A note on the convergence rates in precise asymptotics

Abstract

Abstract Let {X,Xn,n≥1} $\{X, X_{n}, n\geq1\}$ be a sequence of i.i.d. random variables with EX=0 $EX=0$, EX2=σ2 $EX^{2}=\sigma^{2}$. Set Sn=∑k=1nXk $S_{n}=\sum_{k=1}^{n}X_{k}$ and let N ${\mathcal {N} }$ be the standard normal random variable. Let g(x) $g(x)$ be a positive and twice differentiable function on [n0,∞) $[n_{0},\infty)$ such that g(x)↗∞ $g(x)\nearrow\infty $, g′(x)↘0 $g'(x)\searrow0$ as x→∞ $x\to\infty$. In this short note, under some suitable conditions on both X and g(x) $g(x)$, we establish the following convergence rates in precise asymptotics limε↘0[∑n=n0∞g′(n)P{|Sn|σn≥εgs(n)}−ε−1/sE|N|1/s]=γ−η, $$ \lim_{\varepsilon\searrow0}\Biggl[ \sum_{n=n_{0}}^{\infty} g'(n)P\biggl\{ \frac{|S_{n}|}{\sigma\sqrt{n}}\geq \varepsilon g^{s}(n) \biggr\} -{\varepsilon}^{-1/s}E|{\mathcal {N}}|^{1/s}\Biggr]=\gamma -\eta, $$ where γ=limn→∞(∑k=n0ng′(k)−g(n)) $\gamma=\lim_{n\to\infty}(\sum_{k=n_{0}}^{n}g'(k)-g(n))$, η=∑n=n0∞g′(n)P{Sn=0} $\eta=\sum_{n=n_{0}}^{\infty}g'(n)P\{S_{n}=0\}$. It can describe the relations among the boundary function, weighting function, convergence rate and the limit value in studies of complete convergence. The result extends and generalizes the corresponding results of Gut and Steinebach (Ann. Univ. Sci. Budapest. Sect. Comput. 39:95–110, 2013), Kong (Lith. Math. J. 56(3):318–324, 2016), Kong and Dai (Stat. Probab. Lett. 119(10):295–300, 2016).

Keywords

• Convergence rates
• Precise asymptotics
• Complete convergence