7 research outputs found
Uncomputably noisy ergodic limits
V'yugin has shown that there are a computable shift-invariant measure on
Cantor space and a simple function f such that there is no computable bound on
the rate of convergence of the ergodic averages A_n f. Here it is shown that in
fact one can construct an example with the property that there is no computable
bound on the complexity of the limit; that is, there is no computable bound on
how complex a simple function needs to be to approximate the limit to within a
given epsilon
Uncomputably noisy ergodic limits
V'yugin has shown that there are a computable shift-invariant measure on
Cantor space and a simple function f such that there is no computable bound on
the rate of convergence of the ergodic averages A_n f. Here it is shown that in
fact one can construct an example with the property that there is no computable
bound on the complexity of the limit; that is, there is no computable bound on
how complex a simple function needs to be to approximate the limit to within a
given epsilon
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear
operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n
x. We prove the following variational inequality in the case where T is power
bounded from above and below: for any increasing sequence (t_k)_{k in N} of
natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p,
where the constant C depends only on p and the modulus of uniform convexity.
For T a nonexpansive operator, we obtain a weaker bound on the number of
epsilon-fluctuations in the sequence. We clarify the relationship between
bounds on the number of epsilon-fluctuations in a sequence and bounds on the
rate of metastability, and provide lower bounds on the rate of metastability
that show that our main result is sharp
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Computable Measure Theory and Algorithmic Randomness
International audienceWe provide a survey of recent results in computable measure and probability theory, from both the perspectives of computable analysis and algorithmic randomness, and discuss the relations between them
Mathematical linguistics
but in fact this is still an early draft, version 0.56, August 1 2001. Please d