When to Use a Period

Use a period if the question is actually disguised as a statement that is subtly trying to have a question answered. Like this:

I wonder if I can sit here. – this is asking a question subtly, but it’s being used as a statement, so a period is used instead of a question mark.

Expressing Doubt

You can use an indirect question to express doubt, or you’re uncertain about something. It can also function as an indirect question masked as a statement. Like this:

I don’t know if she went shopping – this functions as an indirect question by expressing doubt.