Geodesic
On the rate of convergence in the conditional invariance principle
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 1, pp. 67-79

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Let $S_n(t)$, $0\le t\le 1$ be a random broken line and $w(t)$ be a standard Wiener process. In this paper, the estimate $O(\log n/\sqrt n)$ is obtained for the distance between the distributions, in the space $C[0,1]$, of the process $S_n(t)$ with the condition $S_n(1)\in(a-\varepsilon,a+\varepsilon)$ and of $w(t)$ with the condition $w(1)\in(a-\varepsilon,a+\varepsilon)$. 
@article{TVP_1978_23_1_a4,
     author = {I. S. Borisov},
     title = {On the rate of convergence in the conditional invariance principle},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {67--79},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_1_a4/}
}
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I. S. Borisov. On the rate of convergence in the conditional invariance principle. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 1, pp. 67-79. http://geodesic.mathdoc.fr/item/TVP_1978_23_1_a4/