We study online fair division, where indivisible goods arrive one at a time and must be allocated immediately and irrevocably among $n\ge2$ agents. Against an oblivious adversary that fixes the input in advance, we ask what fairness guarantees randomized online algorithms can achieve. We present the Deficit Allocation (DA) algorithm, which improves the previously known high-probability approximation to PROP1 from inverse-logarithmic to inverse-doubly-logarithmic in $n/δ$, where $δ$ is the failure probability. DA does not require $δ$ as input. With probability at least $1-δ$, a refined variant achieves an arbitrarily close approximation to proportionality when no single item is too valuable relative to an agent's proportional share; without any restriction on item values, it obtains an arbitrarily close approximation to PROP$k$ for $k$ doubly logarithmic in $n/δ$. For $n2^{-2n}\leδ\le1/2$, our bounds on the approximation factor and $k$ are asymptotically tight for the stronger fractional benchmark that divides each item equally among the agents who value it positively. We also prove that algorithms that use only which agents value each item positively, but not by how much, cannot improve upon the logarithmic dependence of uniformly random allocation. However, other fairness notions admit weaker guarantees. For EF1, every randomized online algorithm has an input on which any positive approximation succeeds with probability at most $e^{-Ω(n)}$. For EFX, this probability is at most $1/n!$ with only $n+1$ items, tight within a factor of $(n+1)/2$; for MMS, it can be at most $5/6$. Finally, allocating each item uniformly among the agents who value it positively gives, with probability at least $1-δ$, an arbitrarily close approximation to EF$k$ for $k$ logarithmic in $n/δ$, and this dependence is necessary for this rule.

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