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Oct 13, 2019 at 14:59 comment added dd Kong It's my fault that I didn't think carefully before posting the question, and this will be the final version; at least the main content won't be altered.
Oct 13, 2019 at 14:11 comment added Yuval Peres PS if you keep changing the question, no one will answer your questions. Better to think hard and post another question after checking many examples.
Oct 13, 2019 at 14:07 history edited Yuval Peres CC BY-SA 4.0
added 140 characters in body
Oct 13, 2019 at 14:06 comment added Nate Eldredge @ddKong: Continuity won't change anything. Replace each of the point masses with a normal distribution centered at the point and having very small variance.
Oct 13, 2019 at 13:57 comment added dd Kong Thanks. I edited the question again, and I want to know what will happen when the random variable is continuous.
Oct 13, 2019 at 13:45 history answered Yuval Peres CC BY-SA 4.0