On the Probabilities of Large Deviations for the Maximum of Sums of Independent Random Variables

Aleshkyavichene, A. K.

doi:10.1137/1124002

Cited by 9 publications

(5 citation statements)

References 7 publications

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“…uniformly for x ∈ [0, o(n 1/6 )). This contrasts with the moderate deviation result for the maximum of partial sums of Aleshkyavichene [1,2], where a finite moment-generating condition is required. However, in view of the result given in (1.1), it is natural to ask whether a finite third moment suffices for (1.2).…”

Section: Introduction and Main Resultscontrasting

confidence: 64%

Self-normalized Cramér type moderate deviations for the maximum of sums

Liu¹,

Shao

Wang

2013

Bernoulli

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Let X1, X2, . . . be independent random variables with zero means and finite variances, and let Sn = n i=1 Xi and V 2 n = n i=1 X 2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max 1≤k≤n S k /Vn is obtained. In particular, for identically distributed X1, X2, . . . , it is proved that P(max 1≤k≤n S k ≥ xVn)/(1 − Φ(x)) → 2 uniformly for 0 < x ≤ o(n 1/6 ) under the optimal finite third moment of X1.

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Section: Introduction and Main Resultscontrasting

confidence: 64%

Self-normalized Cramér type moderate deviations for the maximum of sums

Liu¹,

Shao

Wang

2013

Bernoulli

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“…In [20] this problem was considered in the Cramér zone. Let us briefly discuss the main results of this article.…”

Section: Relation (13) Means Thatmentioning

confidence: 99%

Integro-local and integral theorems for sums of random variables with semiexponential distributions

Боровков

Mogulskii

2006

Sib Math J

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We obtain some integro-local and integral limit theorems for the sums S(n) = ξ(1) + · · · + ξ(n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form P(ξ ≥ t) = e −t β L(t) , where β ∈ (0, 1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x → ∞ of the probabilities P(S(n) ∈ [x, x + Δ)) and P(S(n) ≥ x) in the zone of normal deviations and all zones of large deviations of x: in the Cramér and intermediate zones, and also in the "extreme" zone where the distribution of S(n) is approximated by that of the maximal summand.Keywords: semiexponential distribution, integro-local theorem, Cramér series, segment of the Cramér series, random walk, large deviations, Cramér zone of deviations, intermediate zone of deviations, zone of approximation by the maximal summand § 1. Main Notations. Statement of the Problem Consider independent random variables ξ, ξ(1), ξ(2), . . . with general distribution F(B) = P(ξ ∈ B), mean a := Eξ, variance b 2 := Dξ, and characteristic function f(t) = Ee itξ , t ∈ R.Definition 1.1. Say that the distribution F of a random variable ξ belongs to the class Se of semiexponential distributions if its right tail F + (t) := P(ξ ≥ t) has the formβ ∈ [0, 1], and L(t) is a slowly varying function as t → ∞; if β = 1 then L(t) = o(1). It is assumed thatWithout loss of generality we may consider only two types of distributions (or random variables):[Z] The arithmetic distributions: Now P(ξ ∈ Z) = 1 and for some y 0 ∈ Z such that P(ξ = y 0 ) > 0 the greatest common divisor of all possible values of ξ − y 0 is equal to 1; here Z is the set of integers.For the arithmetic random variables we clearly have f(2πt) = 1 for each integer t.

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“…[43] actually established more general results which improved those by Hu, Shao and Wang (2009) [36]. It should be mentioned that Theorem 6.4 is comparable to the large deviation result for the maximum of partial sum given in [1]. However the latter requires a finite exponential moment condition.…”

Section: Darling-erdós Type Theorem and Maximum Of Self-normalized Summentioning

confidence: 77%

“…Theorem 2.2. S n /V n converge weakly to a random variable Z such that (a) P (|Z| = 1) < 1 if and only if (1) X is in the domain of attraction of a stable law with α ∈ (0, 2];…”

Section: The Central Limit Theorem and Invariance Principlementioning

confidence: 99%

Self-normalized limit theorems: A survey

Shao¹,

Wang²

2013

Probab. Surveys

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Let X 1 , X 2 , . . . , be independent random variables with EX i = 0 and write Sn =This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum Sn/Vn. Other self-normalized limit theorems are also briefly discussed.

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On the Probabilities of Large Deviations for the Maximum of Sums of Independent Random Variables

Cited by 9 publications

References 7 publications

Self-normalized Cramér type moderate deviations for the maximum of sums

Self-normalized Cramér type moderate deviations for the maximum of sums

Integro-local and integral theorems for sums of random variables with semiexponential distributions

Self-normalized limit theorems: A survey

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