Minhaj Ul Arabia Pdf 212
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Minhaj Ul Arabia Pdf 212
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Minhaj Ul Arabia Pdf 212. Minhaj-ul-Quran al-Islam..) \geq 1$, $$\label{eq:tight_FiniteDim}
\begin{aligned}
&2^{d(\alpha-1)}(1-2^{ -2d})^{ -(\alpha+2)} \sum_{n=1}^\infty n^{ -\alpha-2} \\
&\leq \sum_{\substack{(i_1,\ldots,i_{2d})\in {\mathbb{Z}}^{2d} \\ i_1,\ldots,i_{2d}\geq 0}} n^{ -2} \cdot \prod_{k=1}^d \frac{1}{(1-2^{ -2})^{i_k}(1-2^{ -(i_k+1)})^{ -\alpha-1}} \\
&\leq 2^{d(\alpha-1)}(1-2^{ -2d})^{ -(\alpha+2)}.
\end{aligned}$$
This shows the claim in the case where the cardinality of $\mathcal{I}$ is finite. If $\mathcal{I}$ is infinite, we can use the Kolmogorov-Smirnov test on the $\{\mathbf{x}_i\}_{i\in \mathcal{I}}$ to approximate the CDF of the distribution of the $L_2$-norm of the $\{\mathbf{x}_i\}_{i\in \mathcal{I}}$: The statistic of the Kolmogorov-Smirnov test between the empirical CDF of the $L_2$-norm of the $\{\mathbf{x}_i\}_{i\in \mathcal{I}}$ and the CDF of the distribution of the $L_2$-norm of the $\{\mathbf{x}_i\}_{i\in \mathcal{I}}$ given by is asymptotically distributed as a $\chi^2$ random variable. Hence, we can apply the similar method as in the proof of [@Massey1990] (Proposition 3.
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