## Almost everywhere convergence of weighted averages.(English)Zbl 0736.28008

Given a sequence $$(\mu_ n)$$ of probability measures on $$Z$$, and an invertible measure-preserving transformation $$\tau$$ of a probability space $$(X,\beta,m)$$, the averages $$\mu_ nf(x)=\sum^{\infty}_{k=- \infty}\mu_ n(k)f(\tau^ kx)$$ are bounded operators on $$L^ p(m)$$, $$1\leq p\leq\infty$$. Sufficient conditions are given in terms of the Fourier transforms $$(\hat\mu_ n(\gamma))$$ on $$T=\{\gamma:|\gamma|=1\}$$ for the a.e. convergence of $$\mu_ nf(x)$$ for all $$f\in L^ p(m)$$, $$1<p\leq\infty$$. In particular, if $$\lim_{n\to\infty}\sum^{\infty}_{k=-\infty}|\mu_ n(k)-\mu_ n(k-1)|=0$$, then there exists a subsequence $$(\mu_{n_ m})$$ such that for all systems $$(X,\beta,m,\tau)$$ and $$f\in L^ p(m), 1<p\leq\infty$$, $$\lim_{n\to\infty}\mu_{n_ m}f(x)$$ exists a.e. If $$\mu_ n=\mu^ n$$ for some fixed strictly aperiodic finitely supported measure $$\mu$$, then $$(n_ m)$$ can be any subsequence with $$n_{m+1}>n^{\beta}_ m$$ for some $$\beta>1$$. Related theorems, examples, and generalizations to group actions are also described.

### MSC:

 28D05 Measure-preserving transformations
Full Text:

### References:

  Akcoglu, M., del Junco, A.: Convergence of averages of point transformations. Proc. Am. Math. Soc.49, 265-266 (1975) · Zbl 0278.28011  Bellow, A., Losert, V.: The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences. Trans. Am. Math. Soc.288, 307-345 (1985) · Zbl 0619.47004  Bellow, A., Jones, R., Rosenblatt, J.: Convergence for moving averages. Ergodic Theory Dyn. Syst.10, 43-62 (1990) · Zbl 0674.60035  Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Isr. J. Math.61, 39-72 (1988) · Zbl 0642.28010  Calderón, A.P.: Ergodic theory and translation-invariant operators. Proc. Natl. Acad. Sci. USA59, 349-353 (1968) · Zbl 0185.21806  Déniel, Y.: On a.s. Cesàro-? convergence for stationary or orthogonal random variables. J. Theor. Probab.2, 475-485 (1989) · Zbl 0684.60014  Duoandikoetxea, J., Rubio de Francia, J.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math.84, 541-561 (1986) · Zbl 0568.42012  Derriennic, Y.: Personal communication  Emerson, W.R.: The pointwise ergodic theorem for amenable groups. Am. J. Math.96, 472-487 (1974) · Zbl 0296.22009  Foguel, S.R.: On iterates of convolutions. Proc. Am. Math. Soc.47, 368-370 (1975) · Zbl 0299.43004  Foguel, S.R.: Iterates of a convolution on a non-abelian group. Ann. Inst. Henri Poincaré11, 199-202 (1975) · Zbl 0312.60004  Huang, Y.: Random sets for the pointwise ergodic theorem. Ph.d. Thesis, Northwestern University: 1989  Kahane, J.-P.: Some random series of functions, 2nd ed. Cambridge: Cambridge University Press 1985 · Zbl 0571.60002  Pier, J.-P.: Amenable locally compact groups. New York: John Wiley and Sons 1984 · Zbl 0597.43001  Petrov, V.V.: Sums of independent random variables. (Ergeb. Math., Grenzgeb., vol. 82) Berlin Heidelberg New York: Springer 1975 · Zbl 0322.60043  Rosenblatt, J.: Ergodic and mixing random walks on locally compact groups. Math. Ann.257, 31-42 (1981) · Zbl 0459.60006  Rosenblatt, J.: Ergodic group actions. Arch. Math.47, 263-269 (1986) · Zbl 0583.28006  Stein E.M.: On the maximal ergodic theorem. Proc. Natl. Acad. Sci. USA47, 1894-1897 (1961) · Zbl 0182.47102  Tempelman, A.A.: Ergodic theorems for general dynamical systems. Sov. Math. Dokl.8, 1213-1216 (1967) · Zbl 0172.07303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.