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.
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