If a cron job produces any output, cron sends it to you via the local email facility. If a cron job returns a nonzero (i.e. error) status, cron sends you an email about it. There's no robust way to check whether a cron job has finished with no output and a zero (i.e. success) status, or it's still running.
Some distributions don't set up a local email facility in their default installation. It's usually easy to do that: just install one of the MTA packages such as Postfix or Exim; select “local delivery only” or some such when asked how to configure it. If you want the emails to be sent to a remote server over SMTP rather than delivered locally, there are minimal MTAs just for that.
Instead of or in addition to the email, you can arrange for the output of your job to go into a file, by using a redirection on the command line.
50 12 27 4 1 java -jar HelloWorld.jar >~/log/HelloWorld-$(date -d \%Y\%m\%d-\%H\%M\%S).log 2>&1