Canceling your Periscope account is not easy. If you no longer want to use Periscope the best thing to do is to tap the user icon in the bottom right hand corner, then tap the Profile icon in the top right hand corner. From here scroll down and tap Logout. You can then delete the app from your device and never use it again. If logging out is not enough you can have your account deleted by contacting Periscope. Here's how to delete your Periscope account, according to Twitter:
"To disable your Periscope account, please email email@example.com with subject line "Delete Account". Please include your Periscope username, along with the email address or and the phone number associated with your account."
"Please keep in mind that it may take us some time to delete your account.""Please note: You can’t sign up with the same username, phone number or Twitter account once your account is deleted."
If you signed up with your Twitter account and no longer want your Twitter handle associated with Periscope you can edit your account from your Profile page. Just tap the Profile icon from the People tab and select "Edit". From here you can change your user name, @name and add a short description about yourself.