The way these phone work is that the WiFi turns off when NOTHING is using it, and the screen is off.
If there is an app that needs to check for notifications when the screen is off, say FaceBook or Twitter, it will turn the WiFi on to check for notifications, and then back off.
If you are not in range or connected to a WiFi network, it then pulls those notifications through the cell network (3g/4G). There is no reason the WiFi HAS to be on for you to get notifications. You should get notifications in your car, when you are MILES from any WiFi network.
If you are not getting notifications, there is a different problem going on.