What key activities do you do to validate problem statements?

I recommend thinking about these questions:

1) Is this worth solving?

• What is the problem statement? Who are the users?
• Is this a real problem?
• What is the TAM? What can we influence? 
• What is the definition of success

2) Why us? Why now?

• Are you the right team/org/company to solve this problem?
• Should you work on it now? What happens if you do not?

3) How Might We break up the problem? What sub-problems should we go after(repeat steps 1-2) ? 

In general, the main thing I look for when validating any problem or solution is data. As a PM, it is important for you to figure out where to get the right data for what you are validating.

For example, if it's a problem statement around a user need or pain point, the first step would be to run user research panels to get data directly from users or potential users. I recommend you augment that with market research data such as independent analyst reports, research reports published by competitors of their user base, and related user data available to you such as broad user surveys.