An Asymptotic Representation of the Sample Distribution Function
An Asymptotic Representation of the Sample Distribution Function
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样本分布函数的渐近表示
DOI：
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发表时间：
1969
期刊：
影响因子：
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通讯作者：
D. Brillinger
中科院分区：
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作者：
D. Brillinger
设X_1，···，X_n是[0，L]上均匀分布的独立观测值。设Fn(X)~xj^x的比例。我们将证明定理。存在随机函数{Gn(X)；Orgarrgl}，其分布与{Fn(X)；0^xg1}相同，且存在布朗运动W，使得对布朗函数B(X)=n~W(Nx)sup|ni*[Gn(X)x[B(X)xB(L)](1)°M L，几乎必然为n-*<；*>；该定理对于研究{Fn(X)；0^#^L}的泛函，特别是依赖于？的泛函的渐近行为是有用的。2.构造Gn(X)，设Yu F2，··是均值为1的独立指数变量，设S(K)=Fi+···+F*tk=1，2，···，设5(0)=0。设Gn(X)？k/nHs(K)/S(n+1)^x<；S(k+1)/S(n+1)。对于每个n，这个{Gn(X)；0̂x^1}的分布与{Fn(X)；0 GB x s1}相同。我们现在记录一系列引理。引理1.存在布朗运动W使得
BY DAVID R. BRILLINGER Communicated by David Blackwell, January 10, 1969 1. Let Xi, • • • , Xn be independent observations from the uniform distribution on [0, l ] . Let Fn(x)~the proportion of the Xj^x. We will prove THEOREM. There is a random function {Gn(x); Orgarrgl}, with the same distribution as {Fn(x) ; 0 ^ x g 1} for each n, and there is a Brownian motion W, such that for the Brownian B(x) =n~W(nx) sup | ni*[Gn(x) x [B(x) xB(l)] (1) ° M l Ofri-^Oog w)flog log n)<*] almost surely as n—*<*>. This theorem is of use in the investigation of the asymptotic behavior of functionals of {Fn(x); 0 ^ # ^ l } , especially functionals dependent on ». 2. We construct Gn(x) as follows; let Yu F2, • • • be independent exponential variables with mean 1. Let S(k) = Fi + • • • +F* t k = 1, 2, • • • and let 5(0) =0. Set Gn(x) « k/n HS(k)/S(n + 1) ^ x < S(k + 1)/S(n + 1). This {Gn(x) ; 0 ̂ x ^ 1} has the same distribution as {Fn(x) ; 0 £x S1} for each n. We now record a series of lemmas. LEMMA 1. There is a Brownian motion W such that