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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....

A recursion formula for the moments of the gaussian orthogonal ensemble

M. Ledoux (2009)

Annales de l'I.H.P. Probabilités et statistiques

We present an analogue of the Harer–Zagier recursion formula for the moments of the gaussian Orthogonal Ensemble in the form of a five term recurrence equation. The proof is based on simple gaussian integration by parts and differential equations on Laplace transforms. A similar recursion formula holds for the gaussian Symplectic Ensemble. As in the complex case, the result is interpreted as a recursion formula for the number of 1-vertex maps in locally orientable surfaces with a given number of...

A recursive robust Bayesian estimation in partially observed financial market

Jianhui Huang (2007)

Applicationes Mathematicae

I propose a nonlinear Bayesian methodology to estimate the latent states which are partially observed in financial market. The distinguishable character of my methodology is that the recursive Bayesian estimation can be represented by some deterministic partial differential equation (PDE) (or evolution equation in the general case) parameterized by the underlying observation path. Unlike the traditional stochastic filtering equation, this dynamical representation is continuously dependent on the...

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