3 Answers

Information here.http://www.schizophrenic.com/articles/schizophrenia/schizophrenic-behavior

Folks addicted to illegal drugs lie and steal to support their addiction. Schizophrenics if untreated may behave compulsively but usually don't steal. I have known hoarders who would steal something small to add to their collection of things to remember you by or in thinking of another as a gift. My parents stole my money to give to charities  and to prevent me from buying drugs when I was not involved with any illegal drugs whatsoever.

It is easy to "generalize" about illness, but each case is unique in some way.  My experience with schizophrenics has been minimal, but one man I know would never steal or lie about anything, period. 
The other guy would steal and lie without provocation.