Self-normalized Cramér moderate deviations for a supercritical Galton–Watson process
نویسندگان
چکیده
Abstract Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds estimator. The results are believed to optimal or near-optimal.
منابع مشابه
Self-normalized Moderate Deviations and Lils
Let fXn;n 1g be i.i.d. R d-valued random variables. We prove Partial Moderate Deviation Principles for self-normalized partial sums subject to minimal moment assumptions. Applications to the self-normalized law of the iterated logarithm are also discussed.
متن کاملProcess Level Moderate Deviations for Stabilizing Functionals
Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which is a level-3 result for empirical point fields as well as a level2 result for empirical point measures. The level-3 rate function coincides with the so-called specific information. We s...
متن کاملCramér Type Moderate deviations for the Maximum of Self-normalized Sums
Let {X ,X i , i ≥ 1} be i.i.d. random variables, Sk be the partial sum and V 2 n = ∑n i=1 X 2 i . Assume that E(X ) = 0 and E(X )<∞. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max1≤k≤n Sk ≥ x Vn)/(1−Φ(x))→ 2 uniformly in x ∈ [0, o(n)).
متن کاملSelf-normalized Cramér type Moderate Deviations for the Maximum of Sums
Let X1, X2, . . . be independent random variables with zero means and finite variances, and let Sn = ∑n i=1Xi and V 2 n = ∑n i=1X 2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1, X2, · · · , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1 − Φ(x)) → 2 uniformly for 0 < x ≤ o(n1/6) under the opt...
متن کاملModerate Deviations for a Diffusion Type Process in Random Environment
Let σ(u), u ∈ R be an ergodic stationary Markov chain, taking a finite number of values a1, . . . , am, and b(u) = g(σ(u)), where g is a bounded and measurable function. We consider the diffusion type process dX t = b(X ε t /ε)dt+ ε σ `
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Applied Probability
سال: 2023
ISSN: ['1475-6072', '0021-9002']
DOI: https://doi.org/10.1017/jpr.2022.134