Given a sequence of n real numbers {Si}i≤n, we consider the longest weakly increasing subsequence, namely i1<i2<⋯<iL
Nearby in the stack
with
Sik≤Sik+1
and
L
maximal. When the elements
Si
are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that
EL=(2+o(1))n
. We consider the case when
{Si}i≤n
is a random walk on
R
with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies
EL≥cn
. Our main result is an upper bound
EL≤n1/2+o(1)
, establishing the leading asymptotic behavior. If
{Si}i≤n
is a simple random walk on
Z
, we improve the lower bound by showing that
EL≥cnlogn
. We also show that if
{Si}
is a simple random walk in
Z2
, then there is a subsequence of
{Si}i≤n
of expected length at least
cn1/3
that is increasing in each coordinate. The above one-dimensional result yields an upper bound of
n1/2+o(1)
. The problem of determining the correct exponent remains open.