By K. Patterson

This booklet offers an authoritative review of the literature on non-stationarity, integration and unit roots, offering path and counsel. It additionally offers precise examples to teach how the ideas will be utilized in functional events and the pitfalls to avoid.

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**Sample text**

Consider the variance of S2 : var(S2) = var(y1 + y2) = var(y1) + var(y2) + 2cov(y1, y2); if y1 and y2 are independent or there is no (serial) correlation, which is the weaker assumption, then cov(y1, y2) = 0, and, hence, var(S2) = 2y2. In general, var(St) = ty2, so that the variance of the terms in the partial sum process are not constant and increase linearly with time. 9. 2 Continuous random variables The aim in this section is to generalise the concepts of independence, dependence and conditional expectation to the case of continuous random variables.

39) −∞ where G(Z) is the distribution function of z and, as usual, F(X) is the distribution function of x, see Rao (1973, p. 93) and Billingsley (1995, p. 274). ) is a linear function, see L1 and L3. In some cases we can say something about E[g(x)] from knowledge of g(x). If g(x) is a convex function then from Jensen’s inequality, see, for example, Rao (1973), then E[g(x)] ≥ g[E(x)]. A convex function requires that the second derivative of g(x) with respect to x is positive; for example, for positive x, the slope of g(x) increases with x.

Such an example occurs when the distribution of a test statistic has a degrees of freedom effect. In this case we interpret the sample space of interest as being that for each xj, rather than the sequence as a whole. In the case of a stochastic process, the sample space is the space of a sequence of length n (or T in the case of a random variable with an inherent time dimension). 2. If the n tosses of the coin are consecutive, then the sample space, of dimension 2n, is denoted ⍀n, where the generic element of ⍀n, i, refers to an n-dimensional ordered sequence.