Definition and Relationship to Other Modes of Convergence
Almost sure convergence is one of the most fundamental concepts of convergence in probability and statistics. A sequence of random variables (X n ) n ≥ 1, defined on a common probability space (Ω, \(\mathcal{F}\), P), is said to converge almost surely to the random variable X, if
Commonly used notations are \(X_n \mathop \to \limits^{a.s.} X\) or lim n → ∞ X n = X (a. s. ). Conceptually, almost sure convergence is a very natural and easily understood mode of convergence; we simply require that the sequence of numbers (X n (ω)) n ≥ 1 converges to X(ω) for almost all ω ∈ Ω. At the same time, proofs of almost sure convergence are usually quite subtle.
There are rich connections of almost sure convergence with other classical modes of convergence, such as convergence in probability, defined by lim n → ∞ P( | X n − X | ≥ ε) = 0 for all ε>...
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Aaronson J, Burton RM, Dehling H, Gilat D, Hill T, Weiss B (1996) Strong laws for L- and U-statistics. Trans Am Math Soc 348:2845–2866
Baum LE, Katz M (1965) Convergence rates in the law of large numbers. Trans Am Math Soc 120:108–123
Birkhoff GD (1931) Proof of the Ergodic theorem. Proc Nat Acad Sci USA 17:656–660
Borel E (1909) Les probabilits dnombrables et leurs application arithmtique. Rendiconti Circolo Mat Palermo 27: 247—271
Brosamler G (1988) An almost everywhere central limit theorem. Math Proc Cambridge Philos Soc 104:561–574
Cantelli FP (1933) Sulla determinazione empirica della leggi di probabilita. Gior Ist Ital Attuari 4:421–424
Csörgo M, Révész P (1981) Strong approximations in probability and statistics. Academic, New York
Dehling H, Denker M, Philipp W (1985) Invariance principles for von Mises and U-Statistics. Z Wahrsch verw Geb 67: 139–167
Dehling H (1989) The functional law of the iterated logarithm for von-Mises functionals and multiple Wiener integrals. J Multiv Anal 28:177–189
Dehling H (1989) Complete convergence of triangular arrays and the law of the iterated logarithm for U-statistics. Stat Prob Lett 7:319–321
Doob JL (1953) Stochastic processes. Wiley, New York
Dudley RM (1968) Distances of probability measures and random variables. Ann Math Stat 39:1563–1572
Fisher A (1989) Convex invariant means and a pathwise central limit theorem. Adv Math 63:213–246
Glivenko VI (1933) Sulla determinazione empirica della leggi di probabilita. Gior Ist Ital Attuari 4:92–99
Hartmann P, Wintner A (1941) On the law of the iterated logarithm. Am J Math 63:169–176
Hoeffding W (1961) The strong law of large numbers for U-statistics. University of North Carolina, Institute of Statistics Mimeograph Series 302
Hsu PL, Robbins H (1947) Complete convergence and the law of large numbers. Proc Nat Acad Sci USA 33:25–31
Khintchin A (1924) ber einen Satz der Wahrscheinlichkeitsrechnung. Fund Math 6:9–20
Kingman JFC (1968) The ergodic theory of subadditive stochastic processes. J R Stat Soc B 30:499–510
Kolmogorov AN (1930) Sur la loi forte des grandes nombres. Comptes Rendus Acad Sci Paris 191:910–912
Komlos J, Major P, Tusnady G (1975) An approximation of partial sums of independent RVs and the sample DF I. Z Wahrsch verw Geb 32:111–131
Marcinkiewicz J, Zygmund A (1937) Sur les fonctions indpendantes. Fund Math 29:60–90
Schatte P (1988) On strong versions of the central limit theorem. Math Nachr 137:249–256
Sen PK (1972) Limiting behavior of regular functionals of empirical distributions for stationary mixing processes. Z Wahrsch verw Geb 25:71–82
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Skorohod AV (1956) Limit theorems for stochastic processes. Theory Prob Appl 1:261–290
Stout WF (1974) Almost sure convergence. Academic, New York
Strassen V (1964) An invariance principle for the law of the iterated logarithm. Z Wahrsch verw Geb 3:211–226
Van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge
Wichura MJ (1970) On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann Math Stat 41:284–291
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Dehling, H. (2011). Almost Sure Convergence of Random Variables. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_113
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