# Studying the maxima of columns of a random matrix as a point process

Consider a matrix, $S$, of i.i.d. real RVs : $X_{ij}$ for $1 \leq i \leq s$, $1 \leq j \leq n$. Let $F$ denote the distribution of $X_{ij}$. For $1 \leq j \leq n$, consider $Y_{j}^{(1)} = \max_{i} \{X_{ij}\}$ ($Y_{j}^{(1)}$ is the maximum of the $j$th column).

1. Is there a (general) theory to study $\{Y_{j}^{(1)}\}_{j=1}^{n}$ as points of a point process (on $R$) as $n \rightarrow \infty$?

Now, consider the matrix obtained by deleting, from $S$, the row that contains $\max_{j}Y_{j}^{(1)}$, i.e., if $n^{*} = \arg \max_{j} Y_{j}^{(1)}$ and $s^{*} = \arg \max_{i} X_{in^{*}}$, remove from $S$ the RVs: $\{X_{s^{*}j}\}_{j=1}^{n}$ (i.e., the row $s^{*}$). Call the new matrix $S^{\prime}$. Call $Y_{j}^{(2)} = \max_{i} \{X_{ij}:X_{ij} \in S^{\prime} \}$ (again, the maximum of the $j$th column of the updated matrix $S^{\prime}$).

1. Is there any theory to study $\{Y_{j}^{(2)}\}_{j=1}^{n}$ as a point process on $R$ as $n \rightarrow \infty$? Or even, $\{Y_{j}^{(r)}\}_{j=1}^{n}$, ($1 \leq r \leq s$), in general ?
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It seems the random variables $Y^{(1)}_j$ are i.i.d. with distribution $\mathrm P(Y^{(1)}_j\leqslant y)=[F(y)]^s$. – Did Oct 20 '11 at 5:42
Yes, $Y_{j}^{(1)}$ are indeed i.i.d. with distribution $(F)^{s}$. – A Karthik Oct 20 '11 at 9:30
Then a general theory based on point processes is not needed to study them, is it? – Did Oct 20 '11 at 9:41