Question

Suppose \(\mathbf{X}_{n}\) has a \(N_{p}\left(\boldsymbol{\mu}_{n}, \boldsymbol{\Sigma}_{n}\right)\) distribution. Show that $$ \mathbf{X}_{n} \stackrel{D}{\rightarrow} N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma}) \text { iff } \boldsymbol{\mu}_{n} \rightarrow \boldsymbol{\mu} \text { and } \boldsymbol{\Sigma}_{n} \rightarrow \mathbf{\Sigma} $$

Short answer

The solution hinges upon the convergence in distribution definition and the properties of the multivariate normal distribution. The 'if' condition is fulfilled because if the sequences of means and covariance matrices converge, the multivariate normal distribution, described by these parameters, becomes more and more stable, approaches a limit, hence the sequence of random variable will tend towards the limit distribution. In the opposite direction, if \(\mathbf{X}_{n}\) converges in distribution to \(N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})\), \(\boldsymbol{\mu}_{n}\) must converge to \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}_{n}\) must converge to \(\mathbf{\Sigma}\) to ensure the distribution stays stable and approaches the limit.

Step-by-step solution

Understand the Convergence in Distribution

The convergence in distribution, or convergence in law, means that the cumulative distribution function (CDF) of \(\mathbf{X}_{n}\) converges pointwise to the CDF of \(N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})\) as \(n\) approaches infinity. In other words, \(lim_{n \rightarrow \infty}F_{\mathbf{X}_{n}}(x) = F_{N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})}(x)\) for all \(x\) at which \(F_{N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})}(x)\) is continuous.

Conclude Convergence of Parameters

For a sequence of multivariate normal distributions, their CDFs will converge if and only if the sequences of means and covariance matrices converge, that is, \(lim_{n \rightarrow \infty}\boldsymbol{\mu}_{n} = \boldsymbol{\mu}\) and \(lim_{n \rightarrow \infty}\boldsymbol{\Sigma}_{n} = \mathbf{\Sigma}\). Know that \(\mu_{n}\) and \(\Sigma_{n}\) describe the location and shape of the multivariate normal distribution, hence their convergence ensures the convergence of distribution.

Prove the If and Only If Condition

To prove the if and only if condition, two separate proofs are needed. One proof for the 'if condition', showing that if the sequences of means and covariance matrices converge, then \(\mathbf{X}_{n}\) will converge in distribution to \(N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})\). Another proof for the 'only if condition', showing that if \(\mathbf{X}_{n}\) converges in distribution to \(N_{p}(\boldsymbol{\mu}, \mathbf{\Sigma})\), then the sequences of means and covariance matrices must converge. A detailed proof would involve limit properties and properties of multivariate normal distribution function.

Key concepts

Multivariate Normal Distribution

The multivariate normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions. It's characterized by a mean vector \boldsymbol{\text{\( \mu \)}} and a covariance matrix \boldsymbol{\text{\( \Sigma \)}}. The mean vector represents the expected value of the random variables, and the covariance matrix provides information about the variance of each component and the pairwise covariances between them.

In practical terms, the multivariate normal distribution describes the behavior of random vectors where each element (or variable) could potentially be correlated with others. Understanding this concept is fundamental because it extends to the domain of multivariate statistics, which includes the analysis of several correlated random variables at once.

• It's used in various fields including finance, engineering, and natural sciences.
• The shape of the distribution can be visualized as an ellipse (in 2D) or ellipsoid (in higher dimensions) in the space of the random variables.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a crucial concept in probability theory and statistics. It provides the probability that a random variable will take a value less than or equal to a particular value.

For example, if \textbf{X} is a random variable, its CDF \text{F(\( x \))} is given by \text{\( P(X \leq x) \)}. In the case of the multivariate normal distribution, the CDF becomes a bit more complex due to the multivariate aspect, but it still retains the fundamental property of non-decreasing with increasing \text{\( x \)}. It generalizes to higher dimensions to account for the multivariate aspect of the distribution.

• The CDF is continuous and increases from 0 to 1.
• It provides a full description of the probability distribution for continuous random variables.

Convergence of Mean and Covariance Matrices

Convergence of the mean and covariance matrices are pivotal for proving convergence in distribution for sequences of multivariate normal distributions.

The mean vector \textbf{\( \boldsymbol{\mu} \)} convergence suggests that the average or expected values of the distributions' variables are stabilizing to a fixed vector, which is essential for the distributions to converge. Similarly, convergence of the covariance matrix \textbf{\( \boldsymbol{\Sigma} \)} signifies that not only the variances of the individual variables but also the covariances between any two variables are settling towards a steady state.

• If these parameters don't converge, the shape and orientation of the multivariate normal distribution might keep changing, preventing convergence in distribution.
• The convergence of these matrices is therefore a necessary condition to ensure that a sequence of multivariate normal distributions converges to a particular multivariate normal distribution.