Would it be possible to make a PDA that doesn't actually do any of the processing on its own, it sends input data to a computer which processes it and sends back a signal for the display of the PDA? The PDA would just take input data (such as pressing buttons/touchscreen interaction), send it to a computer to process the data and the program that the PDA is "running" (the computer would actually be doing all of the work, it would launch and run the program, it just gets the command from the PDA), then the computer sends a very simple signal that is purely for display, which is quickly processed and shown on the screen of the PDA. So in a way you could say that it's just having a small wireless computer touchscreen. You'd get the processing power of a computer in a probably very small and light PDA. Why hasn't something like this been made? Is it just because of the (probably) slow response time between sending input data and receiving the effect? Inconsistent wireless connection? Would this become possible in the future if there were wireless coverage nearly everywhere and speeds improved?

Thanks for any answers!