Actually, that is not correct. If the 'stealing' competitor has submitted their site first, Google will probably crawl their site first. Additionally, Google says that you don't need to submit your site (although I always do anyway). When Google crawls your site or attributes content to you depends a lot on the 'authority' Google believes your site has via its algorithm.
At any rate, I'm moving this thread to Internet marketing as it is not about content, but rather about SEO.