1
Pólya’s theorem
2
Random walks
▶
2.1
Random walks on the \(d\)-dimensional integer grid
3
Recurrence and transience
▶
3.1
Basic definition
3.2
Equivalent conditions
4
Occupations and Green’s functions of random walks
▶
4.1
Regularized occupation
▶
Regularized occupation of a walk
Regularized occupation of a random walk
4.2
Green’s function
4.3
Expected occupation from the Green’s function
5
Fourier transform of Green’s function
▶
5.1
Fourier transform of the regularized Green’s function
5.2
Explicit formula for the Fourier transform
5.3
Inversion of the discrete Fourier transform
6
Treatment of the integral in the Fourier inversion
▶
6.1
Decomposition of the integral
6.2
Dominated convergence away from the origin
6.3
Monotone convergence near the origin
6.4
Characterizing finiteness of the integral for simple random walk
Dependency graph
Pólya’s theorem
Alma Nevalainen Niklas Halonen Kalle Kytölä
1
Pólya’s theorem
2
Random walks
2.1
Random walks on the \(d\)-dimensional integer grid
3
Recurrence and transience
3.1
Basic definition
3.2
Equivalent conditions
4
Occupations and Green’s functions of random walks
4.1
Regularized occupation
Regularized occupation of a walk
Regularized occupation of a random walk
4.2
Green’s function
4.3
Expected occupation from the Green’s function
5
Fourier transform of Green’s function
5.1
Fourier transform of the regularized Green’s function
5.2
Explicit formula for the Fourier transform
5.3
Inversion of the discrete Fourier transform
6
Treatment of the integral in the Fourier inversion
6.1
Decomposition of the integral
6.2
Dominated convergence away from the origin
6.3
Monotone convergence near the origin
6.4
Characterizing finiteness of the integral for simple random walk