Compare the concepts of convergence in probability and...

Compare the concepts of convergence in probability and convergence in distribution and give an intuitive explanation to the result $\stackrel{\mathrm{P}}{\rightarrow} \Rightarrow \stackrel{\mathrm{D}}{\rightarrow}$.

Key Concepts

Convergence in Probability

This concept describes a mode of convergence for a sequence of random variables whereby the probability that the absolute difference between the sequence and a given limit exceeds any positive threshold decreases to zero as the sequence progresses. In other words, for every epsilon > 0, the probability that the deviation is greater than epsilon tends to zero as the sample size increases, ensuring that the random variables become increasingly concentrated around the limit in a probabilistic sense.

Convergence in Distribution

This concept, also known as weak convergence, pertains to the convergence of the cumulative distribution functions (CDFs) of a sequence of random variables to the CDF of a limit random variable at every continuity point of the limit CDF. It is a weaker form of convergence than convergence in probability because it only requires that the distributional shape of the random variables approaches that of the limit, without necessitating a closeness in individual outcomes.

Implication of Convergence in Probability to Convergence in Distribution

The result that convergence in probability implies convergence in distribution can be intuitively understood by noting that convergence in probability guarantees that the random variables are increasingly likely to be near the limit value. This concentrated behavior naturally forces the distribution of the random variables to approximate the distribution of the limit, as the probability mass accumulates around the limit. Therefore, even though convergence in distribution is a weaker form, the strong form of convergence in probability suffices to ensure that the distributional profiles also converge.

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