8 2. HISTORY

is finite for all negative λ and for all suﬃciently small positive λ and is infinite

for all suﬃciently large positive λ. The critical value λ0 at which the expectation

switches from being finite to infinite was found by Bass and Chen [2]. It turns out

that the expectation in (2.11) is finite if λ κ(2,

2)−4

and infinite if λ κ(2,

2)−4,

where κ(2, 2) is the best constant in a Gagliardo-Nirenberg inequality; see (1.5).

We mention that our paper [4] has moderate deviations results for the number

of self-intersections of random walks which are similar in form to those of this paper,

although the proofs are different.