  We investigate the statistics of records in a random sequence
$\{x_B(0)=0,x_B(1),\cdots, x_B(n)=x_B(0)=0\}$ of $n$ time steps. The sequence
$x_B(k)$'s represents the position at step $k$ of a random walk `bridge' of $n$
steps that starts and ends at the origin. At each step, the increment of the
position is a random jump drawn from a specified symmetric distribution. We
study the statistics of records and record ages for such a bridge sequence, for
different jump distributions. In absence of the bridge condition, i.e., for a
free random walk sequence, the statistics of the number and ages of records
exhibits a `strong' universality for all $n$, i.e., they are completely
independent of the jump distribution as long as the distribution is continuous.
We show that the presence of the bridge constraint destroys this strong `all
$n$' universality. Nevertheless a `weaker' universality still remains for large
$n$, where we show that the record statistics depends on the jump distributions
only through a single parameter $0<\mu\le 2$, known as the L\'evy index of the
walk, but are insensitive to the other details of the jump distribution. We
derive the most general results (for arbitrary jump distributions) wherever
possible and also present two exactly solvable cases. We present numerical
simulations that verify our analytical results.
