Recent breakthroughs in particle tracking technology have enabled the measurement of fluid particle acceleration statistics in high-Reynolds-number turbulence. The probability density function of each component of acceleration has a highly stretched exponential tail, indicating large acceleration events are much more frequent than for an equivalent Gaussian field. Acceleration statistics of droplets in a wind tunnel show that particle inertia causes the tails of the distribution to become somewhat less stretched. We analyze the influence of particle inertia, in the weak limit, using a perturbation approach. We see that the primary effect of inertia is to bias the sampling of the flow; inertial particles tend to avoid regions with high rotation due to a "centrifuge" effect. Higher-order statistics can be replicated in a model flow consisting of point vortices of random strength and size. Analysis of this flow shows that a second-order effect in the change in the tails of the acceleration distribution of inertial particles is due to the decreasing correlation time of acceleration events of increasing magnitude.