Notifications are sent on events.
If no price change has been made then a notification will be sent when the user changes their intent to renew ('DID_CHANGE_RENEWAL_STATUS') or when the subscription is actually renewed ('INTERACTIVE_RENEWAL').
If the user's credit card is no longer valid Apple tries to get the user to update the information prior to the autorenewal date. If that doesn't work Apple may try again after the renewal date. I do not know if a 'did_change_renewal_status is sent out, but no 'INTERACTIVE_RENEWAL' is sent out.
If the price has changed such that the user must confirm a new price then Apple asks the user to confirm the new price shortly before the subscription expires and again one day after the subscription expires. If the user does not confirm approval I do not know if a 'did_change_renewal_status is sent out, but no 'INTERACTIVE_RENEWAL' is sent out. If the user confirms the price change then an 'INTERACTIVE_RENEWAL' is sent out.
If the user ends autorenewal either by 1) opting out or 2) failing to respond to a price change or 3) their credit card info fails during a renewal and then the user later decides to turn the subscription back on then you get a 'RENEWAL' notification.