By E. J. Hannan
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Another form of sampling which could possibly arise is “periodic nonuniform sampling” (Freeman, 1965, p. 77) where the sample points are oftheformtj nA,j = 1,. . , N , n = 0, f l , . . , a n d where0 5 t j < A. This form of sampling arises, for example, with some types of market, which are open at m irregular arrangement of “days” over a trading period (a week, a month, a year).? We may now know r ( n A ti - t k ) ; j , k = 1 , . . , N . We need only consider the set S of those pairs ( j , k ) for which the corresponding differences ( f j - f k ) do not differ from each other by an integral multiple of A.
The matrix F(3’(1)consists of elements which are continuous, with derivatives which vanish almost everywhere, with respect to Lebesgue measure. It will be difficult to allot to them a physical meaning and we shall often assume that this third component is missing. Indeed, as we shall see below, insofar as this stationary model is an adequate approximation to the complexity which is reality, it is often (but perhaps not always) true that P ( A ) is the only component that can be expected to be present.
Thus y(n) is a stationary discrete sequence. The last integral is where +(A) is the characteristic function of the u,. This is therefore 2 + 2 n j ) 1 2 ~+( ~2nj) - 1+((2j - 1 ) n ) 1 2 ~ ~ (-( 21)n)l. j P(A)= [I+(A -m It is F,at best, which is all we can know from y(n). If F(A) does not increase outside [-n, n ] , then evidently we can obtain F(A)from P(A) if we know P and therefore (6. Otherwise the observed spectrum will need to be interpreted with care. d. (nonnegative) random A CONTINUOUS-TIME PROCESS.