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( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space H p , q to L p , q (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type ( L 1 , L 1 ) . As a consequence we show that the dyadic integral of a ∞ function f L 1 is dyadically differentiable and its derivative is f a.e.

A class of strong limit theorems for countable nonhomogeneous Markov chains on the generalized gambling system

Kangkang Wang (2009)

Czechoslovak Mathematical Journal

In this paper, we study the limit properties of countable nonhomogeneous Markov chains in the generalized gambling system by means of constructing compatible distributions and martingales. By allowing random selection functions to take values in arbitrary intervals, the concept of random selection is generalized. As corollaries, some strong limit theorems and the asymptotic equipartition property (AEP) theorems for countable nonhomogeneous Markov chains in the generalized gambling system are established....

A multiplier theorem for Fourier series in several variables

Nakhle Asmar, Florence Newberger, Saleem Watson (2006)

Colloquium Mathematicae

We define a new type of multiplier operators on L p ( N ) , where N is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on L p ( N ) , to which the theorem applies as a particular example.

A recurrence theorem for square-integrable martingales

Gerold Alsmeyer (1994)

Studia Mathematica

Let ( M n ) n 0 be a zero-mean martingale with canonical filtration ( n ) n 0 and stochastically L 2 -bounded increments Y 1 , Y 2 , . . . , which means that P ( | Y n | > t | n - 1 ) 1 - H ( t ) a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let V 2 = n 1 E ( Y n 2 | n - 1 ) . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. P ( M n [ - c , c ] i . o . | V = ) = 1 for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....

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