Question

Show that strict stationarity of a time series \(\left\\{Y_{j}\right\\}\) means that for any \(r\) we have $$ \operatorname{cum}\left(Y_{j_{1}}, \ldots, Y_{j_{r}}\right)=\operatorname{cum}\left(Y_{0}, \ldots, Y_{j_{r}-j_{1}}\right)=\kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} $$ say. Suppose that \(\left\\{Y_{j}\right\\}\) is stationary with mean zero and that for each \(r\) it is true that \(\sum_{u}\left|\kappa^{u_{1}, \ldots, u_{r-1}}\right|=c_{r}<\infty\) The \(r\) th cumulant of \(T=n^{-1 / 2}\left(Y_{1}+\cdots+Y_{n}\right)\) is $$ \begin{aligned} \operatorname{cum}\left\\{n^{-1 / 2}\left(Y_{1}+\cdots+Y_{n}\right)\right\\} &=n^{-r / 2} \sum_{j_{1}, \ldots, j_{r}} \operatorname{cum}\left(Y_{j_{1}}, \ldots, Y_{j_{r}}\right) \\ &=n^{-r / 2} \sum_{j_{1}=1}^{n} \sum_{j_{2}, \ldots, j_{r}} \kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} \\ &=n \times n^{-r / 2} \sum_{j_{2}, \ldots, j_{r}} \kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}} \\ & \leq n^{1-r / 2} \sum_{j_{2}, \ldots, j_{r}}\left|\kappa^{j_{2}-j_{1}, \ldots, j_{r}-j_{1}}\right| \leq n^{1-r / 2} c_{r} \end{aligned} $$ Justify this reasoning, and explain why it suggests that \(T\) has a limiting normal distribution as \(n \rightarrow \infty\), despite the dependence among the \(Y_{j}\). Obtain the cumulants of \(T\) for the MA(1) model, and convince yourself that your argument extends to the \(\mathrm{MA}(q)\) model. Can you extend the argument to arbitrary linear combinations of the \(Y_{j} ?\)

Short answer

The argument shows that the arithmetic mean \(T\) of a stationary series with certain cumulant conditions becomes normal for \(n \to \infty\). It extends naturally to MA(q) models and general linear combinations.

Step-by-step solution

Understanding Strict Stationarity

Strict stationarity implies that the joint distribution of any collection of time series data points \(\left\{ Y_{j_1}, Y_{j_2}, \ldots, Y_{j_r} \right\}\) does not change when shifted in time. This means the cumulant \(\operatorname{cum}(Y_{j_1}, \ldots, Y_{j_r})\) depends only on time differences \(j_2 - j_1, \ldots, j_r - j_1\). Therefore, we can express it as \(\kappa^{j_2-j_1, \ldots, j_r-j_1}\).

Cumulant of T for Arithmetic Mean

The cumulant of the mean \(T = n^{-1/2}(Y_1 + \cdots + Y_n)\) is calculated from the cumulants of the \(Y_j\). The expression \(\operatorname{cum}\{ n^{-1/2}(Y_1 + \cdots + Y_n) \}\) is expanded as \(n^{-r/2} \sum_{j_1, \ldots, j_r} \operatorname{cum}(Y_{j_1}, \ldots, Y_{j_r})\). With strict stationarity, this becomes \(n^{-r/2} \sum_{j_1 = 1}^{n} \sum_{j_2, \ldots, j_r} \kappa^{j_2-j_1, \ldots, j_r-j_1}\).

Reduction to Summation Over Differences

Simplifying further, with \(j_1\) ranging over all terms and factoring \(n\), the equation becomes \(n \cdot n^{-r/2} \sum_{j_2, \ldots, j_r} \kappa^{j_2-j_1, \ldots, j_r-j_1}\). The assumption \(\sum_u |\kappa^{u_1, \ldots, u_{r-1}}| = c_r < \infty\) ensures convergence.

Cumulant Bound and Limit Analysis

Given the bounded cumulant sum, \(\operatorname{cum}(T)\) is bounded by \(n^{1-r/2} c_r\). As \(n \to \infty\), for \(r > 2\), \(n^{1-r/2} \to 0\), implying higher order cumulants vanish, suggesting that \(T\) converges to a Gaussian distribution.

Application to MA(1) Model

For the MA(1) model, \[ Y_j = e_j + \theta e_{j-1}, \] where \(e_j\) are i.i.d, cumulants are derived from \(e_j\)'s distribution. The structure of \(Y_j\) as a linear function of past innovations results in diminishing cumulants for higher \(r\) as \(n \to \infty\).

Extension to MA(q) Model

In the general MA(q) model, the cumulative contribution of earlier terms forms finite linear combinations of \(e_j\). The argument extends naturally, as the cumulants' structure derives primarily from the finite length dependence inherent in the linear components of the series.

General Linear Combinations

For arbitrary linear combinations of \(Y_j\), similar reasoning applies. The sum of linear processes with bounded cumulants underpins convergence in distribution, as any arbitrary linear combination maintains finite and ultimately diminishing cumulants.

Key concepts

Strict Stationarity

In time series analysis, strict stationarity is a foundational concept that refers to a process where the statistical properties of the series do not change over time. This means that for any collection of time points in the series, the joint distribution remains constant, even if these points are shifted in time.

Mathematically, if you take a set of points such as \( \{ Y_{j_1}, Y_{j_2}, \ldots, Y_{j_r} \} \), strict stationarity implies that the cumulants of these points \( \operatorname{cum}(Y_{j_1}, \ldots, Y_{j_r}) \) only depend on the time differences between them, like \( j_2 - j_1, j_3 - j_1, \ldots \). This can be expressed as \( \kappa^{j_2-j_1, \ldots, j_r-j_1} \).

This property is crucial because it allows for easier modeling and forecasting, as the relationships don't fluctuate unpredictably over time.

Cumulants

Cumulants are a set of statistics that provide insights into the shape and characteristics of a probability distribution. They encompass information about moments like the mean, variance, skewness, and kurtosis of a distribution.

For a series \( T = n^{-1/2}(Y_1 + \cdots + Y_n) \), their calculations become crucial. The cumulant of \( T \) is derived from those of \( \{ Y_j \} \), showing how a weighted sum of random variables leads to conclusions about overall behavior.

By expressing the cumulant of \( T \) as borrowed from the cumulants of the individual \( Y_j \) values, each shifted by their respective time differences, this structured approach allows us to calculate the cumulative impact of multiple time points on the mean. This is beneficial when examining how the distribution of cumulated time series data behaves under strict stationarity.

Moving Average Model

The Moving Average (MA) model is a simple yet powerful tool in time series analysis. It expresses a series as a linear function of errors from past periods. In the case of the MA(1) model, each value in the series \( Y_j \) can be written as \( Y_j = e_j + \theta e_{j-1} \), where \( e_j \) are independent and identically distributed error terms.

This structure implies that the current value not only reflects the immediate random shock but also the weighted impact of the previous one. As for cumulants, their diminishing values for higher orders \( r \) as \( n \to \infty \) means that the influence of past shocks becomes more negligible over time, making it simpler to forecast future movements.

The natural extension to MA(q) models involves more past periods while still maintaining finite dependence, supporting the use of linear processes in capturing broader data dynamics while retaining predictability.

Normal Distribution Convergence

Normal Distribution Convergence, within the context of time series, refers to the behavior of the average of a series of random variables approaching a normal distribution as the sample size grows. This phenomenon is closely tied to the Central Limit Theorem, which states that the distribution of the sum (or average) of a large number of independent and identically distributed (i.i.d) variables becomes approximately normal, regardless of the original distribution, given a calculable mean and variance.

For time series exhibiting strict stationarity and bounded cumulants, a similar effect occurs. As you compute cumulants of \( T = n^{-1/2}(Y_1 + \cdots + Y_n) \) and confirm their bounded nature under this structure, the higher-order cumulants (those beyond the second) shrink to zero as \( n \to \infty \).

This naturally suggests that the distribution of \( T \) is indeed converging to a normal distribution, even in settings where dependencies exist among the \( Y_j \). This principle is immensely valuable because it allows inferential techniques to be applied to time series data that may otherwise seem non-normal.