# Minhaj Ul Arabia Pdf 212

## Minhaj Ul Arabia Pdf 212 [PDF] Minhaj Ul Arabia Pdf 212 [PDF] 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.