Oh gotcha. For a merchant's customers, once a payment in a subscription is approved, if the merchant has configured the appropriate settings in the Merchant Interface, the customer would receive an email receipt. If the payment declines, no customer notification is sent. At that point, the merchant needs to contact the customer to let them know about the decline.
Other than that, there are no other emails that get sent to the customer from Authorize.Net for ARB transactions.
To check to see if an account is configured to send customer emails for ARB transactions, log into the Merchant Interface, click Account from the main toolbar, then Email Receipt under the Transaction Format Settings section. If the box next to Email customer for each successfully authorized ARB transaction is checked, the customer will receive an email after each approved payment is submitted.
Thanks,
Michelle