Definition:Asymptotic Distribution

Definition

For all $n \in \N$, Let $Z_n$ be a random variable.

The asymptotic distribution of $Z_n$ is the limiting probability distribution as $n \to \infty$.

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Examples

Arbitrary Example $1$

Let $X_1, X_2, \ldots, X_n$ be $n$ independent observations from a probability distribution with expectation $\mu$ and finite variance $\sigma_2$.

Then $Y_n = \ds \sum_{k \mathop = 1}^n X_k$ has expectation $n \mu$ and variance $n \sigma_2$.

Both of these tend to infinity, unless $\mu = 0$.

Also see

• Central Limit Theorem

• Results about the asymptotic distribution can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): asymptotic distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): asymptotic distribution