"Why Must the HK inflation rate stay close to the US inflation rate in the long run?" - Ronnie
In the long run, the inflation rate must be similar across any countries with a fixed exchange rate. Consider the alternative: suppose prices in HK were to increase at a much faster rate than in the US? Given enough time, it would be profitable to buy goods in the US, export them to HK, and sell them at a profit. The act of such arbitraging would put upward pressure on prices in the US (since demand in the US would rise), and downward pressure on prices in HK (since supply would increase), thus pushing their inflation rates closer together.
This argument strictly only holds for tradeable goods- for non-tradeables (for example haircuts and apartment rentals), it's impossible to arbitrage away price differences, since there's no gain from renting an apartment in the US if you live in HK, no matter how much cheaper it is! But tradeables are a large enough portion of the total consumption bundble (about 40%-70% of goods in the CPI) that HK and US inflation rates will always tend together in the long run- provided the exchange rate remains fixed.