Let $\{x_{ij}\;|\;i,j=1,2,\ldots\}$ be a double array of i. i. d. random variables and let $\alpha>\frac{1}{2}$, $\beta\ge 0$ and $M>0$ be constants. Then, \begin{align*} \max_{i\le Mn^\beta}\left|\sum_{j=1}^n\frac{x_{ij}-m}{n^\alpha}\right|\to 0\ a.s.\ (n\to\infty) \iff E|x_{11}|^{\frac{1+\beta}{\alpha}}< \infty\ \& \ m=\begin{cases}\displaystyle E x_{11},&(\alpha\le 1),\\ \displaystyle\text{any},&(\alpha>1). \end{cases} \end{align*}
Suppose that $n,p\to\infty$ with $p/n\to c>0$. I am confuse that how to derive law large number from i. i. d $x_j,x_{ij}\sim N_p(0,1)$. I mean I want to derive \begin{align*} \min_{i\le p}\sum_{j=1}^n\frac{x_{ij}^2}{n}\to 1\; a.s.\;\text{and}\;\min_{i\le p}\sum_{j=1}^n\frac{x_jx_{ij}}{n}\to 0\; a.s. \end{align*}
What is $M$ to derive the above convergence?
Thank you in advance.