diff --git a/model/src/pyrenew/process/ar.py b/model/src/pyrenew/process/ar.py
index 7d3d2969..412f05df 100644
--- a/model/src/pyrenew/process/ar.py
+++ b/model/src/pyrenew/process/ar.py
@@ -78,7 +78,7 @@ def _ar_scanner(carry, next):
         )
 
         last, ts = lax.scan(_ar_scanner, inits - self.mean, noise)
-        return (self.mean + ts.flatten(),)
+        return (jnp.hstack([inits, self.mean + ts.flatten()]),)
 
     @staticmethod
     def validate():
diff --git a/model/src/test/test_ar_process.py b/model/src/test/test_ar_process.py
index af658bef..b1a6d9d2 100755
--- a/model/src/test/test_ar_process.py
+++ b/model/src/test/test_ar_process.py
@@ -1,7 +1,6 @@
-# -*- coding: utf-8 -*-
-
 import jax.numpy as jnp
 import numpyro
+from numpy.testing import assert_almost_equal
 from pyrenew.process import ARProcess
 
 
@@ -12,11 +11,29 @@ def test_ar_can_be_sampled():
     """
     ar1 = ARProcess(5, jnp.array([0.95]), jnp.array([0.5]))
     with numpyro.handlers.seed(rng_seed=62):
-        ## can sample with and without inits
+        # can sample with and without inits
         ar1.sample(3532, inits=jnp.array([50.0]))
         ar1.sample(5023)
 
     ar3 = ARProcess(5, jnp.array([0.05, 0.025, 0.025]), jnp.array([0.5]))
     with numpyro.handlers.seed(rng_seed=62):
+        # can sample with and without inits
         ar3.sample(1230)
         ar3.sample(52, inits=jnp.array([50.0, 49.9, 48.2]))
+
+
+def test_ar_samples_correctly_distributed():
+    """
+    Check that AR processes have correctly-
+    distributed steps.
+    """
+    ar_mean = 5
+    noise_sd = jnp.array([0.5])
+    ar_inits = jnp.array([25.0])
+    ar1 = ARProcess(ar_mean, jnp.array([0.75]), noise_sd)
+    with numpyro.handlers.seed(rng_seed=62):
+        # check it regresses to mean
+        # when started away from it
+        long_ts, *_ = ar1.sample(10000, inits=ar_inits)
+        assert_almost_equal(long_ts[0], ar_inits)
+        assert jnp.abs(long_ts[-1] - ar_mean) < 4 * noise_sd
diff --git a/model/src/test/test_random_walk.py b/model/src/test/test_random_walk.py
index 33e4fa96..43dc9043 100755
--- a/model/src/test/test_random_walk.py
+++ b/model/src/test/test_random_walk.py
@@ -1,8 +1,7 @@
-# -*- coding: utf-8 -*-
-
 import jax.numpy as jnp
 import numpyro
 import numpyro.distributions as dist
+from numpy.testing import assert_almost_equal
 from pyrenew.process import SimpleRandomWalkProcess
 
 
@@ -30,15 +29,13 @@ def test_rw_samples_correctly_distributed():
         [0, 2.253, -3.2521, 1052, 1e-6], [1, 0.025, 3, 1, 0.02]
     ):
         rw_normal = SimpleRandomWalkProcess(dist.Normal(step_mean, step_sd))
-
+        init_arr = jnp.array([532.0])
         with numpyro.handlers.seed(rng_seed=62):
-            samples, *_ = rw_normal.sample(n_samples, init=jnp.array([50.0]))
+            samples, *_ = rw_normal.sample(n_samples, init=init_arr)
 
             # diffs should not be greater than
             # 4 sigma
             diffs = jnp.diff(samples)
-            print(samples)
-            print(diffs)
             assert jnp.all(jnp.abs(diffs - step_mean) < 4 * step_sd)
 
             # sample mean of diffs should be
@@ -52,3 +49,6 @@ def test_rw_samples_correctly_distributed():
             # should be approximately equal
             # to the step sd
             assert jnp.abs(jnp.log(jnp.std(diffs) / step_sd)) < jnp.log(1.1)
+
+            # first value should be the init value
+            assert_almost_equal(samples[0], init_arr)