diff --git a/src/pyrecest/filters/hyperspherical_ukf.py b/src/pyrecest/filters/hyperspherical_ukf.py
index de26c48215..de2caa9e76 100644
--- a/src/pyrecest/filters/hyperspherical_ukf.py
+++ b/src/pyrecest/filters/hyperspherical_ukf.py
@@ -201,10 +201,13 @@ def predict_nonlinear_arbitrary_noise(  # pylint: disable=too-many-locals
                 "noise_samples and noise_weights must contain the same number "
                 "of samples."
             )
+        if noise_weights.shape[0] == 0:
+            raise ValueError("noise_weights must contain at least one sample.")
         if not all(isfinite(noise_weights)):
             raise ValueError("noise_weights must be finite.")
         if not all(noise_weights > 0):
             raise ValueError("noise_weights must be strictly positive.")
+        noise_weights = noise_weights / noise_weights.max()
         noise_weights = noise_weights / noise_weights.sum()
 
         mu = reshape(asarray(self._filter_state.mu, dtype=float64), (-1,))
diff --git a/tests/filters/test_hyperspherical_ukf_arbitrary_noise_normalization.py b/tests/filters/test_hyperspherical_ukf_arbitrary_noise_normalization.py
index d65a6b4e75..a5de6079da 100644
--- a/tests/filters/test_hyperspherical_ukf_arbitrary_noise_normalization.py
+++ b/tests/filters/test_hyperspherical_ukf_arbitrary_noise_normalization.py
@@ -8,11 +8,11 @@
 from pyrecest.filters.hyperspherical_ukf import HypersphericalUKF
 
 
+@unittest.skipIf(
+    pyrecest.backend.__backend_name__ in ("pytorch", "jax"),
+    reason="Arbitrary-noise prediction is not supported on this backend",
+)
 class HypersphericalUKFArbitraryNoiseNormalizationTest(unittest.TestCase):
-    @unittest.skipIf(
-        pyrecest.backend.__backend_name__ in ("pytorch", "jax"),
-        reason="Arbitrary-noise prediction is not supported on this backend",
-    )
     def test_radial_model_scale_does_not_create_spurious_covariance(self):
         ukf = HypersphericalUKF(dim=2, alpha=1.0)
         noise_samples = np.array([[0.0, 1.0]])
@@ -36,3 +36,37 @@ def scaled_same_direction(_x, v):
             np.zeros((2, 2)),
             atol=1e-12,
         )
+
+    def test_maximum_finite_weights_do_not_overflow(self):
+        ukf = HypersphericalUKF(dim=2, alpha=1.0)
+        noise_samples = np.array([[0.0, 1.0]])
+        noise_weights = np.full(2, np.finfo(float).max)
+
+        def fixed_direction(_x, _v):
+            return array([1.0, 0.0])
+
+        with np.errstate(over="raise", invalid="raise", divide="raise"):
+            ukf.predict_nonlinear_arbitrary_noise(
+                fixed_direction, noise_samples, noise_weights
+            )
+
+        npt.assert_allclose(
+            np.asarray(ukf.filter_state.mu, dtype=float),
+            np.array([1.0, 0.0]),
+            atol=1e-12,
+        )
+        npt.assert_allclose(
+            np.asarray(ukf.filter_state.C, dtype=float),
+            np.zeros((2, 2)),
+            atol=1e-12,
+        )
+
+    def test_rejects_empty_noise_support(self):
+        ukf = HypersphericalUKF(dim=2, alpha=1.0)
+
+        with self.assertRaisesRegex(ValueError, "at least one sample"):
+            ukf.predict_nonlinear_arbitrary_noise(
+                lambda x, _v: x,
+                np.empty((1, 0)),
+                np.empty((0,)),
+            )