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Article
A discrete fractal in \({\mathbb{Z}}\) related to Pascal’s triangle modulo 2
For each integer d ≥ 1, let $$\begin{array}{ll}\fancyscript{F}_d = \left\{ k \in \mathbb{Z} : \binom{(2^d+1)k}{k} = 1\quad ({\rm mod}\...
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Article
Tail Properties of Correlation Measures
We study the tail properties of a class of Borel probability measures, called correlation measures. We show that (i) there exist correlation measures with exponentially decaying tail probabilities, and (ii) ro...
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Article
The Length of the Longest Head-Run in a Model with Long Range Dependence
In this paper, we construct stationary sequences of random variables {χ i : i≥0} taking values ±1 with probability 1/2 and we prove an Erdös–Rényi law of large numbers for the le...
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Chapter and Conference Paper
Iterated Brownian Motion and its Intrinsic Skeletal Structure
This is an overview of some recent results on the stochastic analysis of iterated Brownian motion. In particular, we make explicit an intrinsic skeletal structure for the iterated Brownian motion which can be ...
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Article
The uniform modulus of continuity of iterated Brownian motion
LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt≥0, we define theiterated Brownian motion (IBM),Z, by setting
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Article
On the future infima of some transient processes
Let (X(t), t∈S) be a real-valued stochastic process with ℙ(X(0)=0)=1 and $$\mathbb{P} (\mathop {\lim }\limits_{t \to \infty } X(t) = \infty ) ...
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Article
A law of the iterated logarithm for random walk in random scenery with deterministic normalizers
LetX, X i ,i≥1, be a sequence of independent and identically distributed ℤ d -valued random vectors. LetS o=0 and
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Article
A self normalized law of the iterated logarithm for random walk in random scenery
LetX,X i ,i≥1, be a sequence of i.i.d. random vectors in ℤ d . LetS o=0 and, forn≥1, letS n =X 1+...+X ...
