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 to show: two random walks on galton-watson trees have the same distribution
lilly2108
Junior
Dabei seit: 03.05.2021
Mitteilungen: 10
 Themenstart: 2021-06-22
 Hello community, I have two different Random Walks on two Galton-Watson trees and want to show that they have the same distribution. Is it enough to show that they have the same expectation for any bounded, meaurable map $F$? So I want to show: $(\mathcal{B}_{\xi_k}(\mathcal{T}_{*}),(\Psi_{\xi_k}(X_{\tau_k-j}))_{j\leq\tau_k})\stackrel{(d)}{=}(\mathcal{T}_{*}^{\leq\xi_k},(X_j)_{j\leq\tau_k})$ and I have a proof like this: $\mathsf{E}_{*}[F(\mathcal{B}_{\xi_k}(\mathcal{T}_{*}),(\Psi_{\xi_k}(X_{\tau_k-j}))_{j\leq\tau_k})\mathbf{1}_{\{ \xi_k=v,\tau_k<\tau_{*}\}}]=\mathsf{E}_{*}[F(\mathcal{T}^{\leq\bar{v}}_{*}),(X_{j})_{j\leq\tau_k})\mathbf{1}_{\{ \xi_k=\bar{v},\tau_k<\tau_{*}\}}] \\ =\mathsf{E}_{*}[F(\mathcal{T}^{\leq\xi_k}_{*}),(X_{j})_{j\leq\tau_k})\mathbf{1}_{\{ \xi_k=\bar{v},\tau_k<\tau_{*}\}}]$ Is that enough? Thank you in advance.

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