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Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is nothe autocorrelation is zero for all lags since the variables are independent.

Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.

Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but the autocorrelation is zero for all lags since the variables are independent.

2 deleted 4 characters in body
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Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.

Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.

Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.

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Neither is true. Consider the two following examples:

(1) Let $\xi \sim N(0,1)$ and define the stochastic process $X_i=\xi, i=1,2,\dots$. it's easy to check that this process is (strongly) stationary, while at the same time $\mathrm{Cov}(X_i, X_j)=1, \forall i,j$.

(2) Consider a stochastic process consisting of independent Gaussian variables $X_i \sim N(0, i), i=1,2,\dots$. This process is clearly not stationary, but there is no autocorrelation since the variables are independent.