1 Stochastic process: \(y_t = w_{t-1} + w_t\) for \(t = 1, 2, \ldots\), where \(w_t \sim N(0, \sigma^2_w)\).

  • Mean: \(E[y_t] = E[w_{t-1} + w_t] = E[w_{t-1}] + E[w_t] = 0 + 0 = 0\)

  • Variance: \(Var[y_t] = Var[w_{t-1} + w_t] = Var[w_{t-1}] + Var[w_t] = 2\sigma^2_w\)

  • Autocovariance: \(Cov[y_t, y_{t+k}] = Cov[w_{t-1} + w_t, w_{t+k-1} + w_{t+k}]\) for \(k \geq 1\)

    Since \(w_t\) and \(w_{t+k}\) are independent for \(k \geq 1\), the autocovariance function is \(0\) for \(k \geq 1\).

    Therefore, this process is not stationary.

2.Discrete stochastic process: \(y_t; t \in \mathbb{N}\) where \(y_t = A\), and \(A \sim U(3, 7)\).

  • Mean: \(E[y_t] = \frac{3+7}{2} = 5\)

  • Variance: \(Var[y_t] = \frac{(7-3)^2}{12} = \frac{4}{3}\)

  • Autocovariance: Since \(y_t\) is constant, the autocovariance function is \(0\) for \(k \geq 1\).

    Therefore, this process is weak stationary.

3.Discrete stochastic process: \(y_t; t \in \mathbb{N}\) where \(y_t = tA\), and \(A \sim U(3, 7)\).

  • Mean: \(E[y_t] = tE[A] = t \frac{3+7}{2} = 5t\)

  • Variance: \(Var[y_t] = t^2 Var[A] = t^2 \frac{4}{3} = \frac{4}{3}t^2\)

  • Autocovariance: Since the mean and variance are functions of \(t\), the autocovariance is also a function of \(t\), hence not stationary.

    Therefore, this process is not stationary.

4. Stochastic process: \(y_0 = \delta \leq \infty\) and \(y_t = y_{t-1} + w_t\) for \(t = 1, 2, \ldots\), where \(w_t \sim N(0, \sigma^2_w)\).

  • Mean: \(E[y_t] = E[y_{t-1}] + E[w_t] = E[y_{t-1}] = \delta\) (since the increments have zero mean)

  • Variance: \(Var[y_t] = Var[y_{t-1}] + Var[w_t] = Var[y_{t-1}] + \sigma^2_w\)

  • Autocovariance: \(Cov[y_t, y_{t+k}] = Cov[y_{t-1} + w_t, y_{t+k-1} + w_{t+k}]\) for \(k \geq 1\)

    \(= Cov[y_{t-1}, y_{t+k-1}] = Cov[y_{t}, y_{t+k}]\) (since \(w_t\) and \(w_{t+k}\) are independent)

    \(= Cov[y_{t}, y_{t+k}]\) (since the increments are identically distributed)

    This implies that the autocovariance is independent of time, hence stationary.

    Therefore, this process is strong stationary.