In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property
$$
P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right],
$$
where $a,t\ge 0$. We noticed some interesting properties that follow, and wonder if the above property is known in the community. Notice that in the case of equality, this is equivalent to the memoryless property.



The weak version of the memoriless property
By googling 



> 
> "decreasing conditional survival" function
> 
> 
> 


we find this term used e.g. in [this paper](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwiMwLPCrcjhAhVLs6wKHVpiAiAQFjAAegQIABAB&url=http%3A%2F%2Fshodhganga.inflibnet.ac.in%2Fbitstream%2F10603%2F41218%2F5%2Fchapter%25201.pdf&usg=AOvVaw2lx6iRTEXtOmXrDDlp3W9F) and a few others.


