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We present new high order approximations schemes for the Cox-Ingersoll-Ross(CIR) process that are obtained by using a recent technique developed byAlfonsi and Bally (2021) for the approximation of semigroups. The idea consistsin using a suitable combination of discretization schemes calculated ondifferent random grids to increase the order of convergence. This techniquecoupled with the second order scheme proposed by Alfonsi (2010) for the CIRleads to weak approximations of order $2k$, for all $k\in\mathbb{N}^*$. Despitethe singularity of the square-root volatility coefficient, we show rigorouslythis order of convergence under some restrictions on the volatility parameters.We illustrate numerically the convergence of these approximations for the CIRprocess and for the Heston stochastic volatility model and show thecomputational time gain they give.

High order approximations of the Cox-Ingersoll-Ross process semigroup using random grids