Geodesic
Weak Convergence of a Certain Functional
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 779-784 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the functional $T_n=(S_1^2+\dots+S_n^2)/(nV_n^2)$ derived from a sequence $\{X_n\}_{n\ge 1}$ of independent identically distributed random variables, where $S_k=X_1+\dots+X_k$, $V_n^2=X_1^2+\dots+X_n^2$. Let $G$ be the distribution function of the random variable $\int_{0}^{1}W^2(t)\,dt$, where $W(t)$, $t\in [0,1]$, is a Wiener process. We show that the distribution function $T_n$ weakly converges to $G$ as $n\to\infty$ if and only if the distribution function of the random variable $X_1$ belongs to the attraction domain of the normal law and $\mathbf{E}X_1=0$.
Keywords: weak convergence, convergence in probability, random variable, distribution function. 
@article{TVP_2001_46_4_a9,
     author = {V. M. Kruglov and G. N. Petrovskaya},
     title = {Weak {Convergence} of a {Certain} {Functional}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {779--784},
     year = {2001},
     volume = {46},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a9/}
}
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V. M. Kruglov; G. N. Petrovskaya. Weak Convergence of a Certain Functional. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 779-784. http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a9/