Digital Gaussian White noise signal generation in C++
If you are generating sample of white noise (Normal distribution value) with frequency of 48 kHz, then you already have this in frequency range of 0 - 24 kHz (at least you should if generator is truly random). If you use same samples with different sampling frequency, then they will also be a white noise.

Now if you want to make it a band-limited white noise, then you must use a digital filter. Choice is up to you, but for sake of simplicity I would recommend FIR filter. You must use it to filter in real time your incoming samples. How to do it? Well, that's totally up to you, you can even use short buffer to perform filtering by linear convolution for each generated sample but it won't be very efficient.