Limiting distribution / Asymptotic distribution

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A limiting distribution, also called an asymptotic distribution, is the distribution of a sequence of random variables that converges to a particular distribution as the sample size increases. It refers to the hypothetical distribution or convergence of a sequence of distributions; It is not a distribution in the general sense of the term, but rather a theory which aims to identify a limiting distribution for a series of distributions.

A more formal definition of a limiting distribution is given by Epps [1]. Suppose than a random sequence X_n with cumulative distribution function (CDF) F_n(X_n ), where X is a random variable with CDF F(x). Then,

If F_n converges to F as n > ∞ (for all points where F(x) is continuous), then the distribution of x_n converges to x. This is the limiting distribution of x_n.

In simple terms, we can say that the limiting probability distribution of X_n is the limiting distribution of some function of X_n.

Limiting distribution advantages and disadvantages

Limiting distributions have many benefits for making reliable inferences about populations based on samples. For example, known distributions can approximate the distribution of unknown quantities, helping to decide whether a null hypothesis should be accepted or rejected as well as construct a confidence interval. When it comes to finding appropriate sample sizes, limiting probability distributions are important: the process in basic statistics involves taking a random sample of observations and fitting that data to a known distribution such as the normal or t distribution. However, this isn’t an exact science since fitting data exactly to a distribution is difficult in real life due to limited sample sizes. Consequently, a “best guess” is made based on knowledge about large sample statistics. In such cases, the limiting or asymptotic distribution can be used on small, finite samples to approximate the true distribution of a random variable. Assuming such a distribution exists, when a sample size is sufficiently large, a statistic’s distribution will form a limiting distribution.

However, it is necessary to ensure that the conditions for a limiting distribution are satisfied for accuracy. Therefore, limiting distributions should be approached with awareness of their challenges, but their benefits cannot be overlooked.

Limiting distribution uses

Limiting distributions are a powerful tool for making inferences about populations based on samples. They allow us to use known distributions to approximate the distribution of unknown quantities, such as the sample mean or the sample variance. Some limiting distributions are well-known. For example:

• The t-statistic’s sampling distribution will converge to a standard normal distribution if the sample size is large enough.
• The central limit theorem states that the distribution of the sample mean converges to a normal distribution as the sample size increases. This means that we can use the normal distribution to approximate the distribution of the sample mean, even if the distribution of the population mean is not normal.
• The limiting distribution of the F statistic is given by the limiting distribution of the numerator [2].

In basic cases, an asymptotic distribution exists if the probability distribution of random variables Z_i (i = 1, 2, …) converge to a probability distribution (the asymptotic distribution) as i increases The limiting probability distribution can sometimes be determined by analyzing the behavior of cumulative distribution functions (CDFs) or probability density functions (PDFs). Theorems such as Slutsky’s can be used to study convergence in probability distributions. A special case of a limiting distribution is found when a sequence of random variables is zero (Z_i = 0) as i tends to infinity. In that case, the asymptotic distribution is a degenerate distribution, corresponding to the value zero.

Limiting distributions — stationary distributions that do not change over time — often show up in the study of Markov Chains. The limiting distribution represents the states of the chain as time approaches infinity. In other words, as time n > ∞, a Markov chain has a limiting distribution π = (π_j)_j∈s. To find the limiting distribution, solve the equation π = πP, where π is the initial distribution and P is the transition matrix. The transition matrix is a square matrix that provides the probability of moving from one state to another.

The limiting distribution does not always exist for Markov chains, but when it does, it predicts long-term Markov chain behavior. For example, the limiting distribution for a Markov chain that models the weather can predict future probabilities such as the likelihood of rain. As well as predictions, it can be used for understanding long-term patterns and designing algorithms for Markov chain problem-solving.

References

[1] Epps, T. (2013). Probability and Statistical Theory for Applied Researchers. World Scientific.

[2] Chapter 6: Asymptotic distribution theory. Retrieved May 20, 2023 from: https://www.bauer.uh.edu/rsusmel/phd/sR-9.pdf