Trees Generated by a Simple Branching Process

Hawkes, J. G.

doi:10.1112/jlms/s2-24.2.373

Cited by 93 publications

(85 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, the parabolic capacity arguments rely on relationships between the Hausdorff dimension of random sets and stochastic co-dimension (see §2). The latter is a formalization of a particular application of [23,Theorem 4], which can be found in various forms within the proofs of [1,7,14,19]. We suspect our formulation has other applications.…”

Section: X(t + H) − X(t)| H α/2 Ln(1/h) 2 Dim(ementioning

confidence: 99%

See 1 more Smart Citation

Fast sets and points for fractional Brownian motion

Khoshnevisan

Shi

2000

Lecture Notes in Mathematics

View full text Add to dashboard Cite

Summary. In their classic paper, S. Orey and S.J. Taylor compute the Hausdorff dimension of the set of points at which the law of the iterated logarithm fails for Brownian motion. By introducing "fast sets", we describe a converse to this problem for fractional Brownian motion. Our result is in the form of a limit theorem. From this, we can deduce refinements to the aforementioned dimension result of Orey and Taylor as well as the work of R. Kaufman. This is achieved via establishing relations between stochastic co-dimension of a set and its Hausdorff dimension along the lines suggested by a theorem of S.J. Taylor. Suppose W , W (t); t 0 is standard one-dimensional Brownian motion starting at 0. Continuity properties of the process W form a large part of classical probability theory. In particular, we mention A. Khintchine's law of the iterated logarithm (see, for example, [21, Theorem II.1.9]): for each t 0, there exists a null set N 1 (t) such that for all ω ∈ N 1 (t), lim sup Keywords andLater on, P. Lévy showed that ∪ t 0 N 1 (t) is not a null set. Indeed, he showed the existence of a null set N 2 outside which lim sup The main result of [18] is that with probability one,One can think of this as the multi-fractal analysis of white noise. Above and throughout, "dim(A)" refers to the Hausdorff dimension of A. Furthermore, whenever dim(A) is (strictly) negative, we really mean A = ?. Orey and Taylor's discovery of Eq. (1.3) relied on special properties of Brownian motion. In particular, they used the strong Markov property in an essential way. This approach has been refined in [3,4,11], in order to extend (1.3) in several different directions.Our goal is to provide an alternative proof of Eq. (1.3) which is robust enough to apply to non-Markovian situations. We will do so by (i) viewing F 1 (λ) as a random set and considering its hitting probabilities; and (ii) establishing (within these proofs) links between Eqs. (1.2) and (1.3).To keep from generalities, we restrict our attention to fractional Brownian motion. With this in mind, let us fix some α ∈ ]0, 2[ and define X , X(t); t 0 to be a one-dimensional Gaussian process with stationary increments, mean zero and incremental standard deviation given by,See (1.8) for our notation on L p (P) norms. The process X is called fractional Brownian motion with index α -hereforth written as f BM(α). We point out that when α = 1, X is Brownian motion. Let dim M (E) denote the upper Minkowski dimension of a Borel set E ⊂ R1 ; see references [17,24]. Our first result, which is a fractal analogue of Eq. (1.2), is the following limit theorem: Theorem 1.1. Suppose X is f BM(α) and E ⊂ [0, 1] is closed. With probability one,( 1.4)On the other hand, with probability one,(1.5) SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math. 393-416 (2000) Loosely speaking, when α = 1, Theorem 1.1 is a converse to (1.3). For all λ 0, define F α (λ) to be the collection of all closed sets E ⊂ [0, 1] such that lim supOne can think of the elements of F α (λ) as λ-fast sets. Theorem 1.1 can be...

show abstract

Section: X(t + H) − X(t)| H α/2 Ln(1/h) 2 Dim(ementioning

confidence: 99%

“…This kind of result has been implicitly used in several works. For example, see [7,14,19,20]. To prove it, let us introduce an independent symmetric stable Lévy process S γ , S γ (t); t 0 of index γ ∈ ]0, 1[ .…”

Section: Theorem 22 Suppose E Is a Random Set Then For Allmentioning

confidence: 99%

Fast sets and points for fractional Brownian motion

Khoshnevisan

Shi

2000

Lecture Notes in Mathematics

View full text Add to dashboard Cite

show abstract

“…Many of the martingale and tree ideas used in our proofs can be considered as part of the folklore and have been rediscovered many times in different guises. While it is impossible to give a full list of the relevant publications here, [4,10,11,12,13,16,18,21] represents a good selection of the pioneering papers, from which these ideas have been formed.…”

Section: Resultsmentioning

confidence: 99%

Why study multifractal spectra?

Mörters¹

2009

Trends in Stochastic Analysis

View full text Add to dashboard Cite

show abstract

“…Experts may easily show that this bound for the dimension of C is (a.s.) exact on the event that C 6 = ? (see 131,183,313]). …”

Section: General Labyrinthsmentioning

confidence: 99%

“…When the scheme incorporates a randomised step, then the ensuing set may be termed a`random fractal'. Such sets may be studied in some generality (see 131,153,183,313]), and properties of fractal dimension may be established. The following simple example is directed at a`percolative' property, namely the possible existence in the random fractal of long paths.…”

Section: General Labyrinthsmentioning

confidence: 99%

Percolation and disordered systems

Grimmett

1997

Lecture Notes in Mathematics

View full text Add to dashboard Cite

PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematical theory of percolation and related topics. The rst nine chapters summarise rigorous results in percolation theory, with special emphasis on results obtained since the publication of my book 156] entitled`Percolation', and sometimes referred to simply as G] in these notes. Following this core material are chapters on random walks in random labyrinths, and fractal percolation. The nal two chapters include material on Ising, Potts, and random-cluster models, and concentrate on a`percolative' approach to the associated phase transitions.The rst target of this course is to draw a picture of the mathematics of percolation, together with its immediate mathematical relations. Another target is to present and summarise recent progress. There is a considerable overlap between the rst nine chapters and the contents of the principal reference G]. On the other hand, the current notes are more concise than G], and include some important extensions, such as material concerning strict inequalities for critical probabilities, the uniqueness of the in nite cluster, the triangle condition and lace expansion in high dimensions, together with material concerning percolation in slabs, and conformal invariance in two dimensions. The present account di ers from that of G] in numerous minor ways also. It does not claim to be comprehensive. A second edition of G] is planned, containing further material based in part on the current notes.A special feature is the bibliography, which is a fairly full list of papers published in recent years on percolation and related mathematical phenomena. The compilation of the list was greatly facilitated by the kind responses of many individuals to my request for lists of publications.Many people have commented on versions of these notes, the bulk of which have been typed so superbly by Sarah Shea-Simonds. I thank all those who have contributed, and acknowledge particularly the suggestions of Ken Alexander, Carol Bezuidenhout, Philipp Hiemer, Anthony Quas, and Alan Stacey, some of whom are mentioned at appropriate points in the text. In addition, these notes have bene ted from the critical observations of various members of the audience at St Flour.Members of the 1996 summer school were treated to a guided tour of the library of the former seminary of St Flour. We were pleased to nd there a copy of the Encyclop edie, ou Dictionnaire Raisonn e des Sciences, des Arts et des M etiers, compiled by Diderot and D'Alembert, and published in Geneva around 1778. Of the many illuminating entries in this substantial work, the following de nition of a probabilist was not overlooked. 3 PROBABILISTE, s. m. (Gram. Th eol.) celui qui tient pour la doctrine abominable des opinions rendues probables par la d ecision d'un casuiste, & qui assure l'innocence de l'action faite en cons equence. Pascal a foudroy e ce systême, qui ouvroit la porte au crime en accordant a l'autorit e les pr erogatives de la certitude, a l'opinion & la s ec...

show abstract

Trees Generated by a Simple Branching Process

Cited by 93 publications

References 0 publications

Fast sets and points for fractional Brownian motion

Fast sets and points for fractional Brownian motion

Why study multifractal spectra?

Percolation and disordered systems

Contact Info

Product

Resources

About