theorem [Ka] (see also [LLR] on the weak convergence of a point process PN to the Poisson process P. Applying his theorem in our situation, weak convergence holds whenever (i) For all intervals (a; b] 2 R fu ln ;N (X )2(a;b]g Ke ; as N 1: ii) For all nite unions of half open disjoint ....

O. Kallenberg, Random measures, 4th edition, Akademie Verlag, Berlin, 1986.

....the bivariate J function in terms of the conditional intensity of X. This provides explicit formulae for JX;Y in many examples. The Papangelou conditional intensity X (u) of a point process X at a location u is described informally in Kallenberg (1984) and defined rigorously in Glotzl (1980a,b) Kallenberg (1983), Kozlov (1976) Nguyen and Zessin (1976) Intuitively we may regard X (u) du as the conditional probability that there will be a point of the process X in the neighbourhood du of a location u 2 R d , given full information about the process X outside this neighbourhood, and full information ....

O. Kallenberg. Random measures. Akademie Verlag/Academic Press, Berlin/New York, third edition, 1983.

.... = B p t p cannot be realized as stopping times and therefore the strong Markov property cannot be applied. We use the idea of Palm distributions to overcome this diculty. Palm distributions are also a common tool in other branches of probability such as queuing theory or point processes, see [OK83] for a general reference and [UZ88] PZ94] MP98] for applications in fractal geometry. De nition: Denote by M(IR d ) the Polish space of all locally nite Borel measures on IR d equipped with the vague topology and by d the Lebesgue measure on IR d . A stationary quasidistribution ....

O. Kallenberg. Random Measures. (Akademie-Verlag, Berlin, 1983).

....first see, why we can restrict our attention to the behaviour of the occupation measure at the origin. An elegant approach to this problem is the idea of Palm distributions, which also appears in many other branches of probability like queuing theory or point processes. For a general reference see Kallenberg (1983), for an account 4 from the point of view of fractal geometry see U. Zahle (1988) or Patzschke and M. Zahle (1992, 1993) The following definition is based on a well known characterization theorem of Mecke. Definition Denote by M(IR d ) the Polish space of all locally finite Borel measures ....

....i e GammaY i i = i 1 1 j i 1 1 j and, for k arbitrary, say with s n Gamma1 k s n , we can repeat previous arguments and sandwich the sum between two sums which both converge to the same limit. This finishes the proof. By the continuity theorem for Laplace transforms (see e.g. Kallenberg (1983), 15.5.2) we infer from the case r = 1 in Lemma 4.1 that the distribution of the density function at the origin with respect to 1 or 2 converges almost surely to a standard exponential distribution and that the distribution with respect to converges almost surely to the distribution of the ....

O. Kallenberg. Random Measures. (Akademie-Verlag, Berlin, 1983).

....(1) relates tangent measure distributions to Palm distributions. Palm distributions originate from queuing theory and they are nowadays widely used in the theory of point processes where they play the r ole of conditional distributions of stationary point processes given a point at the origin (see [10]) A probability distribution P on M(IR) is a Palm distribution if there is a stationary nite measure Q on M(IR) with nite intensity 0 and Z M (B) dQ( Z B P (T u ) M) du for all M M(IR) B IR Borel. The link between our theorem and Palm distributions is the following ....

O. Kallenberg, \Random Measures", Akademie-Verlag, Berlin, 1983.

....Poisson point measure. We have (2.1) iff P k : 1 X i=1 ffl ( i d(k) X (k) i ) X m ffl (t m ;j m ) P1 ; k 1; 2.4) in M p ( 0; 1) Theta [ Gamma1; 1] n f0g) where for a nice set E , M p (E) denotes the set of point measures on E, topologized by the vague metric. See [26] [17], 23] Note that the limit random point measure P1 is a Poisson random measure on [0; 1) Theta [ Gamma1; 1] n f0g with mean measure L Theta where L is Lebesgue measure. Note also that because of Assumption 1b, we have P1 ( 0; 1) Theta ( Gamma1; 0) 0 almost surely. Choosing a ffi 0 to ....

O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, third edition, 1983.

....of grand canonical Gibbs measures on a metric space X. Our duality formula uses Skorohod type integral operators, and a gradient that acts by finite differences. No differential structure is needed on X. Poisson measures have been characterized using Campbell measures in [8] cf. also [6], and the Wiener measure has been characterized by integration by parts, cf. 16] and [17] Th. 1.2. The characterization of Poisson measures has been extended to Gibbs measures in [10] 7] Gibbs measures on the Wiener pathspace have been considered in [17] in the case of the fixed lattice ZZ ....

O. Kallenberg. Random measures. Akademie-Verlag, Berlin, fourth edition, 1986.

....= B p t p cannot be realized as stopping times and therefore the strong Markov property cannot be applied. We use the idea of Palm distributions to overcome this diOEculty. Palm distributions are also a common tool in other branches of probability such as queuing theory or point processes, see [OK83] for a general reference and [UZ88] PZ94] MP98] for applications in fractal geometry. De nition: Denote by M(R d ) the Polish space of all locally nite Borel measures on R d equipped with the vague topology and by d the Lebesgue measure on R d . A stationary quasidistribution is a ....

O. Kallenberg. Random Measures. (Akademie-Verlag, Berlin, 1983).

....2 A; 0; if x 62 A: We emphasize that we assume that all measures in M p (E) are Radon which means that for any m 2 M p (E) and any compact K ae E; m(K) 1. On the space M p (E) we use the vague metric ae( Delta; Delta) Its properties are discussed for example in Resnick (1987, Section 3. 4) or Kallenberg (1983). Note that a sequence of measures mn 2 M p (E) converge vaguely to m 0 2 M p (E) if for any continuous function f : E 7 [0; 1) with compact support we have mn (f) m 0 (f) where mn (f) R E fdmn : The non negative continuous functions with compact support will be denoted C K (E) A ....

Kallenberg, O., Random Measures, Third edition, Akademie-Verlag, Berlin, 1983.

.... 1. Write Xi (n) f(t n;i ; t n;i 1 Gamma t n;i ) i 1g. Then Xi (n) Xi as n 1. Here and below, we regard Xi and Xi (n) as point processes on [0; 1) Theta (0; 1) and convergence is the natural notion of vague convergence of counting measures on [0; 1) Theta (0; 1) see e.g. [17]. Define fl(n; i) by: v(fl(n; i) is the last vertex in the i Gamma 1 st component of the random graph encountered by breadth first walk. Let C n;i be the size of this i th component. Lemma 8 Let Xi (1) be the point process with points f(l(fl) jflj) fl an excursion of B t g. Let Xi ....

....convergence in the product topology: lim n x (n) i = x i 8i. The set M has its own natural topology pointwise convergence, uniform over compact subsets. These deterministic notions extend in the usual way to notions of convergence in distribution for random elements of the spaces (see e.g. [17] for discussion of convergence in distribution for point processes) The following straightforward lemma provides a connection between these modes of convergence in distribution. Lemma 14 Let Y (n) 2 l 2 ( Gamma n ) for each 1 n 1, and let Xi (n) be the associated SBPP. The following ....

O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1983.

.... ; 2.5) where 2 D, see e.g. KMM78] and [GV68, Chap. III Sec. 4] Remark 2.2 The right hand side of (2. 5) defines, via Minlos theorem, the measure oe on (D 0 ; C oe (D 0 ) but an additional analysis shows that the support of the measure oe is Gamma ae D 0 ; see e.g. Kal74] [Kal83] and [KMM78] hence oe can be considered as a measure on Gamma: Let f : R d Theta Gamma R be such that f 0 and measurable with respect to B(R d ) Theta B( Gamma) Define F (fl) hfl; f ( Delta; fl)i = Z R d f (x; fl) dfl (x) X x2fl f (x; fl) Then oe is ....

.... of t (in this case is called a generalized Poisson process) For infinite measure ae the set of discontinuities of t is locally countable, see e.g. Tak67] and [Kin93] For d 1 the statements 2, 3 follow from the analogous results of the theory of Poisson measures, see e.g. Kal74] [Kal83] and [KMM78] Xi Remark 3.3 Assume that ae is a probability measure on (R;B(R) Let f k ; k 1g be a sequence of independent identically ae distributed random variables and N = fN t ; t 0g be the standard Poisson process independent of f k ; k 1g. Then CP is generated by the ....

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O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, third edition, 1983.

....We will learn why subsampling choosing a bootstrap sample of size m = o(n) is required for the bootstrap asymptotics of extremes and heavy tailed phenomena. Let M (E) denote the set of positive Radon measures on a nice locally compact space E. M (E) is metrized by the vague metric (cf. Kallenberg, 1983; Resnick, 1987) which we will denote by ae in the sequel. We denote weak convergence of random elements or probability measures by ) while v denotes vague convergence of measures in M (E) For x 2 E and A ae E define ffl x (A) ae 1; if x 2 A; 0; otherwise. A Radon point measure ....

Kallenberg, O., Random Measures, Third edition, Akademie-Verlag, Berlin, 1983.

....than the actual values, of the s. Writing for the project onto the y axis map (f(s fl ; y fl )g) fy fl g (56) we can recover ord Y from Xi via ord Y = ord ( Xi) Convergence in (57) below is the natural notion of vague convergence of counting measures on [0; 1) Theta (0; 1) see e.g. [10]. Proposition 17 ( 1] Proposition 15) Let Y (n) 2 l 2 ( Gamma n ) for each 1 n 1, and let Xi (n) be the associated SBPP. Suppose Xi (n) d Xi (1) 57) where Xi (1) is a point process satisfying supfs : s; y) 2 Xi (1) for some yg = 1 a.s. 58) if (s; y) 2 Xi ....

O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1983.

....(Resnick, 1986, 1987, 1991) and will not be emphasized here beyond some reminders about notation and one central result. In what follows M (E) is the set of positive Radon measures on a nice locally compact space E; M p (E) is the set of point measures. M (E) is metrized by the vague metric (cf. Kallenberg, 1983; Resnick, 1987; Neveu, 1976) We denote weak convergence of random elements or probability measures by ) and v denotes vague convergence of measures in M (E) For x 2 E and A ae E define ffl x (A) ae 1; if x 2 A; 0; otherwise. A Radon point measure with points in E is denoted P ....

Kallenberg, O., Random Measures, Third edition, Akademie-Verlag, Berlin, 1983.

....A; 0; if x = 2 A: We emphasize that we assume that all measures in M p (E) are Radon which means that for any m 2 M p (E) and any compact K ae E, m(K) 1. On the space M p (E) we use the vague metric ae( Delta; Delta) Its properties are discussed for example in Resnick (1987, Section 3. 4) and Kallenberg (1983). Note that a sequence of measures mn 2 M p (E) converge vaguely to m 0 2 M p (E) if for any continuous function f : E 7 [0; 1) with compact support we have mn (f) m 0 (f) where mn (f) R E f dmn . The non negative continuous functions with compact support will be denoted by C K (E) A ....

Kallenberg, O., Random Measures, Third edition, Akademie-Verlag, Berlin, 1983.

....A; 0; if x = 2 A: We emphasize that we assume that all measures in M p (E) are Radon which means that for any m 2 M p (E) and any compact K ae E, m(K) 1. On the space M p (E) we use the vague metric ae( Delta; Delta) Its properties are discussed for example in Resnick (1987, Section 3. 4) and Kallenberg (1983). Note that a sequence of measures mn 2 M p (E) converge vaguely to m 0 2 M p (E) if for any continuous function f : E 7 [0; 1) with compact support we have mn (f) m 0 (f) where mn (f) R E f dmn . The non negative continuous functions with compact support with be denoted by C K (E) A ....

Kallenberg, O., Random Measures, Third edition, Akademie-Verlag, Berlin, 1983.

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O. Kallenberg, \Random Measures", Akademie-Verlag, Berlin, 1983.

.... a random measure MN on R converges to M, if for any k 2 N, any continuous function F : R R, and any collection of continuous, integrable functions f 1 ; f k on R , F MN (dx)f 1 (x) MN (dx)f k (x) F M(dx)f 1 (x) M(dx)f k (x) 1:35) see e.g. [Ka,Re]) By standard approximation arguments, if M has moments of all orders, it is enough to consider for f i all monomial functions f(x) x 1 : x , and F (y) y 1 : y k . Now let W ;N be any process of the form (1.24) Then 1 : w i l l : w i m 2 ; N ....

.... 1 (w) dw)h j (w) 2:26) Now clearly ;N (dw)h i (w) N (dy)h i ( x) 2:27) We will use the fact that convergence in distribution of a sequence P N is equivalent to almost sure convergence of a sequence P N that for each N have the same distribution as P N (see e.g. [Ka,RE]) We can of course replace P N by P N in (2.26) without changing anything. We have already shown that the denominators in the arguments of h i converge in distribution, i.e. converges a.s. to . The same argument applies to the numerators, i.e. for xed x the ....

O. Kallenberg, Random measures, 4th edition, Akademie Verlag, Berlin, 1986.

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O. Kallenberg. Random Measures. (Akademie-Verlag, Berlin, 1983).

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O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, (1983).

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O. Kallenberg. Random Measures. Akademie Verlag, Berlin, 1983. 15

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O. Kallenberg. Random Measures. (Akademie-Verlag, Berlin, 1983).

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O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1983.

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O. Kallenberg. Random measures. Akademie-Verlag, Berlin, fourth edition, 1986.

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O. Kallenberg, Point Processes. Akademie-Verlag, Berlin, 1983.

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Kallenberg, O., Random Measures, 3rd edn, Akademie-Verlag, Berlin, 1983.

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O. Kallenberg, Random Measures, 3rd Edition, Akademie Verlag/Academic Press, Berlin/New York, 1983.

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Kallenberg, O., Random Measures, Akademie--Verlag, Berlin, 1986(4th edition).

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O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1983.

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O. Kallenberg. Random Measures. Akademie-Verlag, Berlin, 1983.

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O. Kallenberg, Random Measures, Akademie-Verlag, Berlin, 1983.

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O. Kallenberg. Random Measures. Akademie Verlag, Berlin, 1983.

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