Consider the differentiability of the stochastic process {X (t)} of Example 4.9, which has X (t) equal to some Aj in each finite time interval, and with the Aj identically distributed, mean-zero, and independent.

Question

Consider the differentiability of the stochastic process [latex] \{X (t)\}[/latex] of Example 4.9, which has [latex] X (t)[/latex] equal to some [latex] A_{j} [/latex] in each finite time interval, and with the [latex] A_{j} [/latex] identically distributed, mean-zero, and independent.

Accepted Answer

Taking the expected value gives [latex] \mu_{X}(t) \mu_{A}[/latex], so we can say that [latex] \mu_{\dot{X} }(t)=0 [/latex], and in Example 4.9 we found the autocorrelation function for [latex] \{ X(t)\} [/latex], which can be written as

[latex] \phi _{XX}(t,s)=E(A^{2})\sum\limits_{j}{(U[t-j\Delta ]-U[t-(j+1)\Delta ])(U[s-j\Delta ]-U[s-(j+1)\Delta ])}[/latex]

This function is discontinuous on the boundaries of the dimension [latex] \Delta [/latex] squares along the diagonal. Thus, the [latex] \phi_{X \dot{X} }(t,s)[/latex] cross-correlation function is infinite for some [latex] (t,s)[/latex] values:

[latex] \phi_{X \dot{X} }(t,s)=E(A^{2}) \sum\limits_{j}{(U[t-j\Delta ]-U[t-(j+1)\Delta ])(\delta[s-j\Delta ]-\delta[s-(j+1)\Delta ])}[/latex]

In particular, it is infinite for [latex] s= k \Delta [/latex] for any integer [latex] k [/latex], if [latex] (k-1) \Delta \lt t \lt (k+1) \Delta [/latex] (i.e., along the boundaries of the dimension [latex] \Delta [/latex] squares along the diagonal). Similarly, [latex] \phi _{X\dot{X} }(t,s)[/latex] is infinite along all the [latex]t=k \Delta [/latex] boundaries of the same squares along the diagonal. Everywhere, except on the boundaries of these squares, one obtains [latex] \phi _{X\dot{X} }(t,s)=0[/latex]and [latex] \phi _{\dot{X} \dot{X} }(t,s)=0[/latex], reflecting the fact that this process has a zero derivative except at certain discrete times at which the derivative is infinite. This agrees with the behavior of the time histories of the process.

Question 4.18: Consider the differentiability of the stochastic process {X ...

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