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We consider the simple random walk (or P\'olya walk) on the one-dimensionallattice subject to stochastic resetting to the origin with probability $r$ ateach time step. The focus is on the joint statistics of the numbers${\mathcal{N}}_t^{\times}$ of spontaneous returns of the walker to the originand ${\mathcal{N}}_t^{\bullet}$ of resetting events up to some observation time$t$. These numbers are extensive in time in a strong sense: all their jointcumulants grow linearly in $t$, with explicitly computable amplitudes, andtheir fluctuations are described by a smooth bivariate large deviationfunction. A non-trivial crossover phenomenon takes place in the regime of weakresetting and late times. Remarkably, the time intervals between spontaneousreturns to the origin of the reset random walk form a renewal process describedin terms of a single `dressed' probability distribution. These time intervalsare probabilistic copies of the first one, the `dressed' first-passage time.The present work follows a broader study, covered in a companion paper, ongeneral nested renewal processes.

Returns to the origin of the P\'olya walk with stochastic resetting