= Solution
\b[A stationary <causal time series> exists.] The <infinite moving-average representation>
$$
X_t=\sum_{j=0}^\infty 2^{-j}\varepsilon_{t-j}
$$
converges in the sense of <mean-square convergence> because the driving variables are <independent random variables> and $\sum_j4^{-j}<\infty$. Shifting the series gives $X_t=\tfrac12X_{t-1}+\varepsilon_t$. Its <expected value> is zero and its <autocovariance> is
$$
\boxed{\gamma(h)=\frac{4\sigma^2}{3}\,2^{-|h|}.}
$$
Thus it is a <weakly stationary process>; since the driving variables have a <normal distribution>, it is also a <Gaussian process> and a <strictly stationary process>. Throughout the stationarity discussion, take $\sigma^2>0$; degenerate zero noise permits trivial constant solutions.
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