Sharp large deviations for sums of bounded from above random variables

Abstract

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramér type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao [Filomat 2011; 25(2): 197-206].

Cramér moderate deviations for a supercritical Galton-Watson process

“…The proof of Theorem 2.3 is simialr to the proof of Theorem 2.2. However, instead of using Lemma 6.3, we should make use of the following lemma of Fan [7]. The proof of Theorem 2.4 is analogous to the proof of Theorem 2.2.…”

“…After the seminal work of Cramér, a number of Cramér moderate deviations have been established for various settings. See, for instance, Linnik [10] and [7] for independent random variables, Fan, Grama and Liu [4] for martingales, Grama, Liu and Miqueu [8] and Fan, Hu and Liu [5] for a supercritical branching process in a random environment, and Beknazaryan, Sang and Xiao [2] for random fields. In this paper, we are going to establish Cramér moderate deviations for a supercritical Galton-Watson process.…”

Cramér Moderate Deviations for a Supercritical Galton–Watson Process with Immigration

“…Fleischmann and Wachtel [4] explored large deviations for sums of independent and identically distributed random variables for the Galoton-Watson process. Other relevant studies from the literature can be found in [5][6][7][8][9][10][11].…”

Self-normalized Cramér type moderate deviations for stationary sequences and applications

“…Indeed, Linnik [25] proved that for α ∈ (0, 1 6 ], formula (1.2) holds uniformly for 0 ≤ x = o(n α ) as n → ∞ if and only if Ee |X 1 | 4α/(2α+1) < ∞. Following the seminal work of Cramér, various moderate deviation expansions for standardized sums have been obtained by many authors, see, for instance, Petrov [28], Saulis and Statulevičius [36] and [15]. See also Račkauskas [29,30], Grama [19], Grama and Haeusler [20] and [14] for martingales, and Wu and Zhao [38] and Cuny and Merlevède [9] for stationary processes.…”