Abstract
Let
left brace upper U left parenthesis n right parenthesis right brace Subscript n greater than or equals 0
{
U
(
n
)
}
n
≥
0
$\{U(n)\}_{n \geq 0}$
be a sequence of independent random variables such that
U
(
n
) is distributed uniformly on
StartSet 0 comma 1 comma 2 comma ellipsis comma n EndSet
{
0
,
1
,
2
,
…
,
n
}
$\{0, 1, 2, \dots, n\}$
. The Ulam–Kac adder is the history-dependent random sequence defined by
upper X Subscript n plus 1 Baseline equals upper X Subscript n Baseline plus upper X Subscript upper U left parenthesis n right parenthesis
X
n
+
1
=
X
n
+
X
U
(
n
)
$X_{n + 1} = X_{n} + X_{U(n)}$
with the initial condition
upper X 0 equals 1
X
0
=
1
$X_0 = 1$
. We show that for each
m greater than or equals 1
m
≥
1
$m \geq 1$
,
log double struck upper E left bracket upper X Subscript n Superscript m Baseline right bracket divided by StartRoot n EndRoot
log
E
[
X
n
m
]
/
n
$\log \mathbb{E}[X_n^m]/\sqrt{n}$
approaches a constant
c Subscript m
c
m
$c_m$
as
n right arrow normal infinity
n
→
∞
$n \to \infty$
. For a given
m
, a method is provided which allows the high-accuracy numerical computation of
c Subscript m
c
m
$c_m$
. For arbitrary
m
, loose bounds are provided.