NO, it not convection, which is sufficient but not necessary cause. Brownian motion will mix the gases in the absence of convection, wind, turbulance, etc.
If you put the gases in a chamber without convection, etc., they will mix.
Brownian Motion is a result of a molecular-scale twiddling around of thermally vibrating molecules. By contrast, convection happens (sometimes quickly) with rather large volumes of air.
Thunderstorms are a great agent of convection. The typical thunderstorm cell moves air from near the surface up to the tropoause and back, at rates measurable in meters per second. One of those cells last only some 45 minutes, but there are estimated to be thousands going on in the atmosphere at any one time.
There are gentler forms of convection, also going on. Air rises at the equator, for example, and sinks in the Horse Latitudes (about 25-30° north and south of the equator). This goes on constantly, because of differential heating at the Earth’s surface.
The lowest level of the atmosphere is called the troposphere from the Greek root meaning to tun, because it is always turning over. For Brownian Motion fans, I am sorry to say that this is a lot more motion than Brownian Motion ever was dreamed of accomplishing.