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Transition probability estimates for subordinate random walks

Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $ϕ$ we consider a random walk $S^ϕ_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $ϕ$ at zero we establish global estimates for the transition probabilities of the random walk $S^ϕ_n$. The main tools that we apply are the parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.