there several firms out there that require the 10k for collateral against any losses. Some required that if you bring the minimum (usually 5k or 10k), you must trade onsite. however there is echotrade that allows remote with as little as 10k. there are probably more outhere but echo has better reputation.
One thing besides minimum 10k, you must be aware that most firms, if there are part of the pacific, philly, or chica exchanges, hold on to your initial contribution whether 10k or more for at least a year. Which sucks pretty much if the firm starts to crumble, kiss you money goodbye. With the exception of bright who gives you your oney back in a few days if you leave the firm.
Other exceptions to this holding firm rule are nasd firms. Such as worldco. they only hold on to 25% your money for a few months.