BUG: rolling mean and sum not numerical stable for all nan window

doc/source/whatsnew/v1.3.0.rst

@@ -846,6 +846,7 @@ Groupby/resample/rolling
 - Bug in :meth:`GroupBy.cummin` and :meth:`GroupBy.cummax` incorrectly rounding integer values near the ``int64`` implementations bounds (:issue:`40767`)
 - Bug in :meth:`.GroupBy.rank` with nullable dtypes incorrectly raising ``TypeError`` (:issue:`41010`)
 - Bug in :meth:`.GroupBy.cummin` and :meth:`.GroupBy.cummax` computing wrong result with nullable data types too large to roundtrip when casting to float (:issue:`37493`)
+- Bug in :meth:`DataFrame.rolling` returning mean zero for all ``NaN`` window with ``min_periods=0`` if calculation is not numerical stable (:issue:`41053`)
 
 Reshaping
 ^^^^^^^^^

pandas/_libs/window/aggregations.pyx

@@ -170,7 +170,7 @@ cdef inline float64_t calc_mean(int64_t minp, Py_ssize_t nobs,
     cdef:
         float64_t result
 
-    if nobs >= minp:
+    if nobs >= minp and nobs > 0:
         result = sum_x / <float64_t>nobs
         if neg_ct == 0 and result < 0:
             # all positive

pandas/tests/window/test_rolling.py

@@ -1357,3 +1357,49 @@ def test_rolling_std_small_values():
     result = s.rolling(2).std()
     expected = Series([np.nan, 7.071068e-9, 7.071068e-9])
     tm.assert_series_equal(result, expected, atol=1.0e-15, rtol=1.0e-15)
+
+
+@pytest.mark.parametrize(
+    "start, exp_values",
+    [
+        (1, [0.03, 0.0155, 0.0155, 0.011, 0.01025]),
+        (2, [0.001, 0.001, 0.0015, 0.00366666]),
+    ],
+)
+def test_rolling_mean_all_nan_window_floating_artifacts(start, exp_values):
+    # GH#41053
+    df = DataFrame(
+        [
+            0.03,
+            0.03,
+            0.001,
+            np.NaN,
+            0.002,
+            0.008,
+            np.NaN,
+            np.NaN,
+            np.NaN,
+            np.NaN,
+            np.NaN,
+            np.NaN,
+            0.005,
+            0.2,
+        ]
+    )
+
+    values = exp_values + [
+        0.00366666,
+        0.005,
+        0.005,
+        0.008,
+        np.NaN,
+        np.NaN,
+        0.005,
+        0.102500,
+    ]
+    expected = DataFrame(
+        values,
+        index=list(range(start, len(values) + start)),
+    )
+    result = df.iloc[start:].rolling(5, min_periods=0).mean()
+    tm.assert_frame_equal(result, expected)