Examples
Examples
Outline
1
Examples
Arthur Berg
Examples
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Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
E[Xt + Yt ] = µX + µY
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
E[Xt + Yt ] = µX + µY
γx+y (h) = cov(Xt+h + Yt+h , Xt + Yt )
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
E[Xt + Yt ] = µX + µY
γx+y (h) = cov(Xt+h + Yt+h , Xt + Yt )
= cov(Xt+h , Xt ) + cov(Yt+h , Yt ) + 0
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
E[Xt + Yt ] = µX + µY
γx+y (h) = cov(Xt+h + Yt+h , Xt + Yt )
= cov(Xt+h , Xt ) + cov(Yt+h , Yt ) + 0
= γx (h) + γy (h)
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Sum of Uncorrelated Stationary Processes
If {Xt } and {Yt } are uncorrelated stationary processes, i.e. if Xs and Yt are
uncorrelated for every s and t, show that {Xt + Yt } is stationary and compute its
autocovariance function in terms of the autocovariance functions of {Xt } and
{Yt }.
var(Xt + Yt ) = var(Xt ) + var(Yt ) < ∞
E[Xt + Yt ] = µX + µY
γx+y (h) = cov(Xt+h + Yt+h , Xt + Yt )
= cov(Xt+h , Xt ) + cov(Yt+h , Yt ) + 0
= γx (h) + γy (h)
Since the mean and autocovariance functions are free of t, the process {Xt + Yt }
is stationary.
Arthur Berg
Examples
3/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(a) Var[Zt ] = σ 2 (b2 + c2 ) < ∞, E[Zt ] = a, and

2 2
2

σ (b + c ) if h = 0
γZ (h) = bcσ 2
if |h| = 1


0
else
is independent of t.
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(b) Var[Zt ] = b2 σ 2 < ∞, E[Zt ] = a, and γX (h) = b2 σ 2 is independent of t.
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(c) Var[Zt ] = σ 2 (cos2 (ct) + sin2 (ct)) = σ 2 , E[Zt ] = 0, and
γZ (h) = cov(w1 cos(c(t + h)) + w2 sin(c(t + h)), w1 cos(ct) + w2 sin(ct))
= σ 2 (cos(ct) cos(c(t + h)) + sin(ct) sin(c(t + h)))
= σ 2 (cos(c(t + h) − ct)) note: cos(x − y) = cos x cos y + sin x sin y
= σ 2 cos(ch)
is independent of t.
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(d) Not stationary!
var(Zt ) = cos2 (ct)
So, for example, var(Z0 ) 6= var(Z1 ).
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(e)
Var[Zt ] = E[Zt Zt−1 ]2 = σ 4 < ∞
E[Zt ] = 0 and
γZ (h) = cov(Zt+h Zt+h−1 , Zt Zt−1 ) = 0
is independent of t. In particular, this is an example of a dependent white
noise process.
Arthur Berg
Examples
4/ 4
Examples
Now your turn!
Stationary?
Let wt ∼ iid N (0, σ 2 ), t ∈ Z and a, b, c ∈ (0, 1). Which of the following
processes are weakly stationary? For each stationary process specify the mean
and autocovariance function.
(a) Zt = a + bwt + cwt−1
(d) Zt = w0 cos(ct)
(b) Zt = a + bw0
(e) Zt = wt wt−1
(c) Zt = w1 cos(ct) + w2 sin(ct)
(f) Zt = wt cos(ct)+wt−1 sin(ct)
Note: you should show Var[Zt ] < ∞, and E[Zt ] and γZ (h) are independent of t.
(f) var(Zt ) = σ 2 (cos2 (ct) + sin2 (ct)) = σ 2 < ∞.
Computing the autocovariance function at h = 1 gives
cov(wt cos(ct) + wt−1 sin(ct), wt+1 cos(c(t + 1)) + wt sin(c(t + 1)))
= cov(wt cos(ct), wt sin(c(t + 1))) = cos(ct) sin(c(t + 1))
When t = 0, we get cov(Z0 , Z1 ) = sin(c). But when t = −1, we get
cov(Z−1 , Z0 ) = 0 Therefore Zt is not stationary!
Arthur Berg
Examples
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