Sums of Independent Random Variables

By Jiming Jiang. 2010

814 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/886a56ba01d4dd175272b04a6bbaf3acf04a3804

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Citations in the corpus

Spectral Radii of Large Non-Hermitian Random Matrices (2014, background): “By Theorem 1 on page 217 from [40], for any sequence of positive numbers δn such that δn = o((nkn)), P(W j > x) = (1 + o(1))(1 − Φ(x)) (2.”
Robust Graph Convolutional Networks Against Adversarial Attacks (2019, background): “The proof can be found in probability textbooks such as [22] and is omitted for brevity.”
The Brownian map is the scaling limit of uniform random plane quadrangulations (2011, background): “At this point, we will need the following consequence of the local limit theorem of [27], stating that for every a, b ∈ Z and p ≥ 1, there exists a finite constant C > 0 depending only on p such that for every r ≥ 1, √ rqr(a, b) ≤ C 1 + ∣∣∣ b−a r ∣∣∣p .”
Scaling limits of random planar maps with large faces (2011, neutral): “centered random variables [26], Theorem 2.”
Berry–Esseen bounds in the entropic central limit theorem (2011, background): “More precisely (see [19]), the closeness of the distribution function Fn(x) = P{Sn ≤ x} to the standard normal distribution function (x) = 1 √ 2π x ∫”
THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION (2008, neutral): “5) for 0 < x ≤ 1/(18cn), where |θn,x| ≤ 36 and γ(x) is the Cramér–Petrov series (see Petrov [18]) satisfying |γ(x)| ≤ 2x3τns−3 n .”
The circular law for random matrices (2007, positive): “xkεkXk. We use the following inequality for a concentration function of a sum of independent random variables p y(Mbl0) ≤ CMbl0 X k∈∆l 0 λ2 kD(ξgkεk;λk) −1 2 (4.98) with λk ≤ Mbl0. See Petrov [22], p.43, Theorem 3. Put λk = M|xk|. It is straightforward to check that, X k∈∆l 0 λ2 kD(ξgε ;λ ) ≥ pn”
The extremogram: a correlogram for extreme events (2009, methodology): “We apply the central limit theorem for the triangular array of iid mean zero random variables e Sn(Ini), i = 1, . . . , kn. By Gnedenko and Kolmogorov [17], Theorem 3 on p. 101, or Theorem 4.1 in Petrov [ 28 ] and since (6.4) holds, one needs to verify the following condition for any ǫ > 0:”
Limit theorems for nearly unstable Hawkes processes (2013, neutral): “Also, from [38] (Page 214, point 5), we get that if ‖φ‖∞ < ∞ and ∫ +∞ 0 |s| 3φ(s)ds < +∞, then Assumption 2 follows.”
From Stein identities to moderate deviations (2009, neutral): “, Petrov (1975), Chapter 8] based on the conjugated method.”
Consistencies and rates of convergence of jump-penalized least squares estimators (2009, neutral): “random variables for the error we cannot use results as in Shao (1995) to obtain an almost sure bound for this process [cf.”
Non-asymptotic theory of random matrices: extreme singular values (2010, background): “| | − Better estimates would indeed emerge in the form of exponential deviation inequalities [61, 47], but this would only happen when we drop explicit comparisons to the limiting distribution and study the tails of S n by themselves.”