A functional modulus of continuity for Brownian motion

The Uniform Modulus of Continuity Of Iterated Brownian Motion

Let X be a Brownian motion defined on the line (with X(0)=0) and let Y be an independent Brownian motion defined on the nonnegative real numbers. For all t ≥ 0, we define the iterated Brownian motion (IBM), Z, by setting Z t ∆ = X(Y t). In this paper we determine the exact uniform modulus of continuity of the process Z.

A CLT for the L modulus of continuity of Brownian local time

Let {Lt ; (x, t) ∈ R1 ×R1 +} denote the local time of Brownian motion and αt := ∫ ∞ −∞ (Lt ) 2 dx. Let η = N(0, 1) be independent of αt. For each fixed t ∫∞ −∞(L x+h t − Lt )2 dx− 4ht h3/2 L → ( 64 3 )1/2 √ αt η, as h → 0. Equivalently ∫∞ −∞(L x+1 t − Lt )2 dx− 4t t3/4 L → ( 64 3 )1/2 √ α1 η, as t → ∞.

A Clt for the L2 Modulus of Continuity of Brownian Local Time1 By

Let {L x t ; (x, t) ∈ R 1 × R 1 + } denote the local time of Brownian motion, and α t := ∞ −∞ (L x t) 2 dx. Let η = N(0, 1) be independent of α t. For each fixed t, ∞ −∞ (L x+h t − L x t) 2 dx − 4ht h 3/2 L → 64 3 1/2 √ α t η as h → 0. Equivalently, ∞ −∞ (L x+1 t − L x t) 2 dx − 4t t 3/4 L → 64 3 1/2 √ α 1 η as t → ∞. 1. Introduction. In [10] almost sure limits are obtained for the L p moduli o...

On Modulus of Continuity And Adaptability in Nonparametric Functional Estimation

We study adaptive estimation of linear functionals over a collection of finitely many parameter spaces.A between class modulus of continuity, a geometric quantity, is introduced and is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single par...

On L2 modulus of continuity of Brownian local times and Riesz potentials