jax.random.randint cannot generate values equal to DTYPE_MAX
According to randint docstring: `Sample uniform random values in [minval, maxval) with given shape/dtype.`

However because of the conversion here https://github.com/google/jax/blob/5bbb449ae5849c508194c8eb5b10c101f1fa22ae/jax/_src/random.py#L439 passing `maxval={dtype max value + 1}` will generate an all-zero array.
The behavior in numpy is different (and correct):
```
jax.random.randint(jax.random.PRNGKey(0), shape=(10,), minval=0, maxval=256, dtype=jnp.uint8)
>> DeviceArray([0, 0, 0, 0, 0, 0, 0, 0, 0, 0], dtype=uint8)

np.random.randint(size=(10,), low=0, high=256, dtype=np.uint8)
>> array([ 87, 136, 195, 254,  95,  94, 113,   7,  66,  13], dtype=uint8)
```

A workaround is to not pass the dtype to randint (or use a dtype with a higher maximum value), and cast the output:
```
jax.random.randint(jax.random.PRNGKey(0), shape=(10,), minval=0, maxval=256).astype(jnp.uint8)
>> DeviceArray([178, 233,  99, 238,  39, 149,  46, 139,  68, 142], dtype=uint8)
```
diff --git a/jax/_src/numpy/lax_numpy.py b/jax/_src/numpy/lax_numpy.py
index 7e3306743..7d8b4a434 100644
--- a/jax/_src/numpy/lax_numpy.py
+++ b/jax/_src/numpy/lax_numpy.py
@@ -330,6 +330,47 @@ def _promote_args_inexact(fun_name, *args):
   _check_no_float0s(fun_name, *args)
   return _promote_shapes(fun_name, *_promote_dtypes_inexact(*args))
 
+def _convert_and_clip_integer(val, dtype):
+  """
+  Convert integer-typed val to specified integer dtype, clipping to dtype
+  range rather than wrapping.
+
+  Args:
+    val: value to be converted
+    dtype: dtype of output
+
+  Returns:
+    equivalent of val in new dtype
+
+  Examples
+  --------
+  Normal integer type conversion will wrap:
+
+  >>> val = jnp.uint32(0xFFFFFFFF)
+  >>> val.astype('int32')
+  DeviceArray(-1, dtype=int32)
+
+  This function clips to the values representable in the new type:
+
+  >>> _convert_and_clip_integer(val, 'int32')
+  DeviceArray(2147483647, dtype=int32)
+  """
+  val = val if isinstance(val, ndarray) else asarray(val)
+  dtype = dtypes.canonicalize_dtype(dtype)
+  if not (issubdtype(dtype, integer) and issubdtype(val.dtype, integer)):
+    raise TypeError("_convert_and_clip_integer only accepts integer dtypes.")
+
+  val_dtype = dtypes.canonicalize_dtype(val.dtype)
+  if val_dtype != val.dtype:
+    # TODO(jakevdp): this is a weird corner case; need to figure out how to handle it.
+    # This happens in X32 mode and can either come from a jax value created in another
+    # context, or a Python integer converted to int64.
+    pass
+  min_val = _constant_like(val, _max(iinfo(dtype).min, iinfo(val_dtype).min))
+  max_val = _constant_like(val, _min(iinfo(dtype).max, iinfo(val_dtype).max))
+  return clip(val, min_val, max_val).astype(dtype)
+
+
 def _constant_like(x, const):
   return np.array(const, dtype=_dtype(x))
 
diff --git a/jax/_src/random.py b/jax/_src/random.py
index 6df44c3cb..8cdea7600 100644
--- a/jax/_src/random.py
+++ b/jax/_src/random.py
@@ -25,7 +25,7 @@ from jax import numpy as jnp
 from jax import dtypes
 from jax.core import NamedShape
 from jax.api import jit, vmap
-from jax._src.numpy.lax_numpy import _constant_like, asarray
+from jax._src.numpy.lax_numpy import _constant_like, _convert_and_clip_integer, asarray
 from jax.lib import xla_bridge
 from jax.lib import xla_client
 from jax.lib import cuda_prng
@@ -441,20 +441,28 @@ def randint(key: jnp.ndarray,
 def _randint(key, shape, minval, maxval, dtype):
   _check_shape("randint", shape, np.shape(minval), np.shape(maxval))
   if not jnp.issubdtype(dtype, np.integer):
-    raise TypeError("randint only accepts integer dtypes.")
-
-  minval = lax.convert_element_type(minval, dtype)
-  maxval = lax.convert_element_type(maxval, dtype)
+    raise TypeError(f"randint only accepts integer dtypes, got {dtype}")
+
+  minval = _asarray(minval)
+  maxval = _asarray(maxval)
+  if not jnp.issubdtype(minval.dtype, np.integer):
+    minval = minval.astype(int)
+  if not jnp.issubdtype(maxval.dtype, np.integer):
+    maxval = maxval.astype(int)
+
+  # Flag where maxval is greater than the maximum value of dtype
+  # in order to handle cases like randint(key, shape, 0, 256, 'uint8')
+  maxval_out_of_range = lax.gt(
+    maxval, _convert_and_clip_integer(jnp.array(jnp.iinfo(dtype).max, dtype), maxval.dtype))
+
+  minval = _convert_and_clip_integer(minval, dtype)
+  maxval = _convert_and_clip_integer(maxval, dtype)
   minval = lax.broadcast_to_rank(minval, len(shape))
   maxval = lax.broadcast_to_rank(maxval, len(shape))
   nbits = jnp.iinfo(dtype).bits
 
   if nbits not in (8, 16, 32, 64):
-    raise TypeError("randint only accepts 8-, 16-, 32-, or 64-bit dtypes.")
-
-  # if we don't have minval < maxval, just always return minval
-  # https://github.com/google/jax/issues/222
-  maxval = lax.max(lax.add(minval, np.array(1, dtype)), maxval)
+    raise TypeError(f"randint only accepts 8-, 16-, 32-, or 64-bit dtypes, got {dtype}")
 
   # This algorithm is biased whenever (maxval - minval) is not a power of 2.
   # We generate double the number of random bits required by the dtype so as to
@@ -466,6 +474,18 @@ def _randint(key, shape, minval, maxval, dtype):
   unsigned_dtype = _UINT_DTYPES[nbits]
   span = lax.convert_element_type(maxval - minval, unsigned_dtype)
 
+  # Ensure that span=1 when maxval <= minval, so minval is always returned;
+  # https://github.com/google/jax/issues/222
+  span = lax.select(maxval <= minval, lax.full_like(span, 1), span)
+
+  # When maxval is out of range, the span has to be one larger.
+  # If span is already the maximum representable value, this will wrap to zero,
+  # causing remainders below to have no effect, which is the correct semantics.
+  span = lax.select(
+    maxval_out_of_range & (maxval > minval),
+    lax.add(span, lax._const(span, 1)),
+    span)
+
   # To compute a remainder operation on an integer that might have twice as many
   # bits as we can represent in the native unsigned dtype, we compute a
   # multiplier equal to 2**nbits % span. To avoid overflow, we use the identity:
diff --git a/tests/random_test.py b/tests/random_test.py
index 63c5826fa..447940f17 100644
--- a/tests/random_test.py
+++ b/tests/random_test.py
@@ -26,6 +26,7 @@ import scipy.stats
 
 from jax import api
 from jax import core
+from jax import dtypes
 from jax import grad
 from jax import lax
 from jax import numpy as jnp
@@ -983,6 +984,34 @@ class LaxRandomTest(jtu.JaxTestCase):
       api.jit(random.split)(key)
     self.assertEqual(count[0], 1)  # 1 for the argument device_put
 
+  @parameterized.named_parameters(jtu.cases_from_list(
+      {"testcase_name": f"_dtype={dtype}", "dtype": dtype}
+      for dtype in int_dtypes + uint_dtypes))
+  def test_randint_bounds(self, dtype):
+    min = np.iinfo(dtype).min
+    max = np.iinfo(dtype).max
+    key = random.PRNGKey(1701)
+    shape = (10,)
+    if np.iinfo(dtype).bits < np.iinfo(dtypes.canonicalize_dtype(int)).bits:
+      expected = random.randint(key, shape, min, max, dtype)
+      self.assertArraysEqual(expected, random.randint(key, shape, min - 12345, max + 12345, dtype))
+    else:
+      self.assertRaises(OverflowError, random.randint, key, shape, min - 12345, max + 12345, dtype)
+
+  def test_randint_out_of_range(self):
+    key = random.PRNGKey(0)
+
+    r = random.randint(key, (10,), 255, 256, np.uint8)
+    self.assertAllClose(r, jnp.full_like(r, 255))
+
+    r = random.randint(key, (1000,), -128, 128, np.int8)
+    self.assertGreater((r == -128).sum(), 0)
+    self.assertGreater((r == 127).sum(), 0)
+
+    r = random.randint(key, (1000,), -1000, 1000, np.uint8)
+    self.assertGreater((r == 0).sum(), 0)
+    self.assertGreater((r == 255).sum(), 0)
+
 
 if __name__ == "__main__":
   absltest.main(testLoader=jtu.JaxTestLoader())
['jax/_src/numpy/lax_numpy.py', 'jax/_src/random.py', 'tests/random_test.py']