Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 26H (Probability and Measure)

Put $Z_n=|X_n-X|$. If $Z_n\to 0$ in probability, then for $0<\epsilon <1$,Taking the upper limit and then letting $\epsilon \downarrow 0$ proves that the expectations tend to zero. Conversely,so expectation convergence implies convergence in probability. This proves the bounded-metric characterization of convergence in probability.

For the subsequence assertion, choose $n_k$ so thatThe first Borel-Cantelli lemmas then says that only finitely many of these events occur almost surely. Henceeventually almost surely, and $X_{n_k}\to X$ almost surely. This is the almost-sure subsequence from convergence in probability.

Herefor every $0<\epsilon <1$, so $X_n\to 0$ in probability. On the other hand, the events $A_n=\{X_n=1\}$ are independent andThe second Borel-Cantelli lemmas givesThus $X_n$ cannot converge almost surely to zero. This independent rare-event counterexample to almost-sure convergence proves that convergence in probability need not imply almost-sure convergence of the full sequence.

The conclusion printed as convergence in probability is already an assumption; the substantive conclusion is convergence in $L^p$, and we prove that stronger statement.

First, the almost-sure subsequence result from part (a) and Fatou's lemma show that $X\in L^r$:ConsequentlyFix $1\le p<r$ and $\delta >0$. Splitting according to
$|X_n-X|\le \delta$ and applying Hölder's inequality on the complement givesThe second term tends to zero because of convergence in probability and the uniform $L^r$ bound. Taking the upper limit and then $\delta \downarrow 0$ provesThis is convergence in Lp from convergence in probability and an Lr bound.