On Large Deviations of Sums of Independent Random Variables

By Zhishui Hu, V. V. Petrov, John Robinson. 2007

257 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d3a2465ce78c467ea228ccbcc1ccc813ace8b21

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Citations in the corpus

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model (2007, methodology): “As mentioned in the proof of Lemma 4.5 of Tang (2005), using a result of Nagaev (1979) we may prove that for an arbitrarily fixed number v > β, there is some cv > 0 such that for all n = 1, 2, . . . and all x ≥ 0, Pr ( n∑ k=1 Xk > x ) ≤ cvnvF (x).”
On multidimensional branching random walks in random environment (2005, background): “11 from [22] with x = ε √ n 2 , y = n1/4, noting also that, since the number of terms here is √ n, the quantity B2 n of [22] is O( √ n)) to obtain that”
Insensitivity to Negative Dependence of the Asymptotic Behavior of Precise Large Deviations (2006, background): “For classical works of precise large deviations with heavy tails, we refer the reader to Heyde (1967a, 1967b, 1968), A.V. Nagaev (1969a, 1969b, 1969c), S.V. Nagaev (1973, 1979), Pinelis (1985), while for recent works, we refer to Cline and Hsing (1991), Rozovskĭı (1993), Vinogradov (1994), Mikosch ”
Random walks in cones (2011, background): “There are no proofs in [39] and we refer the interested reader to the original paper [23].”
Conditional limit theorems for ordered random walks (2009, unknown): “11 in [13], we have P(Mn > n , An) ≤ exp{−Cn}.”
Asymptotics of randomly stopped sums in the presence of heavy tails (2008, background): “For Weibull-type distributions, the situation is more complicated; see, for example, [28,31,37,38].”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, methodology): “By the Nagaev inequality [57] applied to the split sample in Remark 5, one can show that”
THE FINITE-TIME RUIN PROBABILITY OF THE COMPOUND POISSON MODEL WITH CONSTANT INTEREST FORCE (2005, methodology): “…it is clear that, for each v > 0, We first state the following probabilistic inequality, which can easily be verified using Theorem 1.1 of Nagaev (1979): for a sequence of i.i.d. random variables {X k } ∞ k=1 , with common distribution B and finite moment of order κ ∈ (0, 1), the…”
Performance bounds for parameter estimates of high-dimensional linear models with correlated errors (2016, background): “The latter are generalizations of the classical Nagaev inequality [30] that deals with independent random variables.”
Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models (2007, background): “Nagaev (1973), (1979) studied the large deviation probabilities (1.1) for the claims with regularly varying tails.”
GAUSSIAN APPROXIMATIONS FOR NON-STATIONARY MULTIPLE TIME SERIES (2011, unknown): “8 in Nagaev (1979), P(|SA| ≥ u 2 ) ≤ C4,p m`p/2−1 up f 0 + 2 exp ( − C5,pu 2 mf2 0 ) .”
Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations (2005, result): “Classical results (see for example A.V. Nagaev [26], S.V. Nagaev [27], Cline and Hsing [7]; cf. the surveys in Section 8.6 in [14] and Mikosch and Nagaev [21]) say that for iid (Y t ) and thresholds x n → ∞, the relation P (S n > x n ) ∼ n P (Y > x n ) (1.6) ∼ P (max(Y 1 , . . . , Y n ) > x n ) hold”
Local large deviations and the strong renewal theorem (2016, unknown): “A(x) k . (1.5) Moreover, for some C′ 0 = C0′(J) <∞, P(Sn∈ x+J) ≤ C′ 0 1 an n A(x) . (1.6) The non-local version of (1.4), where Sn ∈ x+ Jis replaced by Sn ≥ x, is known as a Fuk-Nagaev inequality [Nag79]. This is the starting point of our proof of Theorem 1.1, see Section 3. We prove (1.4) throug”
Probability and moment inequalities under dependence (2013, methodology): “In this paper we establish Nagaev and Rosenthal-type inequalities for dependent random variables under easily verifiable dependence conditions. We assume that (Xn) is a stationary causal process of the form Xi = g(· · · , εi−1, εi), (1.5) where εi, i ∈ Z, are i.i.d. random variables. We adopt the fu”
Gaussian Approximation for High Dimensional Time Series (2015, methodology): “By Nagaev (1979), we have the expansion: for y ≥ √ n , Let M n = X 11 + . . . + X n 1 , Z = ( Z 1 , . . . , Z p ) (cid:62) ∼ N (0 , Id p ) and assume Then the Gaussian approximation (7) does not hold .”
On the Bennett–Hoeffding inequality (2009, result): “4) Attempts at refining the Bennett-Hoeffding inequality by taking moments higher than the second ones into consideration were made in [25, 24, 32, 59]; however, in contrast with the Bennett-Hoeffding bounds, the bounds given in [25, 24, 32, 59] were not the best possible in their own terms.”
Conditional distribution of heavy tailed random variables on large deviations of their sum (2009, background): “Interested readers can find reviews on the topic in [13, 15].”