An approximation of partial sums of independent RV'-s, and the sample DF. I

@article{Komlos1975AnAO,
  title={An approximation of partial sums of independent RV'-s, and the sample DF. I},
  author={John Komlos and Péter Major and G{\'a}bor E. Tusn{\'a}dy},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete},
  year={1975},
  volume={32},
  pages={111-131}
}
SummaryLet Sn=X1+X2+⋯+Xnbe the sum of i.i.d.r.v.-s, EX1=0, EX12=1, and let Tn= Y1+Y2+⋯+Ynbe the sum of independent standard normal variables. Strassen proved in [14] that if X1 has a finite fourth moment, then there are appropriate versions of Snand Tn(which, of course, are far from being independent) such that ¦Sn -Tn¦=O(n1/4(log n)1/1(log log n)1/4) with probability one. A theorem of Bártfai [1] indicates that even if X1 has a finite moment generating function, the best possible bound for any… 
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References

SHOWING 1-10 OF 21 REFERENCES
An Asymptotic Representation of the Sample Distribution Function
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
Skorohod embedding of multivariate RV's, and the sample DF
The main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's
On the deviations in the Skorokhod-Strassen approximation scheme
SummaryIn deriving his strong invariance principles, Strassen used a construction of Skorokhod: if the univariate d. f. F has first, second, and fourth moments 0, 1, and Β<t8, respectively, then
Local Limit Theorems for Large Deviations
Let $(X_j ),j = 1,2, \cdots $, be a sequence of independent random variables with the distribution functions $V_j (x)$. We assume the existence of ${\bf D}X_j = \sigma _j^2 ,s_n^2 = \sum\nolimits_{j
A Local Theorem for Densities of Sums of Independent Random Variables
Let $X_1 ,X_2 , \cdots $ be a sequence of independent random variables having mean values 0 and dispersions $\sigma _1^2 ,\sigma _2^2 , \cdots $ and let $p_n (x)$ be the density of the distribution
A new method to prove strassen type laws of invariance principle. 1
SummaryA new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that $${{\mathop {\lim }\limits_{n
Weak convergence and embedding
Studies In The Theory Of Random Processes
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