Stationary Processes. Discrete and Continuous Time

Definition 1. A family $Z = \left\{ Z(B), B \in \mathscr {K}\right\}$ in $L^{2}(\Omega )$ indexed by the sets of a semiring $\mathscr {K}$ is called an orthogonal random measure if

1. $Z(B) - Z(C)$ for $B, C \in \mathscr {K}, B \cap C = \varnothing$, i.e. $(Z(B), Z(C)) = 0$, where $(\cdot , \cdot )$ is the inner product in $L^{2}(\Omega )$;

2. for every $B = \bigcup_{k = 1}^{\infty } B_{k}$, where $B, B_{1}, B_{2}, \ldots \in \mathscr {K}$ and $B_{n} \cap B_{m} = \varnothing$ ($n \neq m$),

   $$Z(B) = \sum _{k = 1}^{\infty } Z\left(B_{k}\right) \text{ a.s., }$$

   and the series converges in mean square. An orthogonal random measure $\mu$ is called centered if $\mathbb {E}\left[Z(B)\right] = 0$ for every $B \in \mathscr {K}$.

Show that the Poisson random measure (see §9 of Chapter I), defined on the semiring of bounded Borel sets in $\mathbb {R}^{n}$, is an orthogonal (noncentered) random measure.

Definition 1. A family $Z = \left\{ Z(B), B \in \mathscr {K}\right\}$ in $L^{2}(\Omega )$ indexed by the sets of a semiring $\mathscr {K}$ is called an orthogonal random measure if

1. $Z(B) - Z(C)$ for $B, C \in \mathscr {K}, B \cap C = \varnothing$, i.e. $(Z(B), Z(C)) = 0$, where $(\cdot , \cdot )$ is the inner product in $L^{2}(\Omega )$;

2. for every $B = \bigcup_{k = 1}^{\infty } B_{k}$, where $B, B_{1}, B_{2}, \ldots \in \mathscr {K}$ and $B_{n} \cap B_{m} = \varnothing$ ($n \neq m$),

   $$Z(B) = \sum _{k = 1}^{\infty } Z\left(B_{k}\right) \text{ a.s., }$$

   and the series converges in mean square.

Let $\left\{ Z_{\lambda }, \lambda \in \mathbb {R}\right\}$ be a complex-valued $L^{2}$-process such that

1. $\mathbb {E}\left[\left|Z_{\lambda }-Z_{\nu }\right|^{2}\right] \rightarrow 0$ for every $\nu \in \mathbb {R}$ as $\lambda \downarrow \nu$,

2. the increments are orthogonal, i.e. for all $\lambda_{1} < \lambda_{2} < \lambda_{3}$

   $$\mathbb {E}\left[\left(Z_{\lambda _{2}}-Z_{\lambda _{1}}\right) \overline{\left(Z_{\lambda _{3}}-Z_{\lambda _{2}}\right)}\right] = 0$$

On the semiring $\mathscr {K} = \left\{ (a, b],-\infty < a \leqslant b < \infty \right\}$, where $(a, a] = \varnothing$, introduce the family of random variables $Z((a, b]): = Z(b)-Z(a)$. Prove that $Z$ is an orthogonal random measure on $\mathscr {K}$.

We introduce the structure function (or structure measure) of the measure $Z$ by setting $\mu (B) = \mathbb {E}\left[\left|Z(B)\right|^{2}\right], B \in \mathscr {K}$. If the measure is generated by the process $Z_{\lambda }, \lambda \in \mathbb {R}$, of Exercise 2, then instead of the integral $\int_{\mathbb {R}} f(\lambda ) Z(d \lambda )$ one also uses the notation $\int_{\mathbb {R}} f(\lambda ) d Z_{\lambda }$.

Let $\left\{ Z_{\lambda }, \lambda \in \mathbb {R}\right\}$ be a centered Gaussian process with orthogonal increments, right-continuous in mean square (find out whether the process must then be left-continuous in mean square). Let, for each $t \in T$, the function $g(t, \cdot ): \mathbb {R} \rightarrow \mathbb {C}$ satisfy the condition

where $\mu$ is the structure measure corresponding to the orthogonal random measure $Z$ generated by the process $\left\{ Z_{\lambda }, \lambda \in \mathbb {R}\right\}$ (see Exercise 2). Prove that then the process

is a centered Gaussian process.

Definition 1. A family $Z = \left\{ Z(B), B \in \mathscr {K}\right\}$ in $L^{2}(\Omega )$ indexed by the sets of a semiring $\mathscr {K}$ is called an orthogonal random measure if

1. $Z(B) - Z(C)$ for $B, C \in \mathscr {K}, B \cap C = \varnothing$, i.e. $(Z(B), Z(C)) = 0$, where $(\cdot , \cdot )$ is the inner product in $L^{2}(\Omega )$;

2. for every $B = \bigcup_{k = 1}^{\infty } B_{k}$, where $B, B_{1}, B_{2}, \ldots \in \mathscr {K}$ and $B_{n} \cap B_{m} = \varnothing$ ($n \neq m$),

   $$Z(B) = \sum _{k = 1}^{\infty } Z\left(B_{k}\right) \text{ a.s., }$$

   and the series converges in mean square. We introduce the structure function (or structure measure) of the measure $Z$ by setting $\mu (B) = \mathbb {E}\left[\left|Z(B)\right|^{2}\right], B \in \mathscr {K}$. An orthogonal random measure $\mu$ is called centered if $\mathbb {E}\left[Z(B)\right] = 0$ for every $B \in \mathscr {K}$.

(change of measure in the integral) Let $Z$ be an orthogonal random measure defined on $\Omega \times \mathscr {G}$ and having a $\sigma$-finite structure function (measure) $\mu$ on the measurable space $(\Lambda , \mathscr {A})$ (the system of sets $\mathscr {G} = \left\{ A \in \mathscr {A}: \mu (A) < \infty \right\}$). Let $h: \Lambda \rightarrow \mathbb {C}, h \in L^{2}(\Lambda , \mathscr {A}, \mu )$. Introduce

and set

Explain why $V$ is an orthogonal random measure (centered if the measure $Z$ is centered) with the $\sigma$-finite structure measure

Prove that if $g: \Lambda \rightarrow \mathbb {C}, g \in L^{2}(\Lambda , \sigma \left\{ \mathscr {H}\right\} , \nu )$, then

Prove that representation (68) holds (a.s.) with independent random variables $z_{k} \sim \mathrm{N}(0,1), k \in \mathbb {Z}$. More precisely, prove that for such a choice of the variables $z_{k}$ ($k \in \mathbb {Z}$) the right-hand side of (68) has a continuous modification which is a centered Gaussian process with covariance function $\min \left\{ s, t\right\} , s, t \in [0,2 \pi ]$.

Formula (68): for the Wiener process $W$ on the interval $[0,2 \pi ]$ the following (Karhunen) representation holds

where $z_{k}$ are orthonormal centered random variables, and the series converges in mean square for each $t \in [0,2 \pi ]$ (for $k = 0$ we set $\left(1-e^{i k t}\right) / i k = -t$).

Let $X = \left\{ X(t), t \in [a, b]\right\}$ be a complex-valued $L^{2}$-process continuous in mean square on the interval $[a, b]$. Prove that this property is equivalent to the function $m(t) = \mathbb {E}\left[X(t)\right]$ being continuous on $[a, b]$ and the covariance function $r(s, t) = \operatorname {cov}(X(s), X(t))$ being continuous on $[a, b] \times [a, b]$.

Prove that the operator $A$ given by formula (70) is compact, i.e. the operator $A$ maps every norm-bounded set into a precompact one (i.e. a set whose closure is compact).

Formula (70): with the covariance function $r = r(s, t)$ continuous on $[a, b] \times [a, b]$ we associate the bounded linear Fredholm operator $A$, which acts in $L^{2}[a, b]$ and is defined as follows:

With the covariance function $r = r(s, t)$ continuous on $[a, b] \times [a, b]$ we associate the bounded linear Fredholm operator $A$, which acts in $L^{2}[a, b]$ and is defined as follows:

Let $\left\{ \phi_{k}, k \in J\right\}$ be the orthonormal system of eigenvectors of the operator $A$ corresponding to its nonzero eigenvalues $\lambda_{k}$, $k \in J$.

Prove that (by the nonnegative definiteness of the function $r$ ) $\lambda_{k} > 0$ for $k \in J$.

With the covariance function $r = r(s, t)$ continuous on $[a, b] \times [a, b]$ we associate the bounded linear Fredholm operator $A$, which acts in $L^{2}[a, b]$ and is defined as follows:

By estimating the quantity

prove that if the conditions of Theorem 19 hold, then in the expansion (76) one can choose as the variables $z_{k}$ the centered orthonormal variables

The integral here is defined (and exists) as the mean-square limit of the corresponding Riemann integral sums. Explain why, for a covariance function $r = r(s, t)$ continuous on $[a, b] \times [a, b]$, the eigenfunctions $\phi_{k} \in C[a, b], k \in J$. Moreover, prove that the mean-square convergence in (76) with these $z_{k}, k \in J$, is uniform in $t \in [a, b]$.

Theorem 19 (Karhunen-Loève). Let $X = \left\{ X(t), t \in [a, b]\right\}$ be a centered $L^{2}$-process, continuous in mean square on $[a, b]$, having covariance function $r = r(s, t)$. Then for each $t \in [a, b]$

where $\lambda_{k}, \phi_{k}$ are the same as in (75), and $\left\{ z_{k}, k \in J\right\}$ are centered orthonormal random variables. For countable $J$ the series (76) converges in mean square.

Theorem 18 (Mercer). Let $r = r(s, t)$ be a covariance function continuous on $[a, b] \times [a, b]$. Then for all $s, t \in [a, b]$

moreover, if $J$ is countable, then the series (75) converges absolutely and uniformly on $[a, b] \times [a, b]$. (As before, $\lambda_{k}$ and $\phi_{k}$ denote the positive eigenvalues and the corresponding eigenfunctions of the operator $A$ from (70).)

Find the Karhunen-Loève expansion on the interval $[0, c]$ for the Ornstein-Uhlenbeck process having the covariance function $r(s, t) = e^{-\alpha \left|s-t\right|}$, where $\alpha > 0, s, t \in \mathbb {R}$.

Let $X = \left\{ X(t), t \in \mathbb {R}\right\}$ be a mean-square continuous process with covariance function $r(s, t) = R(s-t)$. Let $X$ be a periodic process with period $a$, i.e., $X(t+a) = X(t)$ a.s. for every $t \in \mathbb {R}$. Find the Karhunen-Loève expansion for $X$.

Definition 11 of Chapter II. A function $K: T \times T \rightarrow \mathbb {C}$ is called a reproducing kernel of a Hilbert space $H$ consisting of some complex-valued functions defined on a set $T$ if

1. $K(t, \cdot ) \in H$ for every $t \in T$,

2. $\left\langle f, K(t, \cdot ) \right\rangle = f(t)$ for all $f \in H$ and $t \in T$, where $\left\langle \cdot , \cdot \right\rangle$ is the inner product in $H$.

Prove that if the covariance function $r = r(s, t)$ admits the representation (18), then the Hilbert space $H$ with reproducing kernel $r = r(s, t)$ consists of the functions

where $L^{2}[f]$ is the closure in $L^{2}(\Lambda , \mathscr {A}, \mu )$ of the linear span of the functions $f(t, \cdot ), t \in T$. The inner product in $H$ is given by the formula

where $h_{k}$ and $g_{k}, k = 1,2$, are related by (82). Let $Z$ be an orthogonal random measure on $\mathscr {G} \times \Omega$ having the measure $\mu$ as its structure function (see Theorem 2), and let $L^{2}[f] = L^{2}(\Lambda , \mathscr {A}, \mu )$. Explain why, for a process $X = \left\{ X(t), t \in T\right\}$ of the form (19), the process $Y$ appearing in Theorem 6 of Chapter II is given by the formula

where $h$ is determined by $g$ according to (82).

Representation (18):

where $f(t, \cdot ) \in L^{2}(\Lambda ) = L^{2}(\Lambda , \mathscr {A}, \mu )$ for each $t \in T$, and $\mu$ is some $\sigma$-finite measure.

Theorem 2. The stochastic integral (14) is an isometric mapping of $L^{2}(\Lambda , \mathscr {A}, \mu )$ (where $\mu$ is a $\sigma$-finite measure) onto some subspace $L_{Z}^{2} \subset L^{2}(\Omega , \mathscr {F}, \mathbb {P})$. Moreover, $Z$ extends to an orthogonal measure (with structure function $\mu$) on the system of sets $\mathscr {G} = \left\{ B \in \mathscr {A}: \mu (B) < \infty \right\}$ by the formula

For the action of $J$ on functions $f \in L^{2}(\Lambda , \mathscr {A}, \mu )$ the integral notation is usually used

Formula (19):

Theorem 6 of Chapter II (Parzen). Let $X = \left\{ X(t), t \in T\right\}$ be a centered $L^{2}$-process defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$, with zero mean and covariance function $r = r(s, t), s, t \in T$. Let $H$ be the "minimal" Hilbert space with reproducing kernel $r = r(s, t)$ and inner product $\left\langle \cdot , \cdot \right\rangle$. Then on $(\Omega , \mathscr {F}, \mathbb {P})$ there exists a centered $L^{2}$-process $Y = \left\{ Y(h), h \in H\right\}$ such that

where equality of random variables is understood (as always) a.s. and $(\xi , \eta ) = \mathbb {E}\left[\xi \overline{\eta }\right]$ for $\xi , \eta \in L^{2}(\Omega , \mathscr {F}, \mathbb {P})$. Moreover, the spaces $L^{2}[X]$ and $H$ are isometric. Here, for an $L^{2}$-process $X$, $L^{2}[X]$ denotes the mean-square closure of the linear span of the variables $X(t), t \in T$.

Definition 11 of Chapter II. A function $K: T \times T \rightarrow \mathbb {C}$ is called a reproducing kernel of a Hilbert space $H$ consisting of some complex-valued functions defined on a set $T$ if

1. $K(t, \cdot ) \in H$ for every $t \in T$,

2. $\left\langle f, K(t, \cdot ) \right\rangle = f(t)$ for all $f \in H$ and $t \in T$, where $\left\langle \cdot , \cdot \right\rangle$ is the inner product in $H$.

Prove that the (real) Hilbert space $H$ with reproducing kernel $r(s, t) = \min \left\{ s, t\right\} , s, t \in \mathbb {R}_{+}$, which is the covariance function of the Wiener process, consists of the functions

(we integrate with respect to Lebesgue measure). Explain why $L^{2}[W]$, i.e., the closure in $L^{2}(\Omega , \mathscr {F}, \mathbb {P})$ of the linear span of $W(t), t \in \mathbb {R}_{+}$, consists of variables of the form (83), where $Z$ is the orthogonal random measure on $\mathscr {G} \times \Omega$ generated by the Wiener process itself (see Example 1 ), and the function $g$ satisfies condition (84).

Formula (83) from Exercise 12:

Let $X = \left\{ X(t) = \xi e^{i(\eta t+\nu )}, t \in \mathbb {R}\right\}$, where the real random variables $\xi , \eta , \nu$ are such that $\nu$ is uniformly distributed on $[0,2 \pi ]$ and independent of $(\xi , \eta )$, and $\mathbb {E}\left[\xi^{2}\right] < \infty$. Verify that $X$ gives an example of a centered stationary process with covariance function of the form (36), where the measure $G = \mathbb {E}\left[\xi^{2}\right] F$, and $F$ is the probability distribution of the variable $\eta$. Now put $Y = \left\{ Y(t) = \xi \cos (\eta t+\nu ), t \in \mathbb {R}\right\}$. Check that $Y$ is a stationary process. If $X = \left\{ X(t), t \in \mathbb {R}\right\}$ is a stationary complex-valued process, can one assert that $\left\{ \operatorname {Re} X(t), t \in \mathbb {R}\right\}$ and $\left\{ \operatorname {Im} X(t), t \in \mathbb {R}\right\}$ are stationary processes?

Formula (36):

where $G = G(d \lambda )$ is some nonnegative finite ("spectral") measure on $\mathscr {B}(\mathbb {R})$.

(telegraph signal). Let $N = \left\{ N_{t}, t \geqslant 0\right\}$ be a Poisson process with constant intensity $\lambda > 0$. Let the random variable $\zeta$ be independent of the process $N$ and $\mathbb {P}\left(\zeta = -1\right) = \mathbb {P}\left(\zeta = 1\right) = 1 / 2$. Introduce the process called the telegraph signal, or telegraph wave, by setting

Find $\operatorname {cov}(X(s), X(t)), s, t \geqslant 0$ (compare the answer with Exercise 27 of Chapter III).

(generalization of Exercise 15). Let $\left\{ \zeta_{n}, n \geqslant 1\right\}$ be a sequence of independent identically distributed real random variables, independent of the Poisson process $N = \left\{ N_{t}, t \geqslant 0\right\}$, which with probability 1 has unit jumps at the times $\tau_{1} < \tau_{2} < \cdots$ (see Theorem 2 of Chapter II). Put $X(t, \omega ) = \zeta_{k}(\omega )$ for $\tau_{k}(\omega ) \leqslant t < \tau_{k+1}(\omega )$, where $k = 0,1, \ldots \left(\tau_{0} = 0\right)$. If the sequence $\left\{ \tau_{k}(\omega ), k \geqslant 0\right\}$ contains coinciding points (which is possible only with probability 0 ), then let $X(t, \omega ) = 0$ for $t \geqslant 0$. Assuming $\mathbb {E}\left[\zeta_{0}^{2}\right] < \infty$, find the covariance function of the process $X = \left\{ X(t), t \geqslant 0\right\}$.

Theorem 2 of Chapter II (explicit construction of the Poisson process). Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables having the exponential distribution with density

where $\lambda$ is a positive parameter, $i \in \mathbb {N}$. Then the corresponding renewal process (I.10) is a Poisson process with intensity $\lambda$.

For a sequence $\xi_{1}, \xi_{2}, \ldots$ of independent identically distributed nondegenerate nonnegative random variables, the renewal process is defined by the formula

For the shot noise process (see Definition 14 of Chapter VI) with covariance function $r = r(s, t)$, prove that asymptotic stationarity arises in the sense that the limit $\lim_{s \rightarrow \infty } r(s, s+t)$ exists.

Definition 14 of Chapter VI. We say that a stochastic process $X = \left\{ X(t), t \geqslant 0\right\}$ is “shot noise" if

where the bounded Borel function $\phi : \mathbb {R} \rightarrow \mathbb {R}$ has compact support, and the random variables $\tau_{k}$ ($k \in \mathbb {N}$) satisfy condition (VI.84) (which ensures that the series (VI.87) converges for each $t \geqslant 0$ on the set $\Omega_{0}$ of probability one).

Condition (VI.84): on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 13 of Chapter III. The Ornstein-Uhlenbeck process with parameters $\alpha , \beta > 0$ is the process given by the formula

where $W = \left\{ W(t), t \geqslant 0\right\}$ is some standard Brownian motion.

Find the spectral density of the Ornstein-Uhlenbeck process.

Give an example (in connection with Theorem 8) of a nonnegative definite function $R(t), t \in \mathbb {R}$, continuous everywhere except at the point $t = 0$.

Theorem 8 (Bochner-Khinchin). Let $R = R(t), t \in \mathbb {R}$, be a nonnegative definite function continuous at zero. Then for every $t \in \mathbb {R}$

where $G = G(d \lambda )$ is some nonnegative finite ("spectral") measure on $\mathscr {B}(\mathbb {R})$.

Let $h_{1}(t), \ldots , h_{n}(t)$ be complex-valued functions defined on a set $T$. Prove that $r(s, t) = \sum_{j = 1}^{n} h_{j}(s) \overline{h_{j}(t)}, s, t \in T$, is a nonnegative definite function. Give an example of a non-Gaussian process having this covariance function.

Let $\left\{ X_{n}, n \in \mathbb {Z}\right\}$ be a stationary process with mean $a$ and covariance function $R = R(n), n \in \mathbb {Z}$. Prove that as $N \rightarrow \infty$

Suppose that, under the conditions of the preceding exercise,

for some $c, \gamma > 0$ and all $N \in \mathbb {N}$. Prove that then

Show that condition (85) holds if $R(n) = O\left(n^{-\gamma }\right)$ as $n \rightarrow \infty$.

Let $X = \left\{ X_{n}, n \in \mathbb {N}\right\}$ be a centered real Gaussian stationary sequence with covariance function $R = R(n), n \in \mathbb {Z}$. Prove that the condition

is necessary and sufficient for the estimator $\widehat{R}_{N}(m)$ introduced in (48) to be consistent in mean square, i.e., for $m \in \mathbb {Z}$

The estimator (48) of the covariance function $R(m)$ from the observations $X_{0}, X_{1}, \ldots , X_{N-1}$:

Let $X = \left\{ X_{n}, n \in \mathbb {Z}\right\}$ be a moving-average process of the form (39). Assume that $\sum_{k = -\infty }^{\infty }\left|c_{k}\right| < \infty$ and that $\varepsilon_{k}, k \in \mathbb {Z}$, are independent identically distributed random variables with $\mathbb {E}\left[\varepsilon_{k}^{4}\right] < \infty$. Let $\widehat{f}_{N}(\lambda )$ be the periodogram (49). Find $\lim_{N \rightarrow \infty } \operatorname {cov}\left(\widehat{f}_{N}(\lambda ), \widehat{f}_{N}(\nu )\right)$ for frequencies $\lambda , \nu \in [-\pi , \pi ]$.

Formula (39): a moving-average process is a process $X = \left\{ X_{t}, t \in \mathbb {Z}\right\}$ admitting the representation

where $\left\{ c_{k}\right\}_{k \in \mathbb {Z}} \in l^{2}$, i.e., $\sum_{k = -\infty }^{\infty }\left|c_{k}\right|^{2} < \infty$, $\varepsilon = \left\{ \varepsilon_{n}, n \in \mathbb {Z}\right\}$ is white noise, i.e., a centered and orthonormal (in $L^{2}(\Omega )$) process (Definition 10), and the series converges in mean square.

Formula (49): the periodogram (denoted $\widehat{f}_{N}(\lambda )$ in the exercise) based on the observations $X_{0}, X_{1}, \ldots , X_{N-1}$ is

Definition 13 of Chapter III. The Ornstein-Uhlenbeck process with parameters $\alpha , \beta > 0$ is the process given by the formula

where $W = \left\{ W(t), t \geqslant 0\right\}$ is some standard Brownian motion.

Let $Z$ be the orthogonal random measure corresponding to the Ornstein-Uhlenbeck process with parameters $\alpha , \beta > 0$ (see Exercise 27 of Chapter III). Prove that the process

(by continuity, we set $\left(e^{i t \lambda }-1\right) / i \lambda = t$ for $\lambda = 0$ ) is a Wiener process. (The existence of a continuous modification of this process is easily obtained by means of Kolmogorov's theorem (Theorem 18 of Chapter II).)

Theorem 18 of Chapter II (Kolmogorov). Let $X = \left\{ X(t), t \in [a, b]\right\}$ be a real stochastic process such that for some $\alpha , \varepsilon > 0$ and $C = C(\alpha , \varepsilon ) > 0$

Then the process $X$ has a continuous modification.

A matrix function $C(s, t) = \left(C_{j k}(s, t)\right)_{j, k = 1}^{m}, s, t \in T$, with complex-valued entries is called nonnegative (or positive) definite if for all $n \geqslant 1$, any $t_{1}, \ldots , t_{n} \in T$, and column vectors $z_{1}, \ldots , z_{n} \in \mathbb {C}^{m}$

where the symbol * denotes transposition and complex conjugation of all entries.

A vector process $X = \left\{ X(t), t \in T\right\}$ with values in $\mathbb {C}^{m}$ is called an $L^{2}$-process if $\mathbb {E}\left[\left\| X(t)\right\|^{2}\right] < \infty$ for all $t \in T$.

Define the vector mean function and the matrix covariance (correlation) function of the $L^{2}$-process $X$ by setting

Prove that a matrix function $R(s, t), s, t \in T$ (with complex entries) is the covariance function of some vector $L^{2}$-process $X = \left\{ X(t), t \in T\right\}$ if and only if it is nonnegative definite.

Prove that for the convergence of the series (91) in $L^{2}(\Omega , \mathscr {F}, \mathbb {P})$ it suffices that $\sum_{k = -\infty }^{\infty }\left|A_{k}\right|^{2} < \infty$, where $\left|A\right|: = \left(\operatorname {Tr} A A^{*}\right)^{1 / 2} = \left(\sum_{j, q = 1}^{m}\left|a_{j q}\right|^{2}\right)^{1 / 2}$ for $A = \left(a_{j q}\right)_{j, q = 1}^{m}$. Find the covariance function of a vector moving-average process.

A vector moving-average process is introduced by the formula

where $\varepsilon = \left\{ \varepsilon_{n}, n \in \mathbb {Z}\right\}$ is white noise, and $A_{k}, k \in \mathbb {Z}$, are matrices (operators) mapping $\mathbb {C}^{m}$ into $\mathbb {C}^{m}$. The series in (91) is understood as converging in mean square.

A sequence of vectors $\varepsilon_{n}, n \in \mathbb {Z}$, with values in $\mathbb {C}^{m}$ is said to be white noise if in (90)

here $I$ is the identity matrix of order $m$.

Let $T$ be a linear space. An $L^{2}$-process $X = \left\{ X(t), t \in T\right\}$ with values in $\mathbb {C}^{m}$ is called stationary in the wide sense if

where $a(t) = \mathbb {E}\left[X(t)\right] \in \mathbb {C}^{m}$ is the vector mean function and $r(s, t) = \mathbb {E}\left[(X(s)-a(s))(X(t)-a(t))^{*}\right]$ is the matrix covariance function of the $L^{2}$-process $X$, i.e., of a process with values in $\mathbb {C}^{m}$ such that $\mathbb {E}\left[\left\| X(t)\right\|^{2}\right] < \infty$ for all $t \in T$.

Let us introduce the operation of mean-square differentiation. We define the derivative $X^{\prime }(t)$ of a (complex-valued) $L^{2}$-process $X$ given in a neighborhood of a point $t \in \mathbb {R}$ as the derivative of the function $X(t)$ with values in the Banach (Hilbert) space $L^{2}(\Omega ) = L^{2}(\Omega , \mathscr {F}, \mathbb {P})$, i.e., as a random element (denoted $X^{\prime }(t)$ ) such that

The derivative in probability is defined in the same way, except that instead of mean-square convergence (95) only convergence in probability is required.

Prove that for a Poisson process $N = \left\{ N_{t}, t \geqslant 0\right\}$ of constant intensity $\lambda > 0$ the derivative in probability at each point $t \in \mathbb {R}_{+}$ is equal to zero a.s.