For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1$$ d ≥ 1 , we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x ∈ Z d f ( l ( n , x ) ) of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } . Particular cases are the number of (a) visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; (b) $$\alpha $$ α -fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$ f ( i ) = i α ; (c) sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$ f ( i ) = I { i = j } .

中文翻译：

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$${\mathbb {Z}}^d$$ Z d 中瞬态随机游走的本地时间函数的强大数定律

对于任意瞬态随机游走 $$(S_n)_{n\ge 0}$$ ( S n ) n ≥ 0 in $${\mathbb {Z}}^d$$ Z d , $$d\ge 1 $$ d ≥ 1 ，我们证明了空间和 $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$ ∑ x 的强大数定律∈ Z df ( l ( n , x ) ) 局部时间函数 f $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x \}$$ l ( n , x ) = ∑ i = 0 n I { S i = x } 。特殊情况是 (a) 访问站点的数量 [首先由 Dvoretzky 和 ​​Erdős 考虑（Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951）]，它对应于函数 $$f(i )={\mathbb {I}}\{i\ge 1\}$$ f ( i ) = I { i ≥ 1 } ; (b) 随机游走的 $$\alpha $$ α -fold 自相交 [Becker 和 König 研究 (J Theor Probab 22:365–374, 2009)]，对应于 $$f(i)=i ^\alpha $$ f ( i ) = i α ;