The problem with Imran's answer is taxation. If you are a resident of a country without a US tax treaty, you'd be paying too much in tax withholding (30%) and "you" can pay up to 55% in estate taxes after you pass.

Better options are SaxoBank and TD Direct Investing International, as your money will not be held by a US based financial institute. So, no estate taxes. Furthermore, investing in Irish-domiciled funds means utilizing Ireland's 15% taxation rate with the US and reduced rates with other countries. See US-domiciled Vanguard Total World Stock ETF (VT) vs Ireland-domiciled Vanguard FTSE All-World UCITS ETF (VWRL) comparison in the article below.

https://www.bogleheads.org/wiki/Nonresident_alien_with_no_US_tax_treaty_%26_Irish_ETFs
and
https://www.bogleheads.org/wiki/Nonresident_alien_taxation

Hope this helps. Feel free to reply if you have any further questions.