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Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 4, Pages 361–392 (Mi tvp4736)  

This article is cited in 10 scientific papers (total in 10 papers)

Some Limit Theorems for Stationary Processes

I. A. Ibragimov

Leningrad

Abstract: In this paper stationary stochastic processes in the strong sense $\{x_j\}$ are investigated, which satisfy the condition
$$ |\mathbf P(AB)-\mathbf P(A)\mathbf P(B)|\leq\varphi(n)\mathbf P(A),\quad\varphi(n)\downarrow 0, $$
for every $A\in\mathfrak{M}_{-\infty}^0,B\in\mathfrak{M}_n^\infty$, or the “strong mixing condition”
$$ \sup_{A\in\mathfrak{M}_{-\infty}^0,B\in\mathfrak{M}_n^\infty}|\mathbf P(AB)-\mathbf P(A)\mathbf P(B)|\alpha(n)\downarrow0, $$
where $\mathfrak{M}_a^b$ is a $\sigma$-algebra generated by the events
$$ \{(x_{i_1},x_{i_2},…,x_{i_k})\in\mathbf E\},\qquad a \leq i_1<i_2<…<i_k\leq b, $$
$\mathbf E$ being a $k$-dimensional Borel set.
Some limit theorems for the sums of the type
$$\frac{x_1+\cdots+x_n}{B_n}-A_n\quad{or}\quad\frac{f_1+ \cdots+f_n}{B_n }-A_n$$
are established. Here $f_j=T^j f$, and the random variable $f$ is measurable with respect to $\mathfrak{M}_{-\infty}^\infty $.

Full text: PDF file (2597 kB)

English version:
Theory of Probability and its Applications, 1962, 7:4, 349–382

Received: 15.01.1961

Citation: I. A. Ibragimov, “Some Limit Theorems for Stationary Processes”, Teor. Veroyatnost. i Primenen., 7:4 (1962), 361–392; Theory Probab. Appl., 7:4 (1962), 349–382

Citation in format AMSBIB
\Bibitem{Ibr62}
\by I.~A.~Ibragimov
\paper Some Limit Theorems for Stationary Processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1962
\vol 7
\issue 4
\pages 361--392
\mathnet{http://mi.mathnet.ru/tvp4736}
\transl
\jour Theory Probab. Appl.
\yr 1962
\vol 7
\issue 4
\pages 349--382
\crossref{https://doi.org/10.1137/1107036}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. A. Ibragimov, “Dobrushin's works on Markov processes”, Russian Math. Surveys, 52:2 (1997), 239–243  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. K. Budsaba, P. Chen, A. I. Volodin, “Limiting behaviour of moving average processes based on a sequence of $\rho^-$ mixing and negatively associated random variables”, Lobachevskii J. Math., 26 (2007), 17–25  mathnet  zmath
    3. Theory Probab. Appl., 55:3 (2011), 371–394  mathnet  crossref  crossref  mathscinet  isi
    4. Wintenberger O., “Weak Transport Inequalities and Applications To Exponential and Oracle Inequalities”, Electron. J. Probab., 20 (2015), 114  crossref  isi
    5. Feng F., Wang D., Wu Q., “An almost sure central limit theorem for self-normalized weighted sums of the ? mixing random variables”, J. Math. Inequal., 10:1 (2016), 233–245  crossref  mathscinet  zmath  isi  scopus
    6. Wintenberger O., “Exponential inequalities for unbounded functions of geometrically ergodic Markov chains: applications to quantitative error bounds for regenerative Metropolis algorithms”, Statistics, 51:1 (2017), 222–234  crossref  mathscinet  isi  scopus
    7. Rio E., “Asymptotic Theory of Weakly Dependent Random Processes”, Asymptotic Theory of Weakly Dependent Random Processes, Probability Theory and Stochastic Modelling, 80, Springer-Verlag Berlin, 2017, 1–204  crossref  isi
    8. Kanaya Sh., “Convergence Rates of Sums of Alpha-Mixing Triangular Arrays: With An Application to Nonparametric Drift Function Estimation of Continuous-Time Processes”, Economet. Theory, 33:5 (2017), 1121–1153  crossref  isi
    9. Theory Probab. Appl., 63:3 (2019), 479–499  mathnet  crossref  crossref  isi  elib
    10. Ibragimov I.A. Lifshits M.A. Nazarov A.I. Zaporozhets D.N., “On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236  crossref  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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