As I move away from my wireless router my download speed decreases. However, I have read some conflicting arguments where they claim that, at least theoretically, the download speed should not be affected. Can someone elaborate?
If you get further away from your wireless access point things will certainly start to go slower. But with some exceptions.
When a wireless client connects to the AP the two devices negotiate a speed that they can communicate. Wireless have a fixed set of speeds they can communicate at. A range that could be between 2Mb/s up to 600Mb/s on 802.11n gear. You can think of these ranges kind of like gears in a car. If a device can't communicate at the fastest wireless speed it may fall back to a slower speed, if that is better.
Here is where an exception comes in, that lends some truth to what seems to be confusing you. Access points can be configured to not permit you to drop below a certain link speed. This is done, to prevent one device on the network from slowing all devices down. Since a single wireless radio basically has to operate at the speed of the slowest device connected.
But lets say your wireless speed is fixed and unchanging. As you get further away you will probably start dropping/missing IP packets. If and when this happens TCP/IP will slow things down while it waits for re-transmissions and so on.
The download speed will be affected because the signal is weaker. To compensate, your wireless card will throttle down the connection speed to the router to reduce the chance of corruption and dropped packets.
So, worse signal = slower.