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I have a difficult time understanding how can a sequence of random variables, lets say {$z_i$}, converge in probability to a constant.

Is that the mean of the realization of all random variables in the sequence converges to a number? (If yes, isn't it just the central limit theorem?)

Thank you.

Leopold W.
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  • $z_n=0$ with probability $1/n$, z=1 else. What happens to z as n goes to infinity? – VCG Sep 14 '16 at 01:47
  • Thanks for the comment, @VCG. I understand the mechanics of convergence in probability. But I can't follow the intuition. What does $n$ represent exactly? Sample size or just an index? If it is an index, what's the difference between n=5 and n=100000? – Leopold W. Sep 14 '16 at 01:54
  • There are a few interpretations https://www.statlect.com/asymptotic-theory/sequences-of-random-variables – VCG Sep 14 '16 at 02:07
  • Some duplicates: https://stats.stackexchange.com/questions/134701/intuitive-explanation-of-convergence-in-distribution-and-convergence-in-probabil, https://stats.stackexchange.com/questions/337788/why-convergence-in-probability-is-defined-as-convergence-to-r-v, – kjetil b halvorsen Aug 16 '20 at 17:44

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