You might be able to use a Arduino as a control / alert system. GPRS is an option but you need to have a mobile service provider, a M2M contract, and coverage. The arduino would not have to process the video, etc. For example, based on external inputs such as an alarm going off it could simply open the GPRS connection for a standard IP camera to connect to the internet with, transmit the picture to a server, send you an SMS / e-mail with an alert, and close the connection.
I doubt you'll be able to stream video - speeds are usually too low and most M2M contracts that allow a lot of data are expensive. The kind of video you could stream would likely be pretty useless (i.e. little resolution and lots of compression). If high-quality video is what you want, I suggest getting someone nearby to let you use their WiFi network.