  This paper constructs a collection of discrete operators on the
$d$-dimensional lattice $\mathbb{Z}^{d}$, $d\geq 1$, which result from the
conditional expectation of martingale transform in the upper half-space
$\mathbb{R}^{d+1}_{+}$ constructed from Doob $h$-processes. Special cases of
these operators are what we call \textit{the probabilistic discrete Riesz
transforms.} When $d=1,$ they reduce to \textit{the probabilistic discrete
Hilbert transform} used by the first and third authors to resolve the
long-standing open problem concerning the $\ell^p$ norm, $1<p<\infty$, of the
discrete Hilbert transform on the integers $\mathbb{Z}$. The construction for
$d>1$ is motivated by a similar problem, Conjecture 5.5, concerning the norm of
the discrete Riesz transforms arising from discretizing singular integrals on
$\mathbb{R}^d$ as in the original paper of A. P. Calder\'on and A. Zygmund, and
subsequent works of A. Magyar, E. M. Stein, S. Wainger, L. B. Pierce and many
others, concerning operator norms in discrete harmonic analysis. For any $d\geq
1$, it is shown that the probabilistic discrete Riesz transforms have the same
$\ell^p$ norm as the continuous Riesz transforms on $\mathbb{R}^d$ which is
dimension independent and equals the norm of the classical Hilbert transform on
$\mathbb{R}$. Along the way we give a different proof, based on Fourier
transform techniques, of the key estimate used to identify the norm of the
discrete Hilbert transform.
