I. A. Ibragimov, M. A. Lifshits, “On almost sure limit theorems”, Teor. Veroyatnost. i Primenen., 44:2 (1999), 328–350; Theory Probab. Appl., 44:2 (2000), 254

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Teoriya Veroyatnostei i ee Primeneniya, 1999, Volume 44, Issue 2, Pages 328–350
DOI: https://doi.org/10.4213/tvp767 (Mi tvp767)

This article is cited in 33 scientific papers (total in 34 papers)

On almost sure limit theorems

I. A. Ibragimov^a, M. A. Lifshits^b

^a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
^b Saint-Petersburg State University

Full-text PDF (1054 kB) Citations (34)

DOI: https://doi.org/10.4213/tvp767

Abstract: By a sequence of random vectors

{ζ_{k}}

$\{\zeta_k\}$ , we can construct empirical distributions of the type

Q_{n} = (\log n)^{- 1} \sum_{k = 1}^{n} δ_{ζ_{k}} / k

$Q_n = (\log n)^{-1} \sum_{k=1}^n \delta_{\zeta_k}/k$ . Statements on the convergence of these or similar distributions with probability 1 to a limit distribution are called almost sure theorems. We propose several methods which permit us to easily deduce the almost sure limit theorems from the classical limit theorems, prove the invariance principle of the type “almost sure” and investigate the convergence of generalized moments. Unlike the majority of the preceding papers, where only convergence to the normal law is considered, our results may be applied in the case of limit distributions of the general type.

Keywords: limit theorems, convergence almost sure, sums of independent variables, weak dependence, invariance principle.

Received: 12.02.1998

English version:
Theory of Probability and its Applications, 2000, Volume 44, Issue 2, Pages 254–272
DOI: https://doi.org/10.1137/S0040585X97977562

Bibliographic databases:

Language: Russian

Citation: I. A. Ibragimov, M. A. Lifshits, “On almost sure limit theorems”, Teor. Veroyatnost. i Primenen., 44:2 (1999), 328–350; Theory Probab. Appl., 44:2 (2000), 254–272

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tvp767

https://doi.org/10.4213/tvp767

https://www.mathnet.ru/eng/tvp/v44/i2/p328

This publication is cited in the following 34 articles:

Panqiu Xia, Guangqu Zheng, “Almost Sure Central Limit Theorems for Parabolic/Hyperbolic Anderson Models with Gaussian Colored Noises”, J Theor Probab, 38:2 (2025)
István Berkes, Siegfried Hörmann, “Some Optimal Conditions for the ASCLT”, J Theor Probab, 37:1 (2024), 209
A. A. Borovkov, Al. V. Bulinski, A. M. Vershik, D. Zaporozhets, A. S. Holevo, A. N. Shiryaev, “Ildar Abdullovich Ibragimov (on his ninetieth birthday)”, Russian Math. Surveys, 78:3 (2023), 573–583
István Berkes, Endre Csáki, “On the Almost Sure Central Limit Theorem Along Subsequences”, MathPann, 28_NS2:1 (2022), 11
Torrisi G.L., Leonardi E., “Almost Sure Central Limit Theorems in Stochastic Geometry”, Adv. Appl. Probab., 52:3 (2020), 705–734
Azmoodeh E., Nourdin I., “Almost Sure Limit Theorems on Wiener Chaos: the Non-Central Case”, Electron. Commun. Probab., 24 (2019), 9
M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163
Khalil M., Tudor C.A., Zili M., “Spatial Variation For the Solution to the Stochastic Linear Wave Equation Driven By Additive Space-Time White Noise”, Stoch. Dyn., 18:5 (2018), 1850036
L. Pastur, M. Shcherbina, “Szegö-type theorems for one-dimensional Schrödinger operator with random potential (smooth case)”, Zhurn. matem. fiz., anal., geom., 14:3 (2018), 362–388
Khalil M., Tudor C.A., “Correlation Structure, Quadratic Variations and Parameter Estimation For the Solution to the Wave Equation With Fractional Noise”, Electron. J. Stat., 12:2 (2018), 3639–3672
L. PASTUR, M. SHCHERBINA, “Szego-Type Theorems for One-Dimensional Schrodinger Operator with Random Potential (Smooth Case)”, Z. mat. fiz. anal. geom., 14:3 (2018), 362
Zheng G., “Normal Approximation and Almost Sure Central Limit Theorem For Non-Symmetric Rademacher Functionals”, Stoch. Process. Their Appl., 127:5 (2017), 1622–1636
Cenac P., Es-Sebaiy Kh., “Almost Sure Central Limit Theorems For Random Ratios and Applications To Lse For Fractional Ornstein–Uhlenbeck Processes”, Prob. Math. Stat.., 35:2 (2015), 285–300
Shen G., Yan L., Cui J., “Berry-Ess,En Bounds and Almost Sure Clt for Quadratic Variation of Weighted Fractional Brownian Motion”, J. Inequal. Appl., 2013, 275, 1–13
Oprisan A., Korzeniowski A., “Large Deviations via Almost Sure Clt for Functionals of Markov Processes”, Stoch. Anal. Appl., 30:5 (2012), 933–947
Tudor C., “Berry-Esseen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion”, J Math Anal Appl, 375:2 (2011), 667–676
Theory Probab. Appl., 55:2 (2011), 361–367
Bercu B., Nourdin I., Taqqu M.S., “Almost sure central limit theorems on the Wiener space”, Stochastic Process Appl, 120:9 (2010), 1607–1628
Nourdin I., Peccati G., “Universal Gaussian fluctuations of non-Hermitian matrix ensembles: From weak convergence to almost sure CLTs”, Alea-Latin American Journal of Probability and Mathematical Statistics, 7 (2010), 341–375
A. Chuprunov, L. Terekhova, “Almost sure versions of limit theorems for random allocations with the random number of cells”, Lobachevskii J Math, 31:3 (2010), 271

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

Theory of Probability and its Applications

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