獨(dú)立情形下一階矩收斂的精確漸近性的注記

孫曉祥,楊麗娟

(吉林農(nóng)業(yè)科技學(xué)院 文理學(xué)院,吉林 吉林 132101)

目前,關(guān)于隨機(jī)變量列重對(duì)數(shù)律、完全收斂及矩重對(duì)數(shù)律的精確漸近性質(zhì)已有許多研究結(jié)果[1-8].文獻(xiàn)[9]給出了關(guān)于獨(dú)立和NA列部分和精確漸近性的一般形式,揭示了擬權(quán)函數(shù)、邊界函數(shù)、收斂速度和極限狀態(tài)間的密切聯(lián)系.

本文若無(wú)特別說(shuō)明,均以C表示不同的正常數(shù),N表示標(biāo)準(zhǔn)正態(tài)隨機(jī)變量.

定理1[10]對(duì)于任意的d>0和β>0,

成立的充要條件是

EX=0,EX2=σ2.

(1)

本文對(duì)于普通的擬權(quán)函數(shù)和邊界函數(shù),將上述結(jié)果推廣到如下更一般的形式.

定理2設(shè)0

1)g(x)↑∞,x→∞;

則

(2)

成立的充要條件是式(1)成立.

注1由于在上述級(jí)數(shù)中增加或減少有限項(xiàng)不改變結(jié)果,故為簡(jiǎn)便,以下一律從n=1開(kāi)始記,并假設(shè)g(x)和g′(x)(或ψ(x)等)在[1,∞)上有定義并單調(diào),不影響g(x)的普遍性.

注2滿足定理2條件的g(x)有很多種,如xα,logβx,loglogγx,α>0,β>0,γ>0等,但指數(shù)函數(shù)不在此列.

注3在定理2中,令g(x)=(loglogx)(2β+d)/2,s=d/(2β+d)(其中:β>0;d>0),即可得到文獻(xiàn)[10]的定理1,從而推廣了已有的結(jié)果.

令a(ε)=g-1(Mε-1/s),M為任意正實(shí)數(shù),g-1(x)為g(x)的反函數(shù).

要證明式(1)?式(2),此時(shí)不妨假設(shè)σ=1.

命題1在定理2的條件下,有

證明: 由注1知,可以在[0,1)中對(duì)g(x)作適當(dāng)?shù)难a(bǔ)充定義,使g′(x)保持單調(diào)性.做變量代換t=εgs(y),得

由定理2中條件1)和2),有

事實(shí)上,當(dāng)ψ(x)單調(diào)非增時(shí),式(3)顯然成立;當(dāng)ψ(x)單調(diào)非降時(shí),由定理2中條件2)知,對(duì)?δ>0,存在正整數(shù)κ,使得當(dāng)x>κ時(shí),總有ψ(x)≤(1+δ)ψ(x-1),從而

令δ↓0,即得式(3)中右邊不等式.同理可得式(3)左邊不等式.命題1證畢.

命題2在定理2的條件下,有

(4)

由Δn的定義及Markov不等式知

證明: 注意到定理2中條件2),類似命題1,可得

命題4在定理2的條件下,有

(5)

證明: 由引理1,可得

其中T>1/(2s).

首先考慮K2,注意到01/s>1,

其次考慮K1,為方便不妨假設(shè)T=1,

證畢.

下面證明定理2.由命題1～命題4及三角不等式可證得式(1)?式(2).下面證明式(2)?式(1).

首先證明EX=0.由式(2),對(duì)于任意的ε>0,有

因此EY2≤4σ2.從而令K→∞,有EX2<∞.

最后,同理可證

與式(2)相比較,可得EX2=σ2.

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