Geodesic
On the strong law of large numbers for random quadratic forms
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 125-142

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The paper establishes strong laws of large numbers for the quadratic forms $Q_n(X,X)=\sum_{i=1}^n\sum_{j=1}^na_{ij}X_iX_j$ and the bilinear forms $Q_n(X,Y)=\sum_{i=1}^n\sum_{j=1}^na_{ij}X_iY_j$, where $X=(X_n)$ is a sequence of independent random variables and $Y=(Y_n)$ is an independent copy of it. In the case of i.i.d. symmetric $p$-stable random variables $X_n$ we derive necessary and sufficient conditions for the strong laws of $Q_n(X,X)$ and $Q_n(X,Y)$ for a given nondecreasing sequence $(b_n)$ of normalizing constants. For these classes of variables $(X_n)$ the strong laws $\lim b_n^{-1}Q_n(X,X)=0$ a.s. and $\lim b_n^{-1}Q_n(X,Y)=0$ a.s. are shown to be equivalent provided that $a_{ii}=0$ for all $i$.
Keywords: quadratic forms, bilinear forms, strong law of large numbers, Prokhorov-type characterization, tail probabilities.
Mots-clés : p-stable random variables, domains of partial attraction 
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     author = {T. Mikosch},
     title = {On the strong law of large numbers for random quadratic forms},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {125--142},
     publisher = {mathdoc},
     volume = {40},
     number = {1},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a6/}
}
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T. Mikosch. On the strong law of large numbers for random quadratic forms. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 125-142. http://geodesic.mathdoc.fr/item/TVP_1995_40_1_a6/