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We prove strong invariance principle between a transient Bessel process and acertain nearest neighbor (NN) random walk that is constructed from the formerby using stopping times. It is also shown that their local times are closeenough to share the same strong limit theorems. It is shown furthermore, thatif the difference between the distributions of two NN random walks are small,then the walks themselves can be constructed so that they are close enough.Finally, some consequences concerning strong limit theorems are discussed.

Transient nearest neighbor random walk and Bessel process