Strong law of large numbers for a function of the local times of a transient random walk in \({\mathbb{Z}}^d\)

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Abstract: For an arbitrary transient random walk

(S_{n})_{n g e 0}

in

m a t h b b Z^{d}

,

d g e 1

, we prove a strong law of large numbers for the spatial sum

s u m_{x i n m a t h b b Z^{d}} f (l (n, x))

of a function

f

of the local times

l (n, x) = s u m_{i = 0}^{n} m a t h b b I S_{i} = x

. Particular cases are the number of (a) visited sites (first time considered by Dvoretzky and ErdH{o}s), which corresponds to a function

f (i) = m a t h b b I i g e 1

; (b)

a l p h a

-fold self-intersections of the random walk (studied by Becker and K"{o}nig), which corresponds to

f (i) = i^{a} l p h a

; (c) sites visited by the random walk exactly

j

times (considered by ErdH{o}s and Taylor and by Pitt), where

f (i) = m a t h b b I i = j

.

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