$A$ is a doubly stochastic matrix, how about $A^TA$ I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix.
However, the paper says $A^TA$ is symmetric and stochastic. 


*

*Since $A^TA$ is symmetric, if $A^TA$ is stochastic, it must be doubly stochastic.  

*The $(j,j)$ entry of $A^TA$ is exactly $a_{ij}^Ta_{ij}$ with $i=1,\ldots,n$.  
Then is there any other condition I can use to prove this claim is true or false? 

Just prove it from the definition: You have that $[A^TA]_{ij}=\sum_{k}a_{ki}a_{kj}$. To show it's (column) stochastic you need the column-sum to be 1:
$$\sum_{j=1}^n [A^TA]_{ij}=\sum_j\sum_{k}a_{ki}a_{kj}=\sum_k\sum_ja_{ki}a_{kj}=\sum_k a_{kj}\cdot 1=1.$$
