Given a sequence of n real numbers {Si
}
i⩽n
, we consider the longest weakly increasing subsequence, namely i
1 < i
2 < . . . < iL
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
${\mathbb E} L=(2+o(1)) \sqrt{n}$
.
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
${\mathbb E} L\geq c\sqrt{n}$
. Our main result is an upper bound
${\mathbb E} L\leq n^{1/2 + o(1)}$
, establishing the leading asymptotic behavior. If {Si
}
i⩽n
is a simple random walk on ℤ, we improve the lower bound by showing that
${\mathbb E} L \geq c\sqrt{n} \log{n}$
.
We also show that if {
S
i
} is a simple random walk in ℤ2, then there is a subsequence of {
S
i
}
i⩽n
of expected length at least cn
1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n
1/2+o(1). The problem of determining the correct exponent remains open.