I deposited USD in coinbase.com and bought some BTC, which shows in my wallet in coinbase.com. But when I search my address in blockchain.info, I saw no transaction associated with my address. I bought this about 2 month ago so it's not about delay of the transaction to blockchain. Is coinbase.com lying about my transaction? If it's not in blockchain, how can I claim that I own those bitcoin? My addresses are "1KqS8GcH9qpT9sHDuv1JXgTicCPQfBdZho", "1HMP3S6jWWnXc6nJNqPnCB3g9xMPk3iyF2" and "1A5Ttr8pfsmYFLavPjkCoKKUwoMSxbEFNV".
Trades within exchanges do not occur on the blockchain. That would severely bloat the blockchain and produce a lot of unnecessary transactions. These trades are simply done by updating your account's balance within the service's internal database. You are not actually sent Bitcoin and there is no way to prove that you own that much Bitcoin except by showing your account.
The addresses that an exchange gives you are not your addresses. They are addresses to the exchanges wallet and once you send Bitcoin to that address, they will update your account's balance in their internal database system. The exchange is holding your Bitcoin for you, and they can do whatever they want with it. It isn't really yours until you choose to withdraw the Bitcoin. When you withdraw, you likely won't be sent Bitcoin from "your" addresses either, but rather from other addresses in the exchange's wallet and often times your withdrawal will be part of the same transaction as other people's withdrawals.