Canonical Representations of Second-Order Random Processes

Siraya, T. N.

doi:10.1137/1122051

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“…then the representation of their sum, x(t) = x 1 (t) + x 2 (t), is weak-canonical if and only if f 1 (u) = a · f 2 (u), where f 1 (u), f 2 (u) are spectral densities, a = const., but it is not canonical (see [4]). …”

Section: Examplementioning

confidence: 99%

Spectral multiplicity of some stochastic processes

Mitrovic¹

1998

Proc. Amer. Math. Soc.

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Abstract. In this paper we consider the connection between the canonical and the weak-canonical representations for the given second-order stochastic process in a separable Hilbert space and we extend a well-known theorem of H. Cramer concerning sufficient conditions for a process to be of multiplicity one.Let x(t), t ∈ (a, b) ⊂ R, be a second-order real-valued process with Ex(t) = 0 for each t. Let H(x, t) be the linear closure generated by x(s), s ∈ (a, t], in the Hilbert space H of all random variables with finite variance (Ex 2 (t) < ∞). We will suppose that x(t), t ∈ (a, b), is continuous left and purely nondeterministic (i.e. t>a H(x, t) = 0). It is well known (see [1]) that there is a representationwhere:1. The processes z n (u), n = 1, . . . , N, are mutually orthogonal with orthogonal increments such that Ez n (u) = 0 and Ez 2 n (u) = F n (u), where F n (u), n = 1, . . . , N, are nondecreasing functions left continuous everywhere on (a, b).2. The nonrandom functions g n (t, u), u ≤ t, are such that:3. dF 1 > dF 2 > · · · > dF n , where the relation > means absolute continuity between measures.4.H(z n , t), t ∈ (a, b). The expansion (1) satisfying the conditions 1, 2, 3 and 4 is the canonical representation or Cramer representation for the process x(t). The number N (finite or infinite) is called the multiplicity of x(t), and N is uniquely determined by the process x(t). But, the processes z n (u) and the functions g n (t, u) are not uniquely determined.

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Section: Examplementioning

confidence: 99%

Spectral multiplicity of some stochastic processes

Mitrovic¹

1998

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Canonical Representations of Second-Order Random Processes

Cited by 1 publication

References 5 publications

Spectral multiplicity of some stochastic processes

Spectral multiplicity of some stochastic processes

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