As per the definition of an application exception:
The Bean Provider defines application exceptions. Application exceptions that are checked exceptions may be defined as such by being listed in the throws clauses of the methods of the bean�s business interface, home interface, component interface, and web service endpoint. An application exception that is an unchecked exception is defined as an application exception by annotating it with the ApplicationException metadata annotation, or denoting it in the deployment descriptor with the application-exception element.
So you will have to declare IllegalArgumentException in throws clause of the method or defined it as an application exception through the annotation or deployment descriptor.
The latter part of the statement confuses me. So we can have a RuntimeException (which is termed as "unchecked") as an application exception? If this is possible, then i think declaring this IllegalArgumentException (which is an RuntimeException) in the throws clause should have behaved as an application exception.
I am a bit confused right now, but let me do more reading to see what's expected